Ann period Inst period Fourier comma Grenoble \noindent56 commaAnn 7 open . parenthesis Inst . Fourier 2006 closing , Grenoble parenthesis 2437 hyphen 2461 DIFFRACTION .. SPECTRA .. OF WEIGHTED .. DELONE \noindentSETS .. ON56 .. , BETA 7 ( hyphen 2006 ) LATTICES 2437 − 2461 .. WITH .. BETA .. A QUADRATIC .. UNITARY PISOT .. NUMBER \ centerlineby Jean hyphen{DIFFRACTION Pierre GAZEAU\quad ampersandSPECTRA Jean\quad hyphenOF Louis WEIGHTED VERGER\quad hyphenDELONE GAUGRY} open parenthesis asterisk closingAnn . parenthesisInst . Fourier , Grenoble \ centerlinehline {SETS56\quad, 7 ( 2006ON )\ 2437quad - 2461BETA − LATTICES \quad WITH \quad BETA \quad A } Abstract period emdashDIFFRACTION .. The Fourier transform SPECTRA of a weighted OF Dirac WEIGHTED comb of beta hyphenDELONE integers \ centerlineis characterized{QUADRATIC withinSETS the\ frameworkquad ONUNITARY of BETA the theory PISOT - LATTICES of\ Distributionsquad NUMBER WITH comma} in particular BETA A its pure point part which correspondsQUADRATIC to the Bragg UNITARY part of the PISOT diffraction spectrum NUMBER period \noindentThe correspondingby Jeanby intensity− JeanP i e -r function r Pierre e GAZEAU on GAZEAU this $ Bragg\&& $ partJean Jean is computed- Louis− Louis VERGER period VERGER We - deduce GAUGRY− GAUGRY(∗ $) ( \ astthe diffraction) $ spectrum of weighted Delone sets on beta hyphen lattices in the split case for the weight comma when beta is the golden mean period \ [ \ r u l e {3em}{0.4 pt }\ ] Reacutesumeacute periodAbstract emdash . — .. OnThe caracteacuterise Fourier transform ofau a moyen weighted de Dirac la theacuteorie comb of beta - des integers distribu- tions la trans hyphen is characterized within the framework of the theory of Distributions , in particular formeacutee de Fourier d quoteright un peigne de Dirac avec poids comma plus particuli egrave \ centerline { Abstractits pure . −−− point part\quad whichThe corresponds Fourier to the transform Bragg part of the of diffraction a weighted spectrum Dirac . comb of beta − i n t e g e r s } rement la partie The corresponding intensity function on this Bragg part is computed . We deduce purement ponctuelle qui correspond aux pics de Bragg dans le spectre de diffrac hyphen \ centerline { is characterizedthe diffraction spectrum within of weighted the framework Delone sets on of beta the - lattices theory in the of split Distributions case , in particular } t ion period La fonction intensiteacutefor de the ces weight derniers , when est beta donneacutee is the golden d mean quoteright . une mani egrave re explicite period On \ centerline { its pureResum´ pointe´ . — partOn which caract´eriseau corresponds moyen de la th´eoriedes to the distributions Bragg part la trans of - the diffraction spectrum . } en deacuteduit le spectreform´eede de diffraction Fourier d ’ d un quoteright peigne de Dirac ensembles avec poids de Delaunay , plus particuli avec poidse` rement supporteacutes la partie par les beta hyphen reacuteseaux dans le cas o ugrave le poids est factorisable et o ugrave beta est \ centerline {The correspondingpurement ponctuelle intensity qui correspond function aux pics de on Bragg this dans le Bragg spectre depart diffrac is - computed . We deduce } le nombre t ion . La fonction intensit´ede ces derniers est donn´eed ’ une mani e` re explicite . On d quoteright or period \ centerline { the diffractionen d´eduitle spectre spectrum de diffraction of weighted d ’ ensembles deDelone Delaunay sets avec poids on support´esbeta − lattices in the split case } 1 period .. Introductionpar les : beta quasicrystalline - r´eseauxdans motivations le cas o u` le poids est factorisable et o u` beta est le nombre Mainly since the end of the 1 9 th century comma material scientists have been \ centerline { for the weight , when beta is thed ’ or .golden mean . } comprehending the crystalline1 . order Introduction comma owing : to quasicrystalline improvements in observa motivations hyphen tional techniques comma like X minus ray or electron diffraction comma and in quantum me hyphen \ centerline {R\ ’{ e}sumMainly\ ’{ e} since. −−− the \endquad of theOn 1 c 9 a th r a centuryc t \ ’{ e ,} materialrise au scientists moyen have de la been th \ ’{ e} orie des distributions la trans − } chanics open parenthesiscomprehending e period gthe period crystalline Bloch order waves , closing owing to parenthesis improvements .... open in observa square - bracket 1 0 closing square brackettional .... techniques open square , like bracketX− ray 22 or closing electron square diffraction bracket ,period and in Thequantum mathematical me - tools\ centerline which have{form \ ’{ e}e de Fourier d ’ un peigne de Dirac avec poids , plus particuli $ \grave{e} $ rementchanics ( la e . partie g . Bloch} waves ) [ 1 0 ] [ 22 ] . The mathematical tools which have enabled them to classifyenabled and them to improve to classify the and understanding to improve the of such understanding struc hyphen of such struc - tures are mainly lattices comma finite groups open parenthesis point and space groups closing paren- \ centerline {purementtures are ponctuelle mainly lattices qui , finite correspond groups ( point aux and pics space de groups Bragg ) and dans their le spectre de diffrac − } thesis and their representations , and Fourier analysis . So to say , the structural role played representations comma and Fourier analysis period So to say comma the structural role played \ centerlinehline { t ion . La fonction intensit \ ’{ e} de ces derniers est donn \ ’{ e}e d ’ une mani $ \Keywordsgrave{e} : Delone$ re set explicite comma Meyer . On set} comma beta hyphen integer comma beta hyphen lattice comma PV number comma mathematical \ centerline {en d\ ’{ e} duit le spectre de diffraction d ’ ensembles de Delaunay avec poids support \ ’{ e} s } diffraction period Keywords : Delone set , Meyer set , beta - integer , beta - lattice , PV number , mathematical Math period classificationdiffraction : . 52 C 23 comma 78 A 45 comma 42 A 99 period \ centerline { par l e s beta − r \ ’{ e} seaux dans le cas o $ \grave{u} $ le poids est factorisable et o open parenthesis asteriskMath . classification closing parenthesis : 52 C 23 Work , 78 A partially 45 , 42 A supported 99 . by ACINIM 2004 endash 1 54$ \ quotedblleftgrave{u} Case$ beta 1 quotedblright est le nombreCase 2 period} 00 (∗) Work partially supported by ACINIM 2004 – 1 54 “ Numeration . \ centerline {d ’ or . }

\ centerline {1 . \quad Introduction : quasicrystalline motivations }

\ hspace ∗{\ f i l l }Mainly since the end of the 1 9 th century , material scientists have been

\noindent comprehending the crystalline order , owing to improvements in observa −

\noindent tional techniques , like $ X − $ ray or electron diffraction , and in quantum me −

\noindent chanics ( e . g . Bloch waves ) \ h f i l l [ 1 0 ] \ h f i l l [ 22 ] . The mathematical tools which have

\noindent enabled them to classify and to improve the understanding of such struc −

\noindent tures are mainly lattices , finite groups ( point and space groups ) and their

\noindent representations , and Fourier analysis . So to say , the structural role played

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\noindent Keywords : Delone set , Meyer set , beta − integer , beta − lattice , PV number , mathematical diffraction .

\noindent Math . classification : 52 C 23 , 78 A 45 , 42 A 99 .

\noindent $ ( \ ast ) $ Work partially supported by ACINIM 2004 −− 1 54 ‘ ‘ $\ l e f t . Numeration\ begin { a l i g n e d } &’’ \\ &. \end{ a l i g n e d }\ right . $ 2438 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentby the rational2438 integers\quad openJean parenthesis− P i e Z r closingr e GAZEAU parenthesis $ \& was $ essential Jean in− thatLouis process VERGER of math-− GAUGRY ematical \noindentformalizationby period the rational integers $ ( Z ) $ was essential in that process of mathematical formalizationA so hyphen called . quasicrystalline order has been discovered more than twenty years ago open square bracket 38 closing square bracket period The observational techniques for this Anew so state− called of matter quasicrystalline2438 Jean - P i erre GAZEAU order has& Jean been - Louis discovered VERGER - GAUGRY more than twenty yearsare similar ago to [ those 38by ] used . the The for rational crystals observational integers period (Z But) was thetechniques essential absence ofin periodicity thatfor process this open new of mathematical parenthesis state of which matter for- does not mean thatmalization there is a . random distribution of atomic sites closing parenthesis and the \noindentpresence ofare self hyphen similarA similarities so to - called those comma quasicrystalline used which for reveal crystals order to behas algebraic been . But discovered integers the absence openmore square than of twenty bracket periodicity ( which 24 closing square bracketyears open ago square [ 38 ] . bracket The observational 42 closing square techniques bracket for comma this new state of matter \noindentas scaling factorsdoes notdemandare similarmean new that to mathematical those there used for toolsis crystals a open random square . But distribution the bracket absence 2 closing of periodicity of square atomic bracket ( which sites open ) and the square bracket 23 closingdoes square not mean bracket that thereopen issquare a random bracket distribution 25 closing of square atomic bracket sites ) and period the As a matter\noindent presencepresence of self of self− - similaritiessimilarities , which , reveal which to be reveal algebraic to integers be algebraic [ 24 ] [ 42 ] integers , [ 24 ] [ 42 ] , of fact comma it becameas scaling necessary factors to demand conceive new adapted mathematical point sets tools and t [ ilings 2 ] [ as 23 ac ] [ hyphen 25 ] . As a \noindentceptable geometricas scalingmatter supports factors of offact quasicrystals , it demand became comma necessary new mathematical and to also conceive to carry adapted tools out a point specific [ 2 sets ] [ and 23 t ilings] [ 25 ] . As a matter ofspectral fact analysis , it became foras quasicrystalline ac - necessary ceptable geometric diffraction to conceive supports and electronic adaptedof quasicrystals transport point , period and sets also and to carry t ilings out a as ac − ceptableTherefore angeometric appealingspecific mathematical supports spectral analysis of program quasicrystals for quasicrystalline comma combining , and diffraction number also and theory to electronic carry comma out transport a specific spectralfree groups analysis and semi. Therefore hyphen for quasicrystalline groups an appealing comma mathematical measure diffraction theory program and quasi and , combining hyphen electronic harmonicnumber transport theory analysis , . commaTherefore an appealingfree groups mathematical and semi - groups program , measure , theory combining and quasi number - harmonic theory analysis , , freehas been groups developing andhas semi for been the− lastdevelopinggroups two decades ,for measure the period last two The theory decades aim of and this. The quasi paper aim is of− first thisharmonic paper is first analysis to , hasto recall been the developing part ofrecall this theprogram for part the of specifically this last program two devoted decades specifically to the . notions devoted The aim of to sets the of notions this ofpaper sets of isbeta first to recall the part of this program specifically devoted to the notions of sets of beta hyphen integers- integers open(Z parenthesisβ), when β Zis sub a quadratic beta closing Pisot parenthesis - V ij ayaraghavan comma when num - beta is a quadratico f beta Pisot− integers hyphenber V , ij andayaraghavan $ ( associated Z {\ numbetabeta hyphen - lattices} )( or , $β− whenlattices $)\ ,beta and next$ to is present a quadratic Pisot − V $ ijber $ comma ayaraghavan and associatedsome num original beta− results hyphen concerning lattices open the parenthesis Fourier properties or beta and hyphen diffraction lattices spectra closing parenthesis comma andof next Delone to present sets on some beta - integers and on beta - lattices , and more generally \noindentoriginal resultsber concerning , andof weighted associated the Fourier Dirac combs properties beta supported− lattices and diffraction by beta ( or - spectra lattices $ \beta . This approach− $ lattices makes ) , and next to present some originalof Delone sets results on betaprominent concerning hyphen the integers role the played and Fourier on by beta nonclassical hyphen properties lattices numerations andcomma diffraction[ 1 and 6 ] more[ 26 ] generally , associated spectra ofof Delone weighted setsDirac combs onwith beta substitutive supported− integers by dynamical beta hyphen and systems on lattices beta [ 34 period− ] ,lattices and This amounts approach , to and replace makes more the generally tra - of weighted Dirac combs supported by beta − lattices . This approach makes prominent the roleditional played by set nonclassical of rational numerations integers Z by openZβ. squareThis allows bracket computations 1 6 closing square over bracket openprominent square bracket the 26rolethis closing new played setsquare of by numbers bracket nonclassical comma, whereassociated lattices numerations are replaced by [ 1 beta 6 - ] lattices [ 26 . ] , associated with substitutive dynamicalThe systems organisation open ofsquare this paper bracket is as34 follows closing . square Section bracket 2 recalls comma the con and - \noindent with substitutive dynamical systems [ 34 ] , and amounts to replace the tra − amounts to replace thestruction tra hyphen of the set of beta - integers Zβ[8][17][26]( introduced by Gazeau ditionalditional set set of rational ofin rational [ 1 integers 9 ] ) . WeZ integers by describe Z sub beta in $ Section period Z $ 3 Thisby the relevanceallows $ Z computations{\ ofbeta the beta} over -. integers $ This in the allows computations over this new set of numbersdescription comma of where quasicrystalline lattices are structures replaced . by beta More hyphen precisely lattices , these period numbers can \noindentThe organisationthis new ofbe this used set paper as of a issort numbers as of follows universal ,period where support Section lattices for 2model recalls sets are the conor replacedcut hyphen - and - by project beta sets− l a t t i c e s . struction of the set, ofi . beta e . hyphenthose integers point sets Z sub paradigmatically beta open square supposed bracket to 8 closing support square quasicrys bracket - open\ hspace square∗{\ bracketf i l l }The 1talline 7 closing organisation atomic square sites bracketand of their this open diffraction paper square patterns bracket is as . 26 follows We closing then give square. Section in Section bracket 2 4 open . recalls 1 the con − parenthesis introducedtheir by Gazeau Fourier transform , in particular the support of the pure point part of the \noindentin open squarestruction bracketmeasure 1 of 9 closing . the We present set square of in bracket beta Section− closing 4i n. t 2 e parenthesisthe g e r computation s $ Z period{\ of Webeta diffraction describe} [ formulae in Section 8 ] 3[ the 1 relevance 7 of ] theof beta[ weightedhyphen 26 Delone ] integers ($ sets in onthe introducedbyGazeau beta - integers , then on beta - lattices ( Section 4 . description of quasicrystalline3 ) , in the plane structures case withperiod the .. golden More precisely mean for commaβ, in the these case numbers where the weight \noindentcan be usedin as [ a sort1is 9 split of ] universal ) . We . We indicate support describe in for particular model in Section sets how orβ cut− 3lattices hyphen the relevance can and be hyphen viewedproject of as the the most beta − integers in the descriptionsets comma i period ofnatural quasicrystalline e period indexing those point sets sets for structures theparadigmatically numbering . \ ofquad supposed the spotsMore to ( support preciselythe “ Bragg quasicrys peaks , these hyphen ” ) numbers cantalline be atomic used sitesas abeyond and sort their a ofcertain diffraction universal brightness patterns support in periodthe diffraction We for then model pattern give in sets . Section or cut − and − p r o j e c t sets , i . e . those point sets paradigmatically supposed to support quasicrys − 4 period 1 their FourierANNALES transform DE L ’ I NS comma T ITUT FOURIER in particular the support of the pure point part tallineof the measure atomic period sites We present and their in Section diffraction 4 period 2 the patterns computation . We of diffraction then give in Section 4formulae . 1 their of weighted Fourier Delone transform sets on beta , hyphenin particular integers comma the then support on beta of hyphen the pure lattices point part ofopen the parenthesis measure Section . We present 4 period 3 in closing Section parenthesis 4 . 2 comma the incomputation the plane case of with diffraction the golden meanformulae for beta of comma weighted in the case Delone sets on beta − integers , then on beta − l a t t i c e s (where Section the weight 4 . 3 is split) , period in the We plane indicate case in particular with the how goldenbeta hyphen mean lattices for can $ be\beta , $ inviewed the case as the most natural indexing sets for the numbering of the spots whereopen parenthesis the weight the is quotedblleft split . Bragg We indicate peaks quotedblright in particular closing parenthesis how $ \ beyondbeta a− certain$ lattices can be brightnessviewed in as the the diffraction most natural pattern period indexing sets for the numbering of the spots (ANNALES the ‘‘ Bragg DE L quoteright peaks ’’ I NS ) beyond T ITUT aFOURIER certain brightness in the diffraction pattern . \noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2439 \ hspace2 period∗{\ ..f Numeration i l l }DIFFRACTION in base beta OF a BETA Pisot− numberLATTICES and \quad 2439 beta hyphen integers Z sub beta \ centerline2 period 1 period{2 . ..\quad DefinitionsNumeration and notations in base $ \beta $ a Pisot number and } We refer to open square bracket 1 5 closing square bracket .. open square bracket 1 6 closing square bracket\ centerline comma{ Chapterbeta − 7i n in t e open g e r s square $ Z bracket{\beta 26 closing}$ } square bracket comma .. open square bracket 33 closing square bracket .. open square bracket 35 closingDIFFRACTION square bracketOF BETA for - LATTICES the numeration2439 in \ centerlinePisot base comma{2 . 1 and . 2\ toquad . open NumerationDefinitions square bracket and20 in closing base notations squareβ a Pisot bracket} number open square and bracket 24 closing square bracket open square bracket 28 closingbeta -square integers bracketZ openβ square bracket 29 closing Wesquare refer bracket to open [ 1 square 5 ] \quad bracket[ 30 16 closing2 . ] 1. , square Chapter Definitions bracket 7 in open and [ square 26 notations ] bracket, \quad 32[ closing 33 ] square\quad [ 35 ] for the numeration in bracketPisot open base square , and bracket toWe 41 [ refer closing20 to ] [ [ square 1 245 ] ] bracket [ [ 1 286 ] open , ] Chapter [ square 29 ]7 bracket in [ [ 30 26 ] 42 , [ closing 32[ 33 ] ] square [ [ 41 35 bracket ] ] for [ the 42 for ] for an introduction to anmathematical introduction to quasicrystalsnumeration in Pisot ( to base Delone , and to sets [ 20 ] [, 24 Meyer ] [ 28 ] sets [ 29 ] [ . 30 . ] [ . 32 ) ] [. 41 ] [ 42 ] mathematical quasicrystalsfor an introduction open parenthesis to mathematical to Delone sets quasicrystals comma Meyer ( to Delone sets period sets , period Meyer period sets closingNotation parenthesis $\ l e periodf t ...).. 2 . 1\ begin { array }{ c} −−− \\ . \end{ array }For\ right . $ a realNotation number 2 period $ Row x 1\ emdashin R Row− 2 , − period $ − the . a realinteger number part x in R of comma $x$ the integer will part be of x Notation 2.1 F or a real number x ∈ R, the integer part of x will willdenoted be by $ \ l f l o o r x \ rfloor. $ and its fractional part by $ \{ x \} denoted= by x floorleft−be \ xl denoted floor f l o o and r by itsbxx fractionalc and\ rfloor its fractionalpart by. open part$ brace by {x x} closing= x − bbracexc. = x minus floorleft − − − x floor period Definition 2.2 F or β > 1 a real number and z ∈ [0, 1] we DefinitionDefinition 2 period $\ l e f Row t . 2 1 emdash . 2 Row\ begin 2 period{. array . beta}{ c greater} −−− 1 \\ a real. number\end{ array and z in}For open\ right square. \beta > 1 $ a real number and $ z \ in [ 0 , 1 ] $ \quad we denote bracket 0 comma 1 closingdenote square by bracketTβ(z) =.. weβz denotemod 1 the β− transform on [ 0 , 1 ] associated byby T $ sub T beta{\beta openwith parenthesis} β,(and z iz t closing - ) =parenthesis\beta = betaz $ z ..\ modquad 1 themod beta 1 thehyphen $ transform\beta − $ transform on [ 0 , 1 ] associated withj+1 $ \beta j , $ and i t − on open square bracketeratively 0 comma , for 1 closing all integers squarej bracket> 0,Tβ associated(z) := Tβ with(Tβ ( betaz)), where comma by and convention i t hyphen eratively comma for all integers j geqslant 0 comma T sub beta to the power of j plus 1 open \noindent eratively , for all integers $ j \ geqslant 0 , Tˆ{ j + parenthesis z closing parenthesis : = T sub beta open parenthesis0 T sub beta to the power of j open parenthesis1 } {\beta z closing} ( parenthesis z ) closing : parenthesis = T {\ commabetaTβ where= Id.} by( convention T ˆ{ j } {\beta } ( zT )sub beta ) to , the $ power whereby of 0 = Id convention period Definition 2 . 3 . — ( i ) Given Λ ⊂ a discrete set , we say that Definition 2 period 3 period emdash .. open parenthesis i closingR parenthesis .. Given Capital Λ is uniformly dis crete if there exists r > 0 such that k x − y k r for Lambda\ begin { subseta l i g n R∗} a discrete set comma we say that Capital Lambda > all x, y ∈ Λ with x 6= y. Therefore a uniformly discrete subset of can Tis ˆ{ uniformly0 } {\ disbeta crete} if there= exists Id r greater . 0 such that bar x minus y bar geqslant r R be the “ empty set ” , one point sets {x}, or discrete sets of cardinality 2 \endfor{ alla l i x g ncomma∗} y in Capital Lambda with x negationslash-equal y period .. Therefore a uniformly> having this property . discrete subset ( ii ) A subset Λ ⊂ is called re latively dense ( after Besicovitch ) Definitionof R can be 2 the . quotedblleft3 . −−− \quad empty( set i quotedblright ) \quadR Given comma $ one\Lambda point sets open\subset brace xR closing $ a discrete set , we say that if there exists R > 0 such that for all x ∈ , there exists z ∈ Λ such that brace$ \Lambda comma$ or discrete sets of R k x − z k R. A relatively dense set is never empty . ( i ii ) A Delone set iscardinality uniformly geqslant dis 2 crete having this6 if thereproperty exists period $ r > 0 $ such that $ \ parallel of is a su bset of which is uniformly discrete and relatively dense . ( x open− parenthesisy \ parallel ii closingR parenthesis\ g e q s l a n .. tR A subset r $ Capital Lambda subset R is called re latively iv ) A Meyer set Λ is a Delone set such that there exists a finite set F densef o r open a l l parenthesis $ x , after y Besicovitch\ in closing\Lambda parenthesis$ with if there $ x \not= y . $ \quad Therefore a uniformly discrete subset ofexists $R$ R greater can 0 such be the that for‘‘ all empty x in R set comma ’’ there , one exists point z in Capitalsets $Lambda\{ suchx that\} , $ suchthatΛ − Λ ⊂ Λ + F. orbar discrete x minus z sets bar leqslant of R period A relatively dense set is never empty period cardinalityopen parenthesis $ i\ iigeqslant closing( v ) parenthesisA Pisot 2 $ number .. having A Delone ( or this Pisot set of property- RV isij a suayaraghavan bset . of R which number is ,uniformly or PV discrete and number ) β is an algebraic integer > 1 such that all its Galois conjugates are ( irelatively i ) \quad denseA period subsetstrictly less $ than\Lambda1 in modulus\subset . R $ is called re latively dense ( after Besicovitch ) if there e xopen i s t s parenthesis $ R > iv closing0 $ parenthesis such that− .. − A for − Meyer all set $ Capital x \ Lambdain R is a , Delone $ there set such exists that Definition 2.4 Let β > 1 be a real number . A beta - representation there$ z exists\ in a finite\Lambda set F $ such that . $ \ parallel x − z \ parallel \ leqslant R . $ A relatively dense set is never empty . such that Capital Lambda( or β− minusrepresentation Capital Lambda , or representation subset Capital in base Lambdaβ) of plus a real F period number x > 0 ( i i i ) \quad A Delone set of $ R $ is a su bset of $ R $ which is uniformly discrete and open parenthesis vis closing given by parenthesis an infinite .. sequence A Pisot number(xi)i > 0 openand parenthesisan integer ork ∈ PisotZ such hyphen that V ij relatively dense . P+∞ −i+k ayaraghavan number commax = i or=0 PVxiβ number, where closing the parenthesis digits xi belong to a given alphabet (⊂ N). (beta i v )is an\quad algebraicA Meyer integerAssume s greater e t that $ β 1\ such>Lambda1 is that a Pisot$ all its numberis Galois a Delone of conjugates degree set strictly are such greater that than there 1 in the exists a finite set $ Fstrictly $ less than 1sequel in modulus . period Definition 2 period RowAmong 1 emdash all the Row beta 2 -period representations . beta greater of a 1real be number a real numberx > 0, xperiod6= 1, Athere beta hyphen\ [ such representation that exists\Lambda a particular− one \Lambda called R´enyi\subsetβ− expansion\Lambda which is obtained+ F through . \ ] open parenthesis orTOME beta 5 hyphen6 ( 2 0 0 6 representation) , FASCICULE 7 comma or representation in base beta closing paren- thesis of a real number x geqslant 0 ( vis ) given\quad by anA infinite Pisot sequence number open ( or parenthesis Pisot − V x sub $ i ij closing $ ayaraghavan parenthesis i geqslant number 0 and , or an PV number ) integer$ \beta k in Z$ such is that an algebraic integer $ > 1 $ such that all its Galois conjugates are strictlyx = sum sub less i = than 0 to the 1 power in modulus of plus infinity . x sub i beta to the power of minus i plus k comma where the digits x sub i belong to a given alphabet open parenthesis subset N closing parenthesis period DefinitionAssume that $ beta\ l egreater f t . 2 1 is . a Pisot 4\ begin number{ array of degree}{ c} strictly −−− \\ greater. than\end 1{ array in } Let \ right . \betathe sequel> period1 $ be a real number . A beta − representation (Among or $ all\beta the beta hyphen− $ representationsrepresentation of a real , or number representation x geqslant 0 comma in x base equal-negationslash $ \beta 1) comma $ of there a real number $ x \ g e q s l a n t 0 $ exists a particular one called Reacutenyi beta hyphen expansion which is obtained through \noindentTOME 5 6is open given parenthesis by an 2 0 infinite 0 6 closing sequence parenthesis comma $ ( FASCICULE x { i } 7) i \ g e q s l a n t 0 $ and an integer $ k \ in Z $ such that

\noindent $ x = \sum ˆ{ + \ infty } { i = 0 } x { i }\beta ˆ{ − i + k } , $ where the digits $ x { i }$ belong to a given alphabet $ ( \subset N ) . $

Assume that $ \beta > 1 $ is a Pisot number of degree strictly greater than 1 in the s e q u e l .

Among all the beta − representations of a real number $ x \ g e q s l a n t 0 , x \ne 1 , $ t here exists a particular one called R\ ’{ e} nyi $ \beta − $ expansion which is obtained through

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2440 Jean - P i erre GAZEAU & Jean - Louis VERGER - GAUGRY the so - called greedy k k+1 algorithm : in this case , k satisfies β 6 x < β and the digits

x x := bβT i ( )c i = 0, 1, 2, ..., (2.1) i β βk+1

− − − belong to the finite alphabet := {0, 1, 2, ..., bβc}. Notation 2.5 W e Aβ . denote by

hxiβ := x0x1x2...xk.xk+1xk+2... (2.2) the couple formed by the string of digits x0x1x2...xkxk+1xk+2... and the position of the dot , which is at the k th position ( between xk and xk+1). Definition 2 . 6 . — • The integ er part ( in base β) of x is

k X −i+k xiβ . i = 0 • The fra ctional part ( in base β) is

+∞ X −i+k xiβ . i = k + 1 • If a R´enyi β− expansion ends in infinitely many zeros , i t is said to be finite and the ending zeros are omitted . • If i t is periodic after a certain rank , i t is said to be eventually periodic ( the period is the smallest finite string of digits possible , assumed not to be a string of zeros ) . The particular R´enyi β− expansion of 1 plays an important role in the 0 theory . Denoted by dβ(1) it is defined as follows : since β 6 1 < β, the value Tβ(1/β) is set equal to 1 by convention . Then using ( 2 . 1 ) for all i > 1 with x = 1, we have : t1 = bβc, t2 = bβ{β}c, t3 = bβ{β{β}}c, etc . The P+∞ −i equality dβ(1) = 0.t1t2t3... corresponds to 1 = i=1 tiβ . − − − Definition 2.7 T he set . k P −i+k Zβ := {x ∈ R || x | is equal to its integer part ( in base β) xiβ }

i = 0 is called set of beta - integers , or set of β− integers . The set of beta - integers is discrete and locally finite [ 41 ] . It is usual to denote its elements by bn so that 2440 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentthe so hyphen2440 called\quad greedyJean algorithm− P i : in e r this r e case GAZEAU comma $ k\ satisfies& $ Jeanbeta to− theLouis power VERGER of k leqslant− GAUGRY xthe less beta so − tocalled the power greedy of k plus algorithm 1 and : in this case $ , k $ satisfies $ \beta ˆ{ k } \ l ethe q s l digits a n t x < \beta ˆ{ k + 1 }$ and Equation: open parenthesis 2 period 1 closing parenthesis .. x sub i : = floorleftbigg beta T sub beta to\noindent the power ofthe i parenleftbigg d i g i t s x divided by beta to the power of k plus 1 parenrightbigg floorrightbigg i = 0 comma 1 comma 2 comma period period period comma \ beginbelong{ a to l i gthe n ∗} finite alphabet A sub beta : = open brace 0 comma 1 comma 2 comma period period period\ tag ∗{ comma$ ( floorleft 2 . beta 1Z floorβ = ){ closing..., $} b−xn brace, ...,{ b−i period1,} b0, b1:, ...b =n, ...}withb\ l f l o0 o= r 0, b1 \=beta +1, b−n T= − ˆ{bn i } {\beta } ( Notation\ f r a c { 2x period}{\ Rowbeta 1ˆ emdash{ k Row + 2 1 period}} .) denote\ rfloor by i = 0 , 1 , for n 1. 2Equation: , . open . parenthesis . , > 2 period 2 closing parenthesis .. angbracketleft x right angbracket sub beta\end :{ a = l ix g subn ∗} 0 x subANNALES 1 x sub DE L 2 ’ Iperiod NS T ITUT period FOURIER period x sub k period x sub k plus 1 x sub k plus 2 period period period \noindentthe couplebelong formed by to the the string finite of digits alphabet x sub 0 x sub $ 1 A x sub{\ 2beta period} period: period = \{ x sub k0 x sub , k1 plus , 1 x sub 2 k plus, 2 . period . period . period , \ andl f l o the o r \beta \ rfloor \} . $ Notationposition of the $\ l dot e f t comma. 2 which . 5 is\ begin at the{ karray th position}{ c} open −−− parenthesis \\ . \end between{ array x sub}We k\ andright x sub. $ kdenote plus 1 closing by parenthesis period Definition 2 period 6 period emdash .. bullet .. The integ er part open parenthesis in base beta closing\ begin parenthesis{ a l i g n ∗} of x is \ tagLine∗{$ 1 k ( Line 2 2 sum . x sub 2 i beta ) $ to}\ thelangle power of minusx \ irangle plus k period{\ Linebeta 3 i} = 0: = x { 0 } x bullet{ 1 } .. Thex fra{ ctional2 } . part open . parenthesis . x { ink base} . beta closingx { k parenthesis + 1 is} x { k + 2 }Line... 1 plus infinity Line 2 sum x sub i beta to the power of minus i plus k period Line 3 i = k plus 1 \endbullet{ a l i .. g nIf∗} a Reacutenyi beta hyphen expansion ends in infinitely many zeros comma i t is said to be finite and the ending zeros are omitted period \noindentbullet .. Ifthe i t is couple periodic after formed a certain by the rank string comma i oft is saiddigits .. to be $ eventually x { 0 } x { 1 } x { 2 } .periodic . . open x parenthesis{ k } thex period{ k is + the smallest 1 } x finite{ stringk + of digits 2 } possible. .comma . $ and the assumed not to be a string of zeros closing parenthesis period \noindentThe particularposition Reacutenyi of thebeta hyphendot , expansion which is of at 1 plays the an $ important k $ th role position in the ( between $ xtheory{ k period}$ Denoted and $ by x d{ subk beta + open 1 parenthesis} ) . 1 closing$ parenthesis it is defined as follows : since beta to the power of 0 leqslant 1 less beta comma the \ centerlinevalue T sub{ betaDefinition open parenthesis 2 . 6 1 . slash−−− beta \quad closing$ parenthesis\ bullet $ is set\quad equalThe to 1 by integ convention er part ( in base period$ \beta Then using) $ open o f parenthesis $ x $ i 2 s period} 1 closing parenthesis for all i geqslant 1 with x = 1 comma we have : t sub 1 = floorleft beta floor comma t sub 2 = floorleft beta open brace beta\ [ \ begin closing{ a brace l i g n floore d } commak \\ t sub 3 = floorleft beta open brace beta open brace beta closing brace closing\sum bracex floor{ commai }\ etcbeta periodˆ{ The − i + k } . \\ iequality = d 0 sub\end beta{ opena l i g n parenthesis e d }\ ] 1 closing parenthesis = 0 period t sub 1 t sub 2 t sub 3 period period period corresponds to 1 = sum sub i = 1 to the power of plus infinity t sub i beta to the power of minus i period \ centerlineDefinition 2{ period$ \ bullet Row 1 emdash$ \quad Row 2The period fra . set ctional part ( in base $ \beta ) $ i s k} Z sub beta : = braceleftbig x in R bar bar x bar is equal to its integer part open parenthesis in base beta\ [ \ begin closing{ parenthesisa l i g n e d } + sum x\ subinfty i beta\\ to the power of minus i plus k bracerightbig \isum = 0 x { i }\beta ˆ{ − i + k } . \\ iis called = set k of beta+ hyphen 1 \end integers{ a l i g n commae d }\ ] or set of beta hyphen integers period The set of beta hyphen integers is discrete and locally finite open square bracket 41 closing square bracket period It is usual to denote its$ elements\ bullet by$ b sub\quad n so thatI f a R\ ’{ e} nyi $ \beta − $ expansion ends in infinitely many zeros , i t is said to beZ sub finite beta =and open the brace ending period zeros period periodare omitted comma b . sub minus n comma period period period comma b sub minus 1 comma b sub 0 comma b sub 1 comma period period period b sub n comma period$ \ bullet period period$ \quad closingIf brace i t with is b periodic sub 0 = 0 comma after b a sub certain 1 = plus rank 1 comma , i b t sub is minus said n =\quad to be eventually minusperiodic b sub n ( the period is the smallest finite string of digits possible , assumedfor n geqslant not 1to period be a string of zeros ) . ANNALES DE L quoteright I NS T ITUT FOURIER \ hspace ∗{\ f i l l }The particular R\ ’{ e} nyi $ \beta − $ expansion of 1 plays an important role in the

\noindent theory . Denoted by $ d {\beta } ( 1 ) $ it is defined as follows : since $ \beta ˆ{ 0 }\ l e q s l a n t 1 < \beta , $ the

\noindent value $ T {\beta } ( 1 / \beta ) $ is set equal to 1 by convention . Then using ( 2 . 1 ) for all $ i \ g e q s l a n t 1 $

\noindent with $x = 1 ,$ wehave $: t { 1 } = \ l f l o o r \beta \ rfloor , t { 2 } = \ l f l o o r \beta \{\beta \}\ rfloor , t { 3 } = \ l f l o o r \beta \{\beta \{\beta \}\}\ rfloor , $ e t c . The

\noindent e q u a l i t y $ d {\beta } ( 1 ) = 0 . t { 1 } t { 2 } t { 3 } . . .$ correspondsto $1 = \sum ˆ{ + \ infty } { i = 1 } t { i }\beta ˆ{ − i } . $

\ centerline { Definition $\ l e f t . 2 . 7\ begin { array }{ c} −−− \\ . \end{ array }The\ right . $ s e t }

\ [ k \ ]

\ centerline { $ Z {\beta } : = \{ x \ in R \mid \mid x \mid $ is equal to its integer part ( in base $ \beta ) \sum x { i }\beta ˆ{ − i + k }\} $ }

\ [ i = 0 \ ]

\noindent is called set of beta − integers , or set of $ \beta − $ integers . The set of beta − i n t e g e r s i s discrete and locally finite [ 41 ] . It is usual to denote its elements by $ b { n }$ so that

\ [Z {\beta } = \{ . . . , b { − n } ,..., b { − 1 } , b { 0 } , b { 1 } , . . . b { n } ,. .. \} with b { 0 } = 0 , b { 1 } = + 1 , b { − n } = − b { n }\ ]

\noindent f o r $ n \ geqslant 1 . $

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2441 DIFFRACTIONA positive beta OF hyphen BETA − integerLATTICES is also defined\quad open2441 square bracket 1 5 closing square bracket by a finiteA positive sum sum sub beta j =− 0 tointeger the power is of alsoN a sub defined j beta to [ the 1 power 5 ] byof j a finite sum $ \sum ˆ{ N } { j =where 0 } thea integers{ j }\ a subbeta j satisfyˆ{ 0j leqslant}$ a sub j less beta comma together with Line 1 m Line 2 sum a sub j beta to the power of j less beta to the power of m plus 1 comma m = 0 comma\noindent 1 commawhere period the period integers period comma $ a N{ periodj }$ Line s a 3 t ji s = f y 0 $ 0 \ l e q s l a n t a { j } < 2 period\beta 2 period, $ .. togetherBetaDIFFRACTION hyphen with integers OF BETA comma - LATTICES Parry conditions2441 A and positive t ilings beta - integer is also PN j Let beta greater 1defined period The [ 1 5 set ] by Z suba finite beta sum is selfj hyphen=0 ajβ similar and symmetrical with respect to \ [ \ begin { a l i g n e d } m \\ the where the integers aj satisfy 0 6 aj < β, together with \originsum : betaa { Z subj }\ betabeta subsetˆ Z{ subj } beta< comma\beta Z subˆ{ betam = + minus 1 Z} sub, beta m period = 0 , 1Denote , Z. sub . beta . to the , power N of plus. \\ : = Z sub beta cap R to the power ofm plus period The set Z j = 0 \end{ a l i g n e d }\ ] X j m+1 sub beta to the power of plus .... contains openajβ brace< 0 comma β , 1 closing m = 0, brace1, ..., N. and all the positive polynomials in beta for which the coefficients are given by the equations open parenthesis 2 period 1 closing parenthesis period j = 0 \noindent 2 . 2 . \quad Beta − integers , Parry conditions and t ilings Parry open square2 bracket . 2 . 33 Beta closing - integers square bracket , Parry has conditions shown that and the t knowledge ilings Let ofβ d > sub1. The beta Let $ \beta > 1 .$ Theset $Z {\beta }$ i s s e l f − similar and symmetrical with respect to the open parenthesis 1 closingset Z parenthesisβ is self - similar suffices and to symmetrical exhaust all with respect to the the possibilities of such polynomials by the so hyphen called .. quotedblleft Conditions of Parry \ beginopen{ parenthesisa l i g n ∗} CP sub beta closing parenthesis quotedbl : let open parenthesis c sub i closing parenthesiso r i g i n i geqslant : \\\beta 1 in A subZ beta{\ tobeta the power}\ ofsubset N be the followingZ origin{\beta: sequence} :,Z {\beta } = − Z {\beta } . open parenthesis 2 period 3 closing parenthesisβZβ ⊂ Zβ, Zβ = −Zβ. \endc sub{ a l 1i g c n sub∗} 2 c sub 3 times times times = Case 1 t sub 1 t sub 2 t sub 3 times times times if d sub beta open parenthesisDenote 1 closing+ := parenthesis∩ +. =The 0 period set + t subcontains 1 t sub{ 20 times , 1 } timesand all times the positiveis infinite \noindent Denote $ ZZ ˆβ{ + } Z{\β betaR } :Zβ = Z {\beta }\cap R ˆ{ + } comma Case 2 open parenthesispolynomials t sub in β 1 tfor sub which 2 times the times coefficients times t are sub given m minus by the 1 open equations parenthesis ( 2 . 1 t ) sub . . $ Theset $Zˆ{ + } {\beta }$ \ h f i l l c o n t a i n s \{ 0 , 1 \} and all the positive m minus 1 closing parenthesisParry [ 33 closing ] has parenthesis shown that to the the knowledge power of of omegadβ(1) if suffices d sub beta to exhaust open parenthesis all 1 closing parenthesis isthe finite possibilities Case 3 and of = such to 0 polynomials period t sub by 1t the sub so 2 - times called times “ Conditionstimes t sub of m Parrycomma \noindent polynomials in $ \beta $N for which the coefficients are given by the equations ( 2 . 1 ) . where open parenthesis(CPβ closing) ” : let parenthesis (ci)i > 1 ∈ toAβ thebe power the following of omega sequence means that : the word within open parenthesis closing parenthesis( 2 . 3 ) is indefinitely repeated period We have \noindentc sub 1 = tParry sub 1 = [ floorleft 33 ] has beta shown floor period that .. Thenthe knowledge the positive polynomial of $ d sum{\ subbeta i =} 0 to( the 1 ) $ suffices to exhaust all power of v y i beta to the power of minus i plust1 vt2 commat3 ··· ..ifd withβ(1) v = geqslant 0.t1t2 ··· isinfinite,  ω 0 comma y i in N comma belongsc1c2c3 · to ·· Z= sub beta(t1t2 to··· thetm power−1(tm − of1)) plus ififd andβ(1) onlyisfinite if y i in A sub beta and\noindent the followingthe v possibilities plus 1 of such polynomials by the so − c a l l e d \quad ‘‘ Conditions of Parry  and = to0.t1t2 ··· tm, $inequalities ( CP comma{\beta for all} j)$ = 0 comma ’’:let 1 comma $( 2 comma c period{ i } period) period i \ commag e q s l a v n commat 1 are\ in satisfiedA ˆ{ : N } {\wherebeta ()ω}means$ be that the the following word within sequence ( ) is indefinitely : repeated . We have Pv −i+v Equation: open parenthesisc1 = 2t1 period= 4bβ closingc. Then parenthesis the positive .. open polynomial parenthesisi=0 CPyiβ sub beta, closingwith \noindent ( 2 . 3 ) + parenthesis : open parenthesisv > 0, yyi sub∈ jN comma, belongs y sub to Z jβ plusif and 1 comma only if yyi sub∈ j plusAβ and 2 comma the following period periodv +1 period comma y v minusinequalities 1 comma , y for v comma all j = 0, comma1, 2, ..., v, 0 commaare satisfied 0 comma : period period period closing parenthesis\ [ c { 1 prec} openc { parenthesis2 } c c{ sub3 1}\ commacdot c sub\ 2cdot comma c\ subcdot 3 comma= period\ l e f t \{\ periodbegin period{ a l i g n e d } & closingt { 1 parenthesis} t { 2 } t { 3 }\cdot \cdot \cdot i f d {\beta } ( 1where ) quotedblleft = 0 . prec( tCP quotedblβ{):(1 } meansyj,t yj+1 lexicographical{, y2j+2}\, ..., yvcdot− smaller1, yv,\0cdot, 0 period, 0, ...) For≺\cdot(c a1, negativec2, c3,i ... s) polynomial in (2. f4) i n i t e comma, \\ we where “ ≺ ” means lexicographical smaller . For a negative polynomial , we &consider ( the t above{ 1 criterium} t { applied2 }\ tocdot its opposite\cdot period \cdot t { m − 1 } ( consider the above criterium applied to its opposite . t The{ m set} Z − sub beta1 can ) be )viewed ˆ{\ asomega the set} of verticesi f d of{\ the tilingbeta T} sub( beta 1 of the ) is finite \\ The set can be viewed as the set of vertices of the tiling T of the &real line and for which = tothe tiles 0 are .the closed tZβ { intervals1 } t whose{ 2 extremities}\cdot are \cdot \cdotβ t { m } real line for which the tiles are the closed intervals whose extremities are , \twoend successive{ a l i g n e d beta}\ right hyphen. \ integers] period two successive β− integers . In the rest of Section 2 we assume that beta is a quadratic unitary Pisot In the rest of Section 2 we assume that β is a quadratic unitary Pisot number number period Such numbers are of two types open square bracket 8 closing square bracket : if beta . Such numbers are of two types [ 8 ] : if β is a quadratic unitary Pisot is\noindent a quadraticwhere unitary Pisot $ ( ) ˆ{\omega }$ means that the word within ( ) is indefinitely repeated . We have number , of minimal polynomial denoted by P (X) and conjugate denoted number comma of minimal polynomial denoted by P sub beta openβ parenthesis X closing parenthesis by β0, then it satisfies exclusively one of the two following relations ( [ 1 8 ] and\noindent conjugate denoted$ c { 1 } = t { 1 } = \ l f l o o r \beta \ rfloor . $ \quad Then the positive polynomial Lemma 3 p . 72 1 ) : $ \bysum betaˆ{ tov the} power{ i of = prime 0 comma} y then i it satisfies\beta exclusivelyˆ{ − i one +of the v two} following, $ \ relationsquad with Case ( i )β is the dominant root of P (X) = X2 − aX − 1, with a 1, open$ v parenthesis\ g e q s l a open n t $ square bracket 1 8 closing square bracket β > or $Lemma 0 ,3 p period y i 72 1\ closingin parenthesisN ,$ : belongs to $Zˆ{ + } {\beta }$ if and only if $ yCase i.. open\ in parenthesisATOME{\ 5 i 6 closing (beta 2 0 0 6 parenthesis)} ,$ FASCICULE and 7beta the is following the dominant root$v of P + sub beta 1$ open parenthesis Xinequalities,forall closing parenthesis = X to the power $j of = 2 minus 0 aX , minus 1 1 comma , 2 with , . . . , v , $a geqslant are satisfied 1 comma or : TOME 5 6 open parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 \ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 4 ) $} ( CP {\beta } ) : ( y { j } , y { j + } 1 , y { j + } 2 , . . . , y v − 1 , y v ,0,0,0,...) \prec ( c { 1 } , c { 2 } , c { 3 } ,...) \end{ a l i g n ∗}

\noindent where ‘ ‘ $ \prec $ ’’ means lexicographical smaller . For a negative polynomial , we consider the above criterium applied to its opposite .

\ hspace ∗{\ f i l l }The s e t $ Z {\beta }$ can be viewed as the set of vertices of the tiling $ T {\beta }$ o f the

\noindent real line for which the tiles are the closed intervals whose extremities are

\noindent two successive $ \beta − $ i n t e g e r s .

In the rest of Section 2 we assume that $ \beta $ is a quadratic unitary Pisot number . Such numbers are of two types [ 8 ] : if $ \beta $ is a quadratic unitary Pisot

\noindent number , of minimal polynomial denoted by $ P {\beta } (X ) $ and conjugate denoted

\noindent by $ \beta ˆ{\prime } , $ then it satisfies exclusively one of the two following relations ( [ 1 8 ]

\noindent Lemma 3 p . 72 1 ) :

Case \quad ( i $ ) \beta $ is the dominant root of $ P {\beta } ( X ) = X ˆ{ 2 } − aX − 1 , $ with $ a \ geqslant 1 ,$ or

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2442 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentCase open2442 parenthesis\quad ii closingJean parenthesis− P i e r rbeta e GAZEAU is the dominant $ \& root $ ofJean P sub− betaLouis open VERGER parenthesis− GAUGRY X closing parenthesis = X to the power of 2 minus aX plus 1 comma with \ centerlinea geqslant 3{ periodCase ( i i $ ) \beta $ is the dominant root of $ P {\beta } (In X Case ) open = parenthesis X ˆ{ 2 i closing} − parenthesisaX + comma 1 beta , $ to with the power} of prime = minus 1 slash beta comma the alphabet A sub beta is open brace 0 comma 1 comma period period period comma a closing\ [ a brace\ g e with q s l a a n =t2442 floorleft 3Jean beta. -\ P]floor i erre GAZEAU comma & Jean - Louis VERGER - GAUGRY 2 and d sub beta open parenthesisCase ( ii 1 closing)β is the parenthesis dominant = root 0 period of Pβ( aX 1) is= finiteX − periodaX + 1, with In Case .. open parenthesis ii closing parenthesis comma beta to the power of prime = 1 slash beta \ centerline { InCase(i $) , \beta ˆ{\prime } = − 1 / \beta comma A sub beta = open brace 0 comma 1 comma perioda > period3. period comma a minus 1 closing , $ the alphabet $A {\beta }$ i s $ \{ 0 , 1 , . . . , brace with a = floorleft beta floor plus 1 period0 The In Case ( i ), β = −1/β, the alphabet Aβ is {0, 1, ..., a} with a = bβc, a Reacutenyi\} $ with beta hyphen $ a expansion = \ l f l of o o 1 r is d\ subbeta beta open\ rfloor parenthesis, $ 1 closing} parenthesis = 0 and dβ(1) = 0.a1 is finite . period open parenthesis a minus 1 closing parenthesis0 open parenthesis a minus 2 Case 1 omega Case 2 \noindent and $ dIn Case{\beta ( ii} ),( β = 1/β, A )β = ={0, 1, ..., 0 a − .1} with aa 1$= bβ isfinite.c + 1. The period ω In both cases the tilingR´enyi Tβ sub− expansion beta can ofbe 1 obtained is dβ(1) directly = 0.(a − from1)(a a− substitution2) \ centerlinesystem on a{ twoIn hyphen Case \ letterquad alphabet( i i which$ ) is , associated\beta toˆ beta{\prime in a canonical. } = way 1 / \beta In both cases the tiling T can be obtained directly from a substitution system ,Aopen square{\beta bracket} 1= 4 closing\{ square0 ,bracketβ 1 open , square . bracket . . 34 closing , a square− bracket1 \} open$ on a two - letter alphabet which is associated to β in a canonical way [ 1 4 ] [ 34 squarewith bracket $ a 40 = closing\ l f lsquare o o r bracket\beta period\ Hencerfloor comma+ the 1 number . $ of open The parenthesis} noncon- ] [ 40 ] . Hence , the number of ( noncongruent ) t iles of T is 2 in both cases , gruent closing parenthesis t iles of T sub beta is 2 in both β of respective lengths : \noindentcases commaR\ ’ of{ e respective} nyi $ lengths\beta : − $ expansion of 1 is $\ l e f t . d {\beta } Case ( i ) : 1 and 1/β, Case ( ii ) : 1 and 1 − 1/β (Case 1 open ) parenthesis = 0 i .closing ( parenthesis a − : 11 and ) 1 slash ( beta a comma− ..2 Case )\ begin open parenthesis{ a l i g n e d } ii & and is a Meyer set [ 8 ] [ 20 ] . closing\omega parenthesis\\ : 1 andZ 1β minus 1 slash beta 2 . 3 . Averaging sequences of finite approximants of &.and Z sub\end beta{ a isl i g a n Meyer e d }\ right set open. $ square bracket 8 closing square bracket open square bracketZβ 20 An averaging sequence (U )l 0 of finite closing square bracketDefinition period 2 . 8 . — l > approx−i− mants of is given by a closed interval J whose interior contains \noindent2 period 3In period both .. Averaging cases the sequences tilingZ ofβ finite $ T approximants{\beta } of$ Z sub can beta be obtained directly from a substitution the origin , and a real number t > 0, such that : systemDefinition on .. a 2 two period− letter 8 period alphabet emdash An which averaging is sequence associated open parenthesis to $ \beta U sub$ l inclosing a canonical way parenthesis[ 1 4 ] l [ geqslant 34 ] 0[ .. 40 offinite ] . Hence appro to , the the power number of x-i ofhyphen ( noncongruent ) t iles of $T {\beta }$ U = tlJ ∩ , l = 0, 1, ... i smants 2 in of both Z sub beta is given by a closed intervall J whose interiorZβ contains the cases , of respective lengths : origin comma andA a real natural number averaging t greater sequence 0 comma of such that approximants : for the beta - integers is U sub l = t to theyielded power of by l Jthe cap sequence Z sub beta ((−B commaN ) ∪ B lN =)N 0> comma0, where 1 comma period period period \ centerlineA natural averaging{Case(i):1and sequence of such approximants $1 / for\beta the beta hyphen, $ \ integersquad Case ( ii ) : 1 and $ 1is yielded− by1 the / sequence\beta open$ parenthesis} open parenthesis minus B sub N closing parenthesis cup N B sub N closing parenthesis N geqslantB 0N comma= {x ∈ whereZβ | 0 6 x < β },N = 0, 1, ... (2.5) \noindentEquation:and open parenthesis $ Z {\ 2beta period}$ 5 closing is aMeyer parenthesis set .. B [ sub 8 ]N =[ braceleftbig20 ] . x in Z sub beta bar 0 leqslant x less betaIt is to easy the power to check of N that bracerightbig card (BN comma) = cN N, a = number 0 comma in 1 the comma Fibonacci period - period like period\ centerline {2 . 3sequence . \quad definedAveraging by the recurrence sequences of finite approximants of $ Z {\beta }$ } It is easy to check that card open parenthesis B sub N closing parenthesis = c sub N comma a number in the Fibonacci hyphen like c = ac ± c (2.6) \noindentsequence definedD e f i n by i t i the o n recurrence\quad 2 . 8 . −−− NAn+2 averagingN+1 N sequence $ ( U { l } ) l \ g e q s l a n t 0 $ \quad of finite $ appro ˆ{ x−i } − $ Equation: open parenthesiswith c = 21, periodc = a, 6and closing where parenthesis “ + ” .. stands c sub for Ncase plus ( 2 i = ) awhereas c sub N plus “ − 1 plusminuxmants o c f sub $ N Z {\beta0 }$1 is given by a closed interval $ J $ whose interior contains the origin , and a” real is for number $ t > 0 ,$ suchthat : with c sub 0 = 1 commacase ( c i sub i ) 1. = a comma and where .. quotedblleft plus quotedblright .. stands for case open parenthesis i closing parenthesis whereas .. quotedblleft minus quotedblright is for \ [U { l } = t ˆ{ l } J \cap− − −Z {\beta } , l = 0 , 1 case open parenthesisProposition i i closing parenthesis2.9 periodT he cardinal cN of the approximant set BN ,... \ ] . Proposition 2 periodasymp Row - 1 emdash totically Row behaves 2 period as follows . cardinal c sub N of the approximant set B sub N asymp hyphen totically behaves as follows \noindentEquation:A open natural parenthesis averaging 2 period sequence 7 closing parenthesisβ ofN such .. approximants c sub N = beta for to the the power beta of− Ni n t e g e r s is yielded by the sequence $ (c (N = − +Bo(1){ N for}Nlarge) \cup B { N (2}.7) ) divided by gamma plus o open parenthesis 1 closing parenthesisγ for N large Equation: where gamma = N \ geqslant 0 , $ where bracelefttp-braceleftmid-braceleftbt 1 to the power of 1 1 from two-line to(i) minus, to the power of minus CaseCase 1 divided by a beta open parenthesis 1 divided by beta minus 1 closing( parenthesisii). 2 = line-beta beta \ begin { a l i g n ∗} 1 two−line 1 1+ 2 whereγ = {1 1 − 1 β − 1)2 = line − betaβ(1+β) open parenthesis 1 plus beta closing parenthesis to the power of 1− plusaβ ( 2 .. Case to the power of Case sub\ tag open∗{$ parenthesis ( 2 i . i closing 5 parenthesis ) $} B period{ N } to the= power\{ ofx open parenthesis\ in Z i{\ closingbeta parenthesis}\mid comma0 \ l e q s l a n t xANNALES< DE L\ ’beta I NS T ITUTˆ{ FOURIERN }\} , N = 0 , 1 , . . . ANNALES DE L quoteright I NS T ITUT FOURIER \end{ a l i g n ∗}

\noindent It is easy to check that card $ ( B { N } ) = c { N } , $ a number in the Fibonacci − l i k e sequence defined by the recurrence

\ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 6 ) $} c { N + 2 } = a c { N + 1 }\pm c { N } \end{ a l i g n ∗}

\noindent with $ c { 0 } = 1 , c { 1 } = a ,$ andwhere \quad ‘‘ $ + $ ’ ’ \quad stands for case ( i ) whereas \quad ‘ ‘ $ − $ ’ ’ i s f o r

\noindent case ( i i ) .

Proposition $\ l e f t . 2 . 9\ begin { array }{ c} −−− \\ . \end{ array }The\ right . $ c a r d i n a l $ c { N }$ of the approximant set $ B { N }$ asymp − totically behaves as follows

\ begin { a l i g n ∗} \ tag ∗{$ ( 2 . 7 ) $} c { N } = \ f r a c {\beta ˆ{ N }}{\gamma } + o ( 1 ) for N large \\\ tag ∗{$ where \gamma = \{ 1 ˆ{ 1 } 1 ˆ{ two−l i n e } { − ˆ{ − } \ f r a c { 1 }{a{\beta }} ( }\ f r a c { 1 }{\beta } − 1 ) 2 = l i n e −beta {\beta ( 1 + \beta ) }ˆ{ 1 + 2 }$} Case ˆ{ Case }ˆ{ ( i ) , } { ( i i ) . } \end{ a l i g n ∗}

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2443 DIFFRACTIONProofRow 1 emdash OF BETA Row− 2LATTICES period Proof\quad suffices2443 to observe $beta\ l e fto t . the Proof power\ begin of N ={ array b sub}{ c subc} N −−− comma \\ for. N\end geqslant{ array 0 comma} I t \ right .$ suffices to observe and to apply Proposition 5 in open square bracket 1 3 closing square bracket period square \ [ 3\ periodbeta ..ˆ{ CutN hyphen} = and b hyphen{ c { projectN }} schemes, and f o r beta N hyphen\ g e integers q s l a n t 0 , \ ] The most popular geometrical models for quasicrystals are the so hyphen called − − − cut hyphen and hyphenDIFFRACTION project sets OF period BETA .. - These LATTICES sets .. actually2443 P roof were previouslyIt suffices introduced to ob- by \noindentMeyer openandto square bracket apply 28 Proposition closing square bracket5 in .. $ open [ square 1 3 bracket ] 29. .closing\ square square bracket $ serve .. open square bracket 3 1 closing square bracket .. open square bracket 36 closing square bracket in the context\ centerline of Harmonic{3 . Analysis\quad Cut and Num− and hyphen− project schemes and beta − i n t e g e r s } βN = b , forN 0, ber Theory and christened by him model s e ts periodcN First comma> a cut hyphen and hyphen The most popular geometrical models for quasicrystals are the so − c a l l e d projection and to apply Proposition 5 in [13].  cutscheme− and is the− followingproject3 . sets Cut . \ -quad andThese - project s e t s schemes\quad actually and beta were - integers previously introduced by Line 1 R to the powerThe of d most pi sub popular 1 minus geometrical R to the power models of for d timesquasicrystals G pi sub are 2 theright so arrow - called Gcut Line 2\noindent cup Line 3 DMeyer- [ and 28 - ] project\quadsets[ 29. These ] \quad sets[ actually 3 1 ] \ werequad previously[ 36 ] introduced in the contextby of Harmonic Analysis and Num − berwhere Theory pi sub and 1 ....Meyer christenedand pi [ sub 28 ]2 .... by[ 29 are him ] the [ modelcanonical3 1 ] [ s 36 projection e ] ints the . context First mappings of , Harmoniccomma a cut G− isAnalysisand a lo− callyp and r o j e c t i o n schemecompact is abelian the groupfollowingNum comma - ber Theory called and the christenedinternal space by him commamodel R sto e the ts power. First of , adcut is called - and the - physical projection scheme is the following \ [ \spacebegin open{ a l parenthesis i g n e d } R d ˆ geqslant{ d }\ 1 closingpi { parenthesis1 }{ − } commaR ˆ{ D isd a}\ latticetimes comma iG period\ pi e period{ 2 }{\rightarrow } G \\ d d a discrete subgroup of R to the power of d timesR G suchπ1− thatR × Gπ2→ G \opencup parenthesis\\ R to the power of d times G closing parenthesis slash∪ D is compact period The D \end{ a l i g n e d }\ ] projection pi sub 1 bar D is 1 hyphen to hyphen 1 comma and pi subD 2 open parenthesis D closing parenthesis is dense in G period where π1 and π2 are the canonical projection mappings ,G is a lo cally d \noindentLet M = piwhere sub 1 opencompact $ \ parenthesispi abelian{ 1 } Dgroup$ closing\ h , called f i parenthesis l l theandinternal $ and\ pi set space *{ =2 pi, R} sub$is2 called\ circh f i open thel l arephysical parenthesis the canonical projection mappings d pi$ sub , 1 G$ bar D isalo closingspace parenthesis cally(d > 1) to,D theis power a lattice of , minusi . e . 1 :a M discrete right arrow subgroup G period of R The× G setsuch Capital that d Lambda subset R to the(R power× G)/D of dis compact . The projection π1 | D is 1 - to - 1 , and π2(D) is dense \noindentis a modelcompact set if therein G. abelian exist a cut group hyphen , and called hyphen the projection internal scheme space and a relatively $ , R ˆ{ d }$ −1 d iscompact called set the Capital physical OmegaLet M subset= π1(D G) of and non set hyphen∗ = π2 empty◦ (π1 | interiorD) : suchM → thatG. CapitalThe set Lambda Λ ⊂ R = open brace x in M baris x a tomodel the power set ofif there* in Capital exist a Omega cut - and closing - projection brace period scheme and a relatively ∗ \noindentThe set Capitalspace Omegacompact $ ( is called set d Ω a\⊂ windowgeqslantG of non period - empty 1 interior ) , such D$ that isΛ = a{x lattice∈ M | x ,∈ iΩ} .. e . a discrete subgroup of $ RThe ˆ{ followingd }\ wastimes provedThe setG by Ω $ Meyeris called such open a thatwindow parenthesis. Theorem 9 period 1 in open square bracket 30 closing square bracket closingThe parenthesisfollowing was : A proved model by s e Meyer t ( Theorem 9 . 1 in [ 30 ] ) : A model s \noindentis a Meyer s$ e t( periode R t ˆis Conversely{ ad Meyer}\ s iftimes e Capital t . Conversely LambdaG ) if is aΛ / Meyeris a D$ Meyer s e t is scomma e compactt , there there exists exists . The a a model model projection s s e$ t\ Capitalpi { Lambda1 }\ submide t 0 Λ0D $ i s 1 − to − 1 , and $ \ pi { 2 } ( D )$ isdense insuch $ that G Capital . $ Lambda subset Capital Lambda sub 0 period By decoration of a point set Capital Lambda subset R to the power of d comma d geqslant 1 comma suchthatΛ ⊂ Λ . weLet mean $ Mthe new = point\ pi set { 1 } ( D )$ andset $0 ∗ = \ pi { 2 }\ circ ( Capital\ pi Lambda{ 1 }\ plusmid F whereD F is ) a ˆ finite{ − point1 } set containing:M 0\rightarrow such that any elementG of . $ The s e t By decoration of a point set Λ ⊂ d, d 1, we mean the new point set $ \CapitalLambda Lambda\subset plus F hasR a ˆ unique{ d }$ writing in this sum decompositionR > semicolon Capital Lambda Λ + F where F is a finite point set containing 0 such that any element of plusis F a is model a decoration set if there exist a cut − and − projection scheme and a relatively Λ + F has a unique writing in this sum decomposition ; Λ + F is a decoration compactof Capital set Lambda $ \ withOmega F period\subset G $ o f non − empty interior such that $ \Lambda of Λ with F. = In\{ the restx of Section\ in 3M beta\ ismid a quadraticx ˆ{ unitary ∗ } Pisot\ in number\Omega period\} . $ In the rest of Section 3β is a quadratic unitary Pisot number . The3 period s e t 1 $ period\Omega .. One$ hyphen is called dimensional a window model sets . as subsets of beta hyphen integers 3 . 1 . One - dimensional model sets as subsets of beta - integers Let us introduce the alge braic model s e t Let us introduce the alge braic model s e t TheEquation: following open was parenthesis proved 3 period by Meyer 1 closing ( Theorem parenthesis 9 .. . Capital 1 in Sigma [ 30 to ] the) : power A model of Capital s e t Omegais a = Meyer open brace s ex t = . m Conversely plus n beta in if Z open $ \ squareLambda bracket$ is beta a closing Meyer square s e t bracket , there bar x exists to a model s e t $ \Lambda { 0 }$ the power of prime = m plus n beta toΩ the power of prime in Capital0 Omega closing0 brace comma Σ = {x = m + nβ ∈ Z[β] | x = m + nβ ∈ Ω}, (3.1) TOME 5 6 open parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 \ begin { a l i g n ∗} such that \LambdaTOME 5 6 ( 2 0 0\ 6subset ) , FASCICULE 7\Lambda { 0 } . \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }By decoration of a point set $ \Lambda \subset R ˆ{ d } , d \ geqslant 1 , $ we mean the new point set

\noindent $ \Lambda + F $ where $ F $ is a finite point set containing 0 such that any element of

\noindent $ \Lambda + F $ has a unique writing in this sum decomposition $ ; \Lambda + F$ is a decoration

\noindent o f $ \Lambda $ with $ F . $

\ centerline { In the rest of Section $ 3 \beta $ is a quadratic unitary Pisot number . }

\ centerline {3 . 1 . \quad One − dimensional model sets as subsets of beta − i n t e g e r s }

\ centerline { Let us introduce the alge braic model s e t }

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 1 ) $}\Sigma ˆ{\Omega } = \{ x = m + n \beta \ in Z[ \beta ] \mid x ˆ{\prime } = m + n \beta ˆ{\prime } \ in \Omega \} , \end{ a l i g n ∗}

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2444 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindenthline circ 2444 \quad Jean − P i e r r e GAZEAU $ \& $ Jean − Louis VERGER − GAUGRY where Capital Omega subset R is such that the closure Capital Omega is compact and the interior Capital\ [ \ r u Omega l e {3em is}{0.4 pt }\ circ \ ] not empty period Though Z sub beta is symmetrical comma it is not a model set but a Meyer set comma the positive and negative parts of Z sub beta can be interpreted in terms of \noindentmodel setswhere only period2444 $ \OmegaJean - P i erre\subset GAZEAU &RJean $ - Louis is such VERGER that - GAUGRY the closure $ \Omega $ isProposition compact and 3 period the 1 interior period emdash $ \ openOmega square$ bracket i s 8 closing square bracket The algebraic characterizations of Z sub beta to the power of plus and Z sub beta◦ to the power of minus \noindentare the followingnot empty : . Though $ Z {\beta }$ is symmetrical , it is not a model set but a Meyer where Ω ⊂ R is such that the closure Ω is compact and the interior Ω is setCase , open the parenthesis positive i and closing negative parenthesis parts : Equation: of $ open Z {\ parenthesisbeta } 3$ period can 2 be closing interpreted paren- in terms of not empty . Though Zβ is symmetrical , it is not a model set but a Meyer set , thesismodel .. Z sets sub beta only to . the power of plus = Capital Sigma to the power of open parenthesis minus 1 the positive and negative parts of Zβ can be interpreted in terms of model sets comma beta closing parenthesis cap R to the power of plus comma Z sub beta to the power of minus = \ hspace ∗{\ f i l l } Propositiononly . 3 . 1 . −−− [ 8 ] The algebraic characterizations of minus Z sub beta to the power of plus = Capital Sigma to the power of open parenthesis minus+ beta $ Z ˆ{ + } {\beta }Proposition$ and $ 3Z . ˆ 1{ . − — } [{\ 8 ] betaThe algebraic}$ characterizations of Zβ and comma 1 closing parenthesis− cap R to the power of minus comma Equation: open parenthesis 3 period 3 closing parenthesis .. ZZ subβ beta to the power of prime subset open parenthesis minus beta comma beta are the following : closing\noindent parenthesisare andthe Z following sub beta to the : power of prime cap open parenthesis minus 1 comma 1 closing parenthesis = Z open square bracket beta closing square bracket cap open parenthesis minus 1 comma \ begin { a l i g n ∗} 1 closing parenthesis comma Case open parenthesis ii closing parenthesis : Equation:Case(i): open parenthesis 3Case period 4 closing ( i parenthesis ) : ..\\\ Ztag sub∗{ beta$ to( the 3 power . of plus 2 = ) Capital $} Z Sigmaˆ{ + to} {\ the powerbeta of} open= \Sigma ˆ{ ( − 1 , \beta+ (−)1,β}\) cap+ − R ˆ{ ++ } (−,β,1) Z− ˆ{ − } {\beta } square bracket 0 comma beta closing parenthesisZβ = Σ cap∩ RR to, theZβ power= −Z ofβ plus= Σ and Z∩ subR beta, to the(3 power.2) =of minus− =Z Capital ˆ{ + Sigma} {\ tobeta the power} 0= of open\Sigma parenthesisˆ{ ( 0 minus− beta \beta comma 0, closing 1 square ) }\ bracketcap Zβ ⊂ (−β, β) and Zβ ∩ (−1, 1) = Z[β] ∩ (−1, 1), (3.3) Rcap ˆ R{ to− the } power, \\\ oftag minus∗{$ comma ( 3 Equation: . 3 open ) parenthesis $} Z ˆ{\ 3 periodprime 5} closing{\beta parenthesis}\subset .. Z sub Case(ii): beta( subset− \ Capitalbeta Sigma, to\beta the power) of open and parenthesis Z ˆ{\ minusprime beta} {\ commabeta beta}\ closingcap parenthesis( − 1 , 1 ) = Z [ \beta+ ][0,β) \cap+ ( −− 1(−β,0] ,− 1 ) , \\ Case subset Z sub beta plus open brace 0 commaZ plusminux= Σ ∩ 1R dividedand byZ beta=Σ closing∩ braceR , period (3.4) ( i i ) : \\\ tag ∗{$ ( 3β . 4 ) $} Z ˆ{ β+ } {\beta } = \Sigma ˆ{ [ Now comma let us see how the set Z sub beta in Case open parenthesis i closing1 parenthesis open 0 , \beta ) }\cap R ˆ{ + } and⊂ ZΣ ˆ({−β,β −) }⊂ {\+beta{0, ±} }.= \Sigma(3.5) ˆ{ ( parenthesis resp period the decorated set Z-tildewide subZ betaβ def = Zβ β −Z \ subbeta beta plus, open 0 brace ] }\ 0cap commaR plusminux ˆ{ − } 1 divided, \\\ tag by∗{ beta$ ( closing 3 brace . in 5 Case ) open $} Z {\beta } \subset \Sigma ˆ{ ( − \beta , \beta ) }\subset Z {\beta } parenthesis i i closing parenthesisNow closing , let us parenthesis see how the can set beZ consideredβ in Case ( as i ) universal ( resp . thesupport decorated of model set + \{ 0 , \pm \ f r a c { 1 }{\beta }\} . sets like open parenthesisZeβdef= 3 period 1 closing parenthesis period \endLet{ a Capital l i g n ∗} Omega be a bounded1 window in R period It is always possible to find a lambda in Z Zβ + {0, ± β } in Case ( i i ) ) can be considered as universal support of model open square bracket betasets closing like ( 3 square . 1 ) . bracket \ hspaceand an∗{\ integerf i l l j} inNow Z such , let that usCapital see Omega how the subset set open $ parenthesis Z {\beta minus}$ beta in to theCase power ( i of ) j ( resp . the decorated set $ \ widetilde {Z} {\Letbeta Ω be} a boundeddef { window= }$ in R. It is always possible to find a λ in Z[β] comma beta to the powerand anof j integer closingj parenthesisin Z such that plus Ω lambda⊂ (−βj period, βj) + Letλ. Let Capital ∆ = β Delta−j(Ω =− betaλ) ⊂ to the power of minus j open( parenthesis−1, 1). Then Capital , one can Omega easily minus prove lambda that closing parenthesis subset \noindentopen parenthesis$ Z minus{\beta 1 comma} 1+ closing\{ parenthesis0 , period\pm Then\ f r comma a c { 1 one}{\ canbeta easily}\} prove that$ inCapital Case (Sigma i i to ) the ) power can be ofΩ Capital considered Omega = as openj universal−j parenthesis support minus open of0 parenthesis model 0 resp period Σ = (−(resp. + 1)) β {x ∈ Zβ(resp.Zeβ) | x ∈ ∆} + λ . plus 1 closing parenthesis closing parenthesis to the power of j beta to the power of minus j open brace \noindent sets like ( 3 . 1 ) . Ω x in Z sub beta open parenthesisTherefore Z respβ( resp period.Zeβ Z-tildewide) enumerates sub the beta model closing set parenthesisΣ . bar x to the power of prime in Capital Delta closing3 brace . 2 . plus An lambda alternate to the definition power of prime of beta period - integers Let $ \Omega $ be a bounded window in $R . $ It is always possible to find a Therefore Z sub beta open parenthesis resp− − period − Z-tildewide sub beta closing parenthesis+ enumer- Proposition 3.2 T he following characterization of Z holds : ates$ \lambda the model$ set in Capital $ Z Sigma [ to the\beta power. of] Capital $ Omega period β 2 andan3 period integer 2 period .. $ An j alternate $ in definition $Z$ of such beta hyphenthat integers$ \Omega1+β \subset ( − \beta ˆ{ j } Case ( i ) : with γ = β(1+β) , Proposition\beta ˆ{ 3 periodj } Row) +1 emdash\lambda Row 2 period. $ . followingLet $ \ characterizationDelta , = of\beta Z sub betaˆ{ − to thej } power( \Omega of plus holds− : \lambda ) \subset $ 1 1 − β 1 − β n + 1 Case open parenthesis i closing parenthesis+ : with gamma = 1 plus beta to the power of 2 divided Zβ = {bn = γn + − { }, n ∈ N}; (3.6) by\noindent beta open parenthesis$ ( − 1 plus1 beta , closing 1 parenthesis ) .β $1 +sub Thenβ comma ,β onecan1 + β easily prove that Equation: open parenthesis 3 period 6 closing parenthesis .. Z sub beta1 to the power of plus = \ [ \Sigma ˆ{\Omega } = ( − Case( ( ii resp ) : with .γ += 1 − 12 ) )ˆ{ j }\beta ˆ{ − braceleftbigg b sub n = gamma n plus 1 divided by beta 1 minus beta dividedβ , by 1 plus beta minus 1 minusj }\{ beta dividedx \ byin betaZ braceleftbigg{\beta n} plus( 1 divided resp by 1 . plus\ betawidetilde bracerightbigg{Z} {\ commabeta n in} N) bracerightbigg\mid x ˆ{\ semicolonprime }\ in \Delta \} + 1\lambdan ˆ{\prime } . \ ] + = {b = γn + { }, n ∈ }. (3.7) Case open parenthesis ii closing parenthesisZβ : withn gamma =β 1β minus 1 dividedN by beta to the power of 2 sub comma ANNALES DE L ’ I NS T ITUT FOURIER \noindentEquation:Therefore open parenthesis $ Z 3 period{\beta 7 closing} ( parenthesis $ resp .. $ Z . sub beta\ widetilde to the power{Z} of{\ plusbeta = } braceleftbigg) $ enumerates b sub n = the gamma model n plus set 1 divided $ \Sigma by betaˆ{\ braceleftbiggOmega } n divided. $ by beta bracerightbigg comma n in N bracerightbigg period \ centerlineANNALES{ DE3 L. quoteright 2 . \quad I NSAn T alternate ITUT FOURIER definition of beta − i n t e g e r s } \ centerline { Proposition $\ l e f t . 3 . 2\ begin { array }{ c} −−− \\ . \end{ array }The\ right . $ following characterization of $ Z ˆ{ + } {\beta }$ holds : }

\ centerline {Case ( i ) : with $ \gamma = \ f r a c { 1 + \beta ˆ{ 2 }}{\beta ( 1 + \beta ) } { , }$ }

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 6 ) $} Z ˆ{ + } {\beta } = \{ b { n } = \gamma n + \ f r a c { 1 }{\beta }\ f r a c { 1 − \beta }{ 1 + \beta } − \ f r a c { 1 − \beta }{\beta }\{\ f r a c { n + 1 }{ 1 + \beta }\} , n \ in N \} ; \end{ a l i g n ∗}

\ centerline {Case ( ii ) : with $ \gamma = 1 − \ f r a c { 1 }{\beta ˆ{ 2 }} { , }$ }

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 7 ) $} Z ˆ{ + } {\beta } = \{ b { n } = \gamma n + \ f r a c { 1 }{\beta }\{\ f r a c { n }{\beta }\} , n \ in N \} . \end{ a l i g n ∗}

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2445 \ hspaceProofRow∗{\ 1f i emdash l l }DIFFRACTION Row 2 period OF Proof BETA each− LATTICES nonnegative\ nquad comma2445 let rho L open parenthesis n closing parenthesis open parenthesis resp period rho S open parenthesis n closing parenthesis equiv rho open\ hspace parenthesis∗{\ f i l n l } closing$\ l e parenthesis f t . Proof \ closingbegin parenthesis{ array }{ c denote} −−− \\ . \end{ array }For\ right . $ eachthe nonnegativenumber of long open $ n parenthesis , $ resplet period $ \rho short closingL( parenthesis n ) tiles ($resp$. between the n th beta hyphen\rho integerS ( and n ) \equiv \rho ( n ) )$ denote the origin 0 period The length of long tile is 1 whereas theDIFFRACTION length of short OF BETA tile is - LATTICES 2445 \noindent the number of long− (− − resp . short ) tiles between the $ n $ th beta − i n t e g e r and l sub S = 1 slash beta openP parenthesis roof caseF openor each parenthesis nonnegative i closingn, let parenthesisρL(n)( resp closing.ρS(n parenthesis) ≡ ρ(n)) orthe l sub origin S = 1 minus 0 . 1The slash length beta open of parenthesis. long tile case is open 1 whereas parenthesis the ii closing length parenthesis of short closing tile is denote parenthesis$ l { periodS } = From 1the two / equalities\beta ($ case(i))or $l { S } = 1 the number of long ( resp . short ) tiles between the n th beta - integer and the − Equation:1 / open\beta parenthesis( $ 3 period case 8( closing ii ) parenthesis) . From .. the n = two rho Lequalities open parenthesis n closing origin 0 . The length of long tile is 1 whereas the length of short tile is l = 1/β( parenthesis plus rho open parenthesis n closing parenthesis comma Equation: open parenthesisS 3 period case ( i ) ) or l = 1 − 1/β( case ( ii ) ) . From the two equalities 9\ begin closing{ parenthesisa l i g n ∗} .. b sub n = rhoS L open parenthesis n closing parenthesis plus l sub S rho open parenthesis\ tag ∗{$ ( n closing 3 parenthesis . 8 )comma $} n = \rho L ( n ) + \rho ( n ), \\\ tag ∗{$ ( 3 . 9 ) $} b { n } = \rho L ( n ) + we derive the expression for b sub n in terms of n andn = betaρL(n :) + ρ(n), (3.8) l Equation:{ S }\ openrho parenthesis( n 3 period ) , 1 0 closing parenthesis .. b sub n = n plus open parenthesis l sub\end S{ minusa l i g n 1∗} closing parenthesis rho open parenthesisbn = nρL closing(n) + parenthesislSρ(n), period (3.9) Now we know from Proposition 3 period 1 that the sets Z sub beta to the power of plus of nonnegative we derive the expression for b in terms of n and β : beta\noindent hyphen we derive the expression forn $ b { n }$ in terms of $n$ and $ \integersbeta are: also $ defined through the following constraints on their Galois conjugates : Equation: open parenthesis 3 period 1 1 closing parenthesis .. Z sub beta to the power of bn = n + (lS − 1)ρ(n). (3.10) plus\ begin = open{ a l i brace g n ∗} x = r plus s beta in Z open square bracket beta closing square bracket bar x geqslant \ tag ∗{$ ( 3 . 1 0 ) $} b { n } = n + ( l { S } − 1 0 and x to the power of prime = r minus s 1 divided by beta in open parenthesis+ minus 1 comma beta ) \rho ( nNow ) we know . from Proposition 3 . 1 that the sets Zβ of nonnegative beta - closing parenthesis closingintegers brace are case also defined open parenthesis through the i closing following parenthesis constraints comma on their Equation: Galois open parenthesis\end{ a l i g 3 n period∗} 1 2 closing parenthesis .. Z sub beta to the power of plus = open brace x = r plus s beta in Z open square bracket beta closing square bracket bar x geqslant 0 and x to the power of prime \noindent Now we know from Proposition 3 . 1 that the sets $ Z ˆ{ + } {\beta }$ = r plus s 1 divided by beta in open square bracket 0 comma beta closingconjugates parenthesis: closing brace case openof nonnegative parenthesis ii closing beta parenthesis− period + = {x = r + sβ ∈ [β] | x 0and (3.11) integersThere results are for also the conjugate defined of through openZβ parenthesis the following 3 periodZ 1 constraints 0 closing> parenthesis on their the following Galois 1 inequalities obeyed x0 = r − s ∈ (−1, β)} case(i), \ beginby the{ a nonnegative l i g n ∗} integer rho open parenthesis n closingβ parenthesis : conjugates : \\\ tag ∗{$ ( 3+ . 1 1 ) $} Z ˆ{ + } {\beta } = \{ Equation: open parenthesis 3 period 1Z 3 closing= {x = parenthesisr + sβ ∈ Z ..[β n] divided| x 0and by 1 plus beta minus (3.12) beta x = r + s \beta \ in β Z[ \beta ] >\mid x \ g e q s l a n t divided by 1 plus beta less rho open parenthesis n closing parenthesis1 less n divided by 1 plus beta plus 1 0 and \\ x ˆ{\prime } = r − x0 =s r +\ fs r a∈ c {[0,1 β)}{\} casebeta(ii).}\ in ( − divided by 1 plus beta case open parenthesis i closing parenthesisβ comma Equation: open parenthesis 3 period1 , 1 4 closing\beta parenthesis) \} .. n dividedcase by ( beta minus i ) 1 less , rho\\\ opentag parenthesis∗{$ ( 3 n closing . parenthesis 1 2 ) $} Z ˆ{ + } {\beta } = \{ x = r + s \beta \ in Z[ leqslant n divided by betaThere case results open for parenthesis the conjugate ii closing of ( parenthesis3 . 1 0 ) the period following inequalities obeyed \beta ] \mid x \ geqslant 0 and \\ x ˆ{\prime } = r + s Since both intervalsby have the lengthnonnegative equal integer to 1 commaρ(n): we conclude that \ f rEquation: a c { 1 }{\ openbeta parenthesis}\ in 3 period[ 1 5 0 closing , parenthesis\beta ..) rho open\} parenthesiscase ( n closing i i paren- ) thesis. = floorleftbigg n plus 1 divided by beta plus 1 floorrightbigg case open parenthesis i closing \end{ a l i g n ∗} n β n 1 parenthesis comma Equation: open parenthesis− 3 period< ρ(n 1) 6< closing parenthesis+ case .. rho(i), open parenthesis(3.13) n closing parenthesis = floorleftbigg1 n + dividedβ 1 + byβ beta floorrightbigg1 + β 1 case + β open parenthesis ii closing \noindent There results for the conjugaten of ( 3 . 1 0n ) the following inequalities obeyed parenthesis period − 1 < ρ(n) 6 case(ii). (3.14) byCombining the nonnegative open parenthesis integer 3 period $ 1\ 5rho closing( parenthesis nβ ) comma : $ respβ period open parenthesis 3 period 1 6 closing parenthesisSince both with intervals open parenthesis have length 3 periodequal to 1 01 closing, we conclude parenthesis that and using floorleft x\ begin floor ={ xa l minus i g n ∗} open brace x closing brace yield \ tagopen∗{$ parenthesis ( 3 3 . period 1 6 closing 3 ) parenthesis $}\ f r a c { andn open}{ 1 parenthesis + \beta 3 period} 7 − closing \ f r aparenthesis c {\beta }{ 1 + \beta } < \rho ( n ) < \ f r an c {+ 1n }{ 1 + \beta } + \ f r a c { 1 }{ 1 period square ρ(n) = b c case(i), (3.15) + TOME\beta 5 6} opencase parenthesis ( 2 i 0 0 6 ) closing , \\\ parenthesistag ∗{$β comma (+ 1 3 FASCICULE . 1 7 4 ) $}\ f r a c { n }{\beta } − 1 < \rho ( n ) \ l e q s l a n t \ f r an c { n }{\beta } case ( i i ρ(n) = b c case(ii). (3.16) ). β \end{ a l i g n ∗} Combining ( 3 . 1 5 ) , resp . ( 3 . 1 6 ) with ( 3 . 1 0 ) and using bxc = x − {x} \noindent Since bothyield intervals have length equal to 1 , we conclude that ( 3 . 6 ) and (3.7).  TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 \ begin { a l i g n ∗} \ tag ∗{$ ( 3 . 1 5 ) $}\rho ( n ) = \ l f l o o r \ f r a c { n + 1 }{\beta + 1 }\ rfloor case ( i ) , \\\ tag ∗{$ ( 3 . 1 6 ) $}\rho ( n ) = \ l f l o o r \ f r a c { n }{\beta }\ rfloor case ( i i ) . \end{ a l i g n ∗}

\noindent Combining ( 3 . 1 5 ) , resp . ( 3 . 1 6 ) with ( 3 . 1 0 ) and using $ \ l f l o o r x \ rfloor = x − \{ x \} $ y i e l d

\noindent (3.6)and$( 3 . 7 ) . \ square $ TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2446 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindent4 period ..2446 Diffraction\quad spectraJean − P i e r r e GAZEAU $ \& $ Jean − Louis VERGER − GAUGRY Since .. the beta hyphen integers or beta hyphen lattices or some .. decorated version of \ centerlinethem comma{4 besides . \quad their intrinsicDiffraction rich arithmetic spectra and} algebraic properties comma can be seen as a kind of universal support for quasicrystalline structures having Sincefive hyphen\quad orthe ten hyphenbeta − foldintegers or eight hyphen or beta fold− or twelvelattices hyphen or fold some symmetries\quad commadecorated we are version of naturallythem , led besides2446 theirJean intrinsic - P i erre GAZEAU rich& arithmeticJean - Louis VERGER and algebraic - GAUGRY properties , can beto seen fo cusas our a attention kind of on them universal comma4 in support . particular Diffraction for to examine quasicrystalline spectra their diffractive structures having f iproperties v e − or when ten we− assignfoldSince toor each the eight beta point− - in integersfold such sets or or beta atwelve weight - lattices− describingfold or some symmetries to decorated , version we are of naturally led tosome fo extent cus our its quotedblleft attentionthem , besides degree on their them quotedblright intrinsic , in rich particular of o arithmetic ccupation to period and examine algebraic We should properties their insist diffractive here , can on be the factproperties whenseen we as assign a kind to of universal each point support in for such quasicrystalline sets a weight structures describing having five to somethat thisextent concept its of- degree‘‘or ten degree of - fold o ccupation ’’ or eight of o of - ccupation beta fold orhyphen twelve integer . - We fold should sites symmetries encompasses insist , we here are naturally on the fact thatthe .. this quotedblleft conceptled Cut to of and fo degree Project cus our quotedblright attention of o ccupation on them .. approach , inof particular beta which− has tointeger become examine like sites their a paradigm diffractive encompasses for quathe hyphen\quad ‘‘ Cutproperties and Project when we ’’assign\quad to eachapproach point in such which sets ahas weight become describing like to a some paradigm for qua − sicrystalsicrystal experimentalists experimentalistsextent itsperiod “ degree It . is thus It ” of is crucial o ccupation thus to knowcrucial . We in a should comprehensive to know insist in here waya on comprehensive the fact that way thethe diffractive diffractive propertiesthis properties concept of quotedblleft of degree of ‘‘ weighted of weighted o ccupation quotedblright ’’ of\ betaquad .. - beta integerbeta hyphen− sitesintegers integers encompasses by by setting setting the up explicit upformulas explicit , whereas“ Cut the and\ Projectquad ‘‘ ” constant approach which weight has become ’’ \quad likecase a paradigm is already for qua - mathematically wellformulas established comma whereassicrystal [ 27 the experimentalists ] .. . quotedblleft constant . It is thus weight crucial quotedbl to know .. case in a is comprehensive already mathemati- way cally the diffractive properties of “ weighted ” beta - integers by setting up explicit \noindentwell established4 . 1 openformulas . \ squareh f i l , l bracket whereasFourier 27 the closing transform “ constant square bracket weightof a weighted period” case is already Dirac mathematically comb of beta − i n t e g e r s 4 period 1 period ....well Fourier established transform [ 27 ] of . a weighted Dirac comb of beta hyphen integers LetLet us us consider consider the4 the following . 1 following . pure point Fourier pure complex point transform Radon complex measure of a weighted Radon sup hyphen measure Dirac comb sup − of beta - portedported by by the the set set ofintegers beta of hyphen beta integers− i n t e Z g e sub r s beta $ Z= open{\ bracebeta b} sub= n bar\{ n in Zb closing{ n brace}\ :mid n Equation:\ in Z open\} parenthesisLet: $ us 4 consider period 1 the closing following parenthesis pure point .. mu complex = sum Radon w open measure parenthesis sup - n ported closing parenthesis delta sub bby sub the n set comma of beta n in - integers Z Zβ = {bn | n ∈ Z} : \ beginwhere{ wa l openi g n ∗} parenthesis x closing parenthesis is a bounded complex hyphen valued function comma \ tag ∗{$ ( 4 . 1 ) $}\mu = \sum w ( n ) \ delta { b { n }} called weight period In a spe hyphen X , \\ n \ in Z µ = w(n)δ , (4.1) cific context comma .. the latter will be given the meaning of ab siten occupation \end{ a l i g n ∗} probability period In open parenthesis 4 period 1 closing parenthesisn ∈ Z delta sub b sub n denotes the normalized Dirac measure supported \noindentby the singletonwhere openwhere $w bracew(x ( b) sub is x a n bounded closing ) $ brace complex is a period bounded - valued Denote complex function by delta ,− thecalledvalued Diracweight measure function. In open a , called weight . In a spe − parenthesiscific context delta open , spe parenthesis\quad - cificthe context open latter brace , the 0 will closing latter be will brace given be closing given the theparenthesis meaning meaning = of of1 aclosing site a site occupation parenthesis occupation probability.In $( 4 . 1 ) \ delta { b { n }}$ denotes the normalized Dirac measure supported period Clearly commaprobability . In (4.1)δb denotes the normalized Dirac measure supported by by the singleton $ \{ b { n }\}n . $ Denote by $ \ delta $ the Dirac measure the measure open parenthesisthe singleton 4 period{bn}. Denote 1 closing by parenthesisδ the Dirac is measure translation (δ({ bounded0}) = 1). commaClearly i ,period the $ ( \ delta ( \{ 0 \} ) = 1 ) .$ Clearly, e period for all compactmeasure K subset ( 4 R . 1 there ) is translation bounded , i . e . for all compact K ⊂ R there the measure ( 4 . 1 ) is translation bounded , i . e . for all compact $ K exists alpha sub Kexists such thatαK such supremum that sup suba∈ a| inµ R| ( barK + mua) bar6 openαK , parenthesisand so is a Ktempered plus a closing dis - \subset R $ the re R parenthesis leqslant alphatribution sub K .comma Its Fourier and transform so is a temperedµb is also dis a temperedhyphen distribution defined e xtribution i s t s $ period\alpha Itsby Fourier{ K transform}$ such mu-hatwide that $ is\sup also a{ tempereda \ in distributionR }\ definedmid \mu \midby ( K + a ) \ l e q s l a n t \alpha { K } , $ and so is a tempered dis − tributionEquation: open . Its parenthesis Fourier 4 period transform 2 closing $ parenthesis\widehat ..{\ mu-hatwidemu} $ is open also parenthesis a tempered q closing distribution defined −iqx X −iqbn parenthesis = mu parenleftbig e to the powerµb( ofq) minus = µ(e iqx) parenrightbig = w(n)e = sum. w open parenthesis(4.2) n closing\noindent parenthesisby e to the power of minus i qb sub n period n in Z n ∈ Z It may or may not be a measure period \ begin { a l i g n ∗} We know from PropositionIt may or 3 may period not 2 be that a measure b sub n has. the general form b sub n = gamma n plus \ tag ∗{$ ( 4 . 2 ) $}\widehat{\mu} ( q ) = \mu ( e ˆ{ − alpha sub 0 plus alpha openWe know brace from x open Proposition parenthesis 3 . n 2 closing that bn parenthesishas the general closing form bracebn = commaγn+ whereiqx } alpha) sub = 0 =\sum 1 minusw beta ( divided n by ) beta1− eβ open ˆ{ − parenthesisi qb 1 plus{ n beta}} closing. \\ parenthesisn \ in α0 + α{x(n)}, where α0 = β(1+β) α = 1/β in case ( i ), α0 = 0, α = 1 − 1/β Zsub comma alpha = 1 slash beta in case open parenthesis, i closing parenthesis comma alpha sub 0 = 0 in case ( ii ) . It contains a fractional part . Now , for any x ∈ , the “ comma\end{ a alpha l i g n ∗} = 1 minus 1 slash beta R fractional in case open parenthesis ii closing parenthesis period It contains a fractional part period Now comma ANNALES DE L ’ I NS T ITUT FOURIER for\noindent any x in RIt comma may the or .... may quotedblleft not be a fractional measure . ANNALES DE L quoteright I NS T ITUT FOURIER \ hspace ∗{\ f i l l }We know from Proposition 3 . 2 that $ b { n }$ has the general form $ b { n } = \gamma n + $

\noindent $ \alpha { 0 } + \alpha \{ x ( n ) \} , $ where $ \alpha { 0 } = \ f r a c { 1 − \beta }{\beta ( 1 + \beta ) } { , } \alpha = 1 / \beta $ incase(i $) , \alpha { 0 } = 0 , \alpha = 1 − 1 / \beta $

\noindent in case ( ii ) . It contains a fractional part . Now , for any $ x \ in R , $ the \ h f i l l ‘‘ fractional

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2447 DIFFRACTIONpart quotedblright OF BETA .. function− LATTICES x mapsto-arrowright\quad 2447 open brace x closing brace = x minus floorleft x floorpart in open ’ ’ \ squarequad bracketf u n c t i o0 n comma $ x 1 closing\mapsto parenthesis\{ is periodicx \} of period= 1 x and− so is \ l f l o o r x the\ rfloor piecewise continuous\ in [ function 0 e to, the 1 power )$ of minus is periodic i q alpha open of brace period1andso x closing brace period is Let \noindentc sub m openthe parenthesis piecewise q closing continuous parenthesis function = integral $ sub e ˆ0{ to − the poweri q of 1\ ealpha to the power\{ ofx minus\}} i q alpha. $ open Let DIFFRACTION brace x closing OF brace BETA e - toLATTICES the power2447 of minus part 2 ” pi imx function dx Equation:x 7→ {x} open= parenthesis 4 period 3x closing− bxc ∈parenthesis[0, 1) is periodic .. = i e of to period the power 1 and of so minus is iq alpha minus 1 divided by 2 pi\ begin m plus{ a q l alphai g n ∗} = openthe piecewise parenthesis continuous minus 1 closingfunction parenthesise−iqα{x}. Let to the power of m e to the power of minusc { im alpha} divided( q by 2 ) q sinc = open\ int parenthesisˆ{ 1 } alpha{ 0 divided} e by ˆ{ 2 − q plusi m pi q closing\alpha parenthesis\{ x be\}} the coefficientse ˆ{ − of the2 expansion\ pi ofimx e to} thedx power\\\ oftag minus∗{Z$1 i ( q alpha 4 open . brace 3 x ) closing $} = brace i −iqα{x} −2πimx in\ f Fourierr a c { e series ˆ{ − periodi q In open\alpha parenthesis} − 4 period1 }{ 3c2 closingm(q)\ =pi parenthesisem + commae q dx\alpha } = ( − 1 ) ˆ{ m } e ˆ{ − i \ f r a c {\alpha 0 }{ 2 } q } s i n c ( \ f r a c {\alpha }{ 2 } sinc denotes the cardinal sine function− commaiqα sinc open parenthesis x closing parenthesis = sine x q + m \ pi ) e − 1 m −i α q α divided by x sub period Thus = i = (−1) e 2 sinc( q + mπ) (4.3) \endplus{ a infinity l i g n ∗} Equation: open parenthesis2πm 4 period+ qα 4 closing parenthesis ..2 e to the power of minus i q alpha open brace x closingbe the brace coefficients = sum ofc sub the m expansion open parenthesis of e−iqα{x q} closingin Fourier parenthesis series . eIn to ( the4 . 3power ) , of\noindent 2 pi imx mbe minus the infinity coefficients of the expansion of $ esin ˆx{ − i q \alpha sinc denotes the cardinal sine function , sinc (x) = x . Thus \{ wherex the\}} convergence$ in Fourier is punctual series in the usual . In sense ( 4 of .Fourier 3 ) series , for sincpiecewise denotes continuous thefunctions cardinal period sine function , sinc $ ( x ) = \ f r a c {\ sin x }{Givenx } T{ greater. }$ 0 and Thus the set of Fourier coefficients open brace c sub+ m∞ open parenthesis q closing parenthesis bar m in Z closing brace in open parenthesis−iqα{x} 4 periodX 3 closing2πimx parenthesis comma e = cm(q)e (4.4) \ beginlet J{ suba li T g nbe∗} the space of complex hyphen valued functions q mapsto-arrowright g open parenthesis q+ closing\ infty parenthesis\\\ tag such∗{ that$ ( the 4 . 4 ) $} e ˆ{ − mi− ∞ q \alpha \{ x \}}series= \sum c { m } ( q ) e ˆ{ 2 \ pi imx }\\ m − \ infty \end{ a l i g n ∗} where the convergence is punctual in the usual sense of Fourier series for piece- Equation: open parenthesiswise continuous 4 period functions 5 closing . parenthesis .. sum m in Z to the power of c sub m open parenthesis q closing parenthesis g open parenthesis q minus 2 pi divided by T m closing parenthesis e \noindent where theGiven convergenceT > 0 and the is set punctual of Fourier in coefficients the usual{cm(q sense) | m of∈ Z} Fourierin ( 4 . 3 series for to the power of 2 pi imx), piecewiseconverges open continuous parenthesis functions in the punctual . sense closing parenthesis to a function G sub x open let JT be the space of complex - valued functions q 7→ g(q) such that the parenthesis q closing parenthesisseries which is slowly in hyphen \ hspacecreasing∗{\ andf i lo l l cally}Given integrable $ T in q> uniformly0 $ with and respect the set to x : of supremum Fourier sub coefficients x bar G sub x open $ \{ parenthesisc { m } q closing( q parenthesis ) \mid bar m \ in Z \} $ in ( 4 . 3 ) , cm(q)g(q is lo cally integrable and there exists A greater 0 andX nu greater2π 0 such that supremum sub x bar G \noindent l e t $ J { T }$ be the space of− complexm)e2πimx− valued functions(4. $5) q sub x open parenthesis q closing parenthesis bar T m∈ \mapstoless A barg q bar ( to the q power )$ of suchthatthe nu for bar q barZ right arrow infinity period As a matter of fact comma for any T greaterconverges 0 comma ( in all the Fourier punctual expo sense hyphen ) to a function Gx(q) which is slowly in - \noindent s e r i e s nentials e to the powercreasing of i omegaand lo callyq are integrable in J sub T in forq alluniformly omega in with R comma respect by to openx : sup parenthesisx | Gx(q) | 4 period 4 closing parenthesisis lo cally period integrable With these and definitions there exists commaA > 0 we and ν > 0 such that supx | Gx(q) | \ begincan enunciate{ a l i g n ∗} the main< A | resultq |ν for of| thisq |→ section ∞. As period a matter of fact , for any T > 0, all Fourier expo - \ tag ∗{$ ( 4 . 5iωq ) $}\sum { m \ in Z }ˆ{ c { m } ( q ) g Theorem 4 period 1nentials periode emdashare in SupposeJT for that all ω the∈ R Fourier, by ( 4 series . 4 ) . With these definitions , we (Equation: q } − open \ f r parenthesisa ccan{ 2 enunciate\ 4pi period the}{ mainT 6} closing resultm parenthesis of ) this e section ˆ{ ..2 sum . \ inpi n Z backslashimx } open brace 0 closing\end{ a brace l i g n w∗} open parenthesis nTheorem closing parenthesis 4 . 1 . — dividedSuppose by openthat the parenthesis Fourier series minus 2 pi in closing parenthesis e to the power of minus 2 pi inx \noindent converges ( in the punctual sense ) to a function $ G { x } ( converges in the punctual sense to a periodic piecewiseX continuousw(n) −2πinx function qf sub ) $ w open which parenthesis is slowly x closing in − parenthesis comma0} ofperiode 1 comma which has a derivative(4.6) f creasing and lo cally integrable in $ q $(−2πin uniformly) with respect to $ x sub w to the power of prime open parenthesis x closing∈nZ\{ parenthesis continuous bounded on the : \sup { x }\mid G { x } ( q ) \mid $ open set R minus unionconverges of sub in p theopen punctual brace a sensesub p closingto a periodic bracecomma piecewise where continuous the a sub function p quoteright is lo cally integrable and there exists $ A > 0 $ and $ \nu > 0 $ s are the discontinuityf points(x), ofperiod of f sub 1 w , open which parenthesis has a derivative x closingf 0 parenthesis(x) continuous period bounded on the such that $ \supw { x }\mid G { x } ( qw ) \mid $ Let sigma sub p =open f sub set w open−S parenthesis{a }, where a the sub pa plus’ s are 0 closing the discontinuity parenthesis points minus of ff sub(x w). Let open $ < A \mid q \Rmidp ˆ{\p nu }$ f op r $ \mid q \mid \rightarroww parenthesis a sub p minusσp = 0f closingw(ap + parenthesis0) − fw(ap − be0) thebe the jump jump of f of subf w at the singularity aa subp. Suppose p period \ infty . $ As a matter of fact , for any $T > 01 , $ all−β Fourier expo − Suppose further that comma for case open parenthesis i closing parenthesis−iqline− commaparenleftβ theβ+1q function0 γ q further that , for case ( i ) , the function q 7→ e fw( 2π q) arrowright-mapsto e to the power of minus i q sub line-parenleft beta to the power of 1 sub beta0 γ plus \noindent nentialsis in $J eT with ˆ{ i T =\βomega+1/β andq , for}$case are ( inii ) , $ the J function{ T }$q 7→ f ofw r( 2 aπ lq l) 1 to the power of minusis in beta q f sub w to the power of prime parenleftbig gamma divided by 2 pi q parenrightbig$ \omega is\ in R , $ by ( 4 . 4 ) . With these definitions , we in J sub T with T = beta plus 1 slash beta and comma for case open parenthesis ii closing parenthesis \noindent can enunciate the main result of this section . comma the function q mapsto-arrowright f sub wJT towithT the power= β − of1 prime/β.T hen, parenleftbig gamma divided by 2 pi q parenrightbig is in \ centerlineJ sub T with{Theorem T =TOME beta minus5 4 6 ( . 2 0 1 01 6 slash. ) ,−−− FASCICULE betaSuppose period 7 Then that comma the Fourier series } TOME 5 6 open parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 \ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 6 ) $}\sum {\ in { n } Z \setminus \{} 0 \} \ f r a c { w ( n ) }{ ( − 2 \ pi in ) } e ˆ{ − 2 \ pi inx } \end{ a l i g n ∗}

\noindent converges in the punctual sense to a periodic piecewise continuous function

\noindent $ f { w } ( x ) , $ ofperiod 1 , which has a derivative $ f ˆ{\prime } { w } ( x ) $ continuous bounded on the

\noindent open s e t $ R − \bigcup { p }\{ a { p }\} , $ where the $ a { p }$ ’ s are the discontinuity points of $ f { w } ( x ) . $ Let $ \sigma { p } = f { w } ( a { p } + 0 ) − f { w } ( a { p } − 0 )$ bethejumpof $f { w }$ at the singularity $ a { p } . $ Suppose further that , for case ( i ) , the function $ q \mapsto e ˆ{ − i q { l i n e −p a r e n l e f t \beta }ˆ{ 1 }ˆ{ − \beta } {\beta + 1 } q } f ˆ{\prime } { w } ( \ f r a c {\gamma }{ 2 \ pi } q ) $ i s in $ J { T }$ with $ T = \beta + 1 / \beta $ and , for case ( ii ) , the function $ q \mapsto f ˆ{\prime } { w } ( \ f r a c {\gamma }{ 2 \ pi } q ) $ i s in

\ begin { a l i g n ∗} J { T } with T = \beta − 1 / \beta . Then , \end{ a l i g n ∗}

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2448 Jean - P i erre GAZEAU & Jean - Louis VERGER - GAUGRY ( i ) the Fourier transform µb of µ is the sum of a pure point tempered distribu tion µbpp and a regular tempered distribu tion µrb as follows :

µb = µbpp + µrb with : d X γ m ) µr(q) = m(q)(w(0) + f 0 ( q − ), (4.7) b w 2π κ m∈Z

dm(q) 2π X 2π m µ (q) = σ δ(q − ( + a )), (4.8) bpp γ p γ κ p m,p∈Z where κ = 1 + β in case ( i ) or κ = β in case ( ii ) , and

 iπ q(1 − β) parenleft−lineβ2 +β  cm(q)e 1 (mβ − ) case(i), dm(q) = 2π  cm(q) case(ii),

( ii ) the pure point part µbpp is supported by the scaled finite union of

translatesofZ[β]: 2π ({a , a , ..., a } + [β]) (4.9) γ 1 2 N Z

where a1, a2, ..., aN are the discontinuities of fw in [ 0 , 1 ) . Proof . — By Proposition 3 . 2 , for n > 0, we express the Fourier exponen - tial in ( 4 . 2 ) as follows

−iqb −iq 1−β −inγq −iq β−1 n + 1 e n = e β(1+β) e e β { } case(i), (4.10) 1 + β −iqb −inγq −iq 1 n e n = e e β { } case(ii). (4.11) β Due to the periodicity of the “ fractional part ” function x → {x}, of period 1 , we expand the last factor in ( 4 . 1 0 ) and ( 4 . 1 1 ) in Fourier series :

n+1 −iqbn −iq 1−βline−parenleft −inγq X 2πim{ } e = e β1+β e cm(q)e 1+β case(i), (4.12) m ∈ Z

n −iqbn −inγq P 2πim{ } (4.13) e = e cm(q)e β case ( i i ) .

m ∈ Z Since 2448 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentopen parenthesis2448 \ iquad closingJean parenthesis− P i .. e the r r e Fourier GAZEAU transform $ \& mu-hatwide $ Jean − ofLouis mu is theVERGER sum of− aGAUGRY pure point tempered \ hspacedistribu∗{\ tionf i mu-hatwidel l }( i ) \ subquad pp andthe a Fourierregular tempered transform distribu $ tion\widehat hatwide-mu{\mu r} as$ follows o f : $ \mu $ ismu-hatwide the sum of = hatwide-mu a pure point sub pp tempered plus mu-hatwide r with : Equation: open parenthesis 4 period 7 closing parenthesis .. mu-hatwide r open parenthesis q closing parenthesis = sum m in Z to the power of\ hspace d m open∗{\ parenthesisf i l l } distribu q closing tion parenthesis $ \widehat open parenthesis{\mu} w{ openpp parenthesis}$ and a 0 closing regular parenthesis tempered distribu tion 2πim{x} 2πimx −2πimbxc 2πimx plus$ \widehat f sub w to{\ themu power} r of $ prime as open follows parenthesise : gamma= e dividede by 2= pie q minus, m divided by kappa to the power of closingwe parenthesis get the following closing Fourier parenthesis expansions comma : Case Equation: ( i ) . open parenthesis 4 period 8\ begin closing{ parenthesisa l i g n ∗} .. mu-hatwide sub pp open parenthesis q closing parenthesis = 2 pi divided by gamma\widehat sum{\ mmu comma} = p in Z\ towidehat the power{\mu of} d sub{ mpp open} parenthesis+ \widehat q closing{\mu parenthesis} r \\ sigmawith sub : \\\ tag ∗{$ ( 4 . 7 ) $}\widehat{\mu} β r (1 q )γ = m \sum { m p delta parenleftbigg q minus−iqb 2 pin dividedX by gamma−i1−qline open−parenleftβ parenthesis m divided−2 byπin kappa( q− plus) a sub e = cm(q)e 1+β 2eπim e 2π 1+β p\ in closingZ parenthesis}ˆ{ d } parenrightbiggm(q)(w(0)+fˆ comma 1 + β {\prime } { w } ( \ f r a c {\gamma }{ 2 \ pi } q − \ f r a c { m }{\kappa }ˆ{ ) } ), \\\ tag ∗{$ ( where kappa = 1 plus beta in case open parenthesis i closing parenthesis or kappa =m beta∈ Z in case open4 parenthesis . 8 ) ii $closing}\widehat parenthesis{\mu comma} { pp and } ( q ) = \ f r a c { 2 \ pi }{\gamma } \sumd sub{ mm open , parenthesis pANNALES\ in qDE closing L ’ IZ NS} T parenthesisˆ ITUT{ d FOURIER{ m =} Case( 1 c sub q m ) open}\ parenthesissigma { qp closing}\ paren-delta thesis( q e to− the \ powerf r a c { of2 parenleft-line\ pi }{\ betagamma 2 sub} 1 sub( plus\ f r a beta c { m to}{\ the powerkappa of} i pi+ parenleftbig a { p m} beta)), minus q open parenthesis 1 minus beta closing parenthesis divided by 2 pi parenrightbig case open parenthesis\end{ a l i g i n closing∗} parenthesis comma Case 2 c sub m open parenthesis q closing parenthesis case open parenthesis ii closing parenthesis comma \ centerlineopen parenthesis{where ii closing $ \kappa parenthesis= .. the 1 pure + point\beta part mu-hatwidest$ incase( sub ipp )or is supported $ \kappa by =the scaled\beta finite$ union in case of ( ii ) , and } translates of Z open square bracket beta closing square bracket : Equation: open parenthesis 4 period\ [ d 9{ closingm } parenthesis( q ).. 2 pi = divided\ l e f t by\{\ gammabegin { parenleftbiga l i g n e d } open& brace c { am sub} 1( comma q a sub ) 2e comma ˆ{ p a periodr e n l e f period t −l i n periode \beta comma a2 sub{ N1 closing}ˆ{ bracei \ pluspi } Z open{ + square\beta bracket}} beta( closing m \beta −square \ f r bracket a c { q parenrightbig ( 1 − \beta ) }{ 2 \ pi } ) case ( i ) , \\ &where c a sub{ m 1 comma} ( a subq 2 ) comma case period (period ii period ) comma , \end a sub{ a N l i are g n e the d }\ discontinuitiesright . \ ] of f sub w in open square bracket 0 comma 1 closing parenthesis period Proof period emdash By Proposition 3 period 2 comma for n geqslant 0 comma we express the Fourier\ hspace exponen∗{\ f i lhyphen l }( i i ) \quad the pure point part $ \widehat{\mu} { pp }$ is supported by the scaled finite union of tial in open parenthesis 4 period 2 closing parenthesis as follows \ beginEquation:{ a l i g open n ∗} parenthesis 4 period 1 0 closing parenthesis .. e to the power of minus iq b sub n = e totranslates the power of minus of i q 1 Z minus [ beta\beta divided by]: beta open\\\ tag parenthesis∗{$ ( 1 plus 4 beta . closing 9 parenthesis) $}\ f r a c { 2 e\ pi to the}{\ powergamma of minus} ( in gamma\{ a q e{ to1 the} power, of a minus{ 2 i} q beta, minus . 1 . divided . by , beta a open{ N } brace\} n+ plus Z 1 divided [ by\beta 1 plus beta]) closing brace case open parenthesis i closing parenthesis comma Equation:\end{ a l i g open n ∗} parenthesis 4 period 1 1 closing parenthesis .. e to the power of minus iq b sub n = e to the power of minus in gamma q e to the power of minus i q 1 divided by beta open brace n divided by\ centerline beta closing{ bracewhere case $ open a parenthesis{ 1 } , ii closing a { parenthesis2 } , period . . . , a { N }$ areDue the to thediscontinuities periodicity of the of.. quotedblleft $ f { w fractional}$ in part [ 0 quotedblright , 1 ) . } .. function x right arrow open brace x closing brace comma of \ hspaceperiod∗{\ 1 commaf i l l } weProof expand . −−− the lastBy factor Proposition in open parenthesis 3 . 2 , 4 periodfor $ 1 0 n closing\ g e parenthesis q s l a n t and 0 open, $ parenthesis we express 4 period the 1 Fourier 1 closing parenthesis exponen − in Fourier series : Equation: open parenthesis 4 period 1 2 closing parenthesis .. e to the power of minus i qb sub n = e\noindent to the powertial of minus in i ( q 4 1 minus . 2 ) beta as line-parenleft follows sub beta 1 plus beta e to the power of minus in gamma q sum c sub m open parenthesis q closing parenthesis e to the power of 2 pi im open brace n plus\ begin 1 divided{ a l i g byn ∗} 1 plus beta closing brace case open parenthesis i closing parenthesis comma m in Z \ tagopen∗{$ parenthesis ( 4 4 .period 1 1 3 0closing ) parenthesis $} e ˆ{ − e to thei q power b of{ minusn }} i qb= sub n e = ˆ e{ to − the poweri q of\ f minus r a c { in1 gamma− q \ sumbeta c sub}{\ mbeta open parenthesis( 1 q closing + \ parenthesisbeta ) e}} to thee power ˆ{ − of 2in pi im open\gamma braceq } ne divided ˆ{ − by betai closing q \ f r brace a c {\ ..beta case open− parenthesis1 }{\ ibeta i closing}}\{\ parenthesisf r perioda c { n + 1 }{ 1 + m\ inbeta Z }\} case ( i ) , \\\ tag ∗{$ ( 4 . 1 1 ) $} e ˆ{ − i qSince b { n }} = e ˆ{ − in \gamma q } e ˆ{ − i q \ f r a c { 1 }{\beta }} \{\e tof the r a c power{ n }{\ of 2 pibeta im open}\} brace xcase closing brace ( = i i e to ) the power . of 2 pi imx e to the power of minus\end{ 2a pil i g im n ∗} floorleft x floor = e to the power of 2 pi imx comma we get the following Fourier expansions : DueCase to open the parenthesis periodicity i closing of parenthesis the \quad period‘‘ fractional part ’’ \quad f u n c t i o n $ x \rightarrowLine 1 e to the power\{ ofx minus\} i qb, sub $ n = o f sum c sub m open parenthesis q closing parenthesis e to theperiod power of 1 minus , we i expand 1 minus q the sub line-parenleftlast factor beta in sub (4 1 plus . 1 beta 0 ) to and the power ( 4 . of 1 beta 1 )2 e in pi im Fourier 1 series : divided by 1 plus beta e to the power of minus 2 pi in open parenthesis gamma divided by 2 pi q minus m\ begin divided{ a by l i g 1 n plus∗} beta closing parenthesis Line 2 m in Z \ tagANNALES∗{$ ( DE 4 L quoteright . 1 I 2 NS T ) ITUT $} e FOURIER ˆ{ − i qb { n }} = e ˆ{ − i q 1 − \beta { l i n e −p a r e n l e f t } {\beta 1 + \beta }} e ˆ{ − in \gamma q }\sum c { m } ( q ) e ˆ{ 2 \ pi im \{\ f r a c { n + 1 }{ 1 + \beta }\}} case ( i ) , \\ m \ in Z \end{ a l i g n ∗}

\noindent $ ( 4 . 1 3 ) e ˆ{ − i qb { n }} = e ˆ{ − in \gamma q }\sum c { m } ( q ) e ˆ{ 2 \ pi im \{\ f r a c { n }{\beta } \}}$ \quad case ( i i ) .

\ [ m \ in Z \ ]

\ centerline { Since }

\ [ e ˆ{ 2 \ pi im \{ x \}} = e ˆ{ 2 \ pi imx } e ˆ{ − 2 \ pi im \ l f l o o r x \ rfloor } = e ˆ{ 2 \ pi imx } , \ ]

\noindent we get the following Fourier expansions : Case ( i ) .

\ [ \ begin { a l i g n e d } e ˆ{ − i qb { n }} = \sum c { m } ( q ) e ˆ{ − i 1 − { q { l i n e −p a r e n l e f t \beta }}ˆ{\beta } { 1 + \beta }} 2 { e }\ pi im \ f r a c { 1 }{ 1 + \beta } e ˆ{ − 2 \ pi in ( \ f r a c {\gamma }{ 2 \ pi } q − \ f r a c { m }{ 1 + \beta } ) }\\ m \ in Z \end{ a l i g n e d }\ ]

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA - LATTICES 2449 Case ( ii ) .

X −2πin( γ m e−iqbn = c (q)e 2 q − ), (4.14) m π β m ∈ Z For both cases we adopt the generic expansion

X −2πin( γ m e−iqbn = d (q)e 2 q − ), (4.15) m π κ m ∈ Z where κ = 1 + β( case ( i ) ) or κ = β( case ( ii ) ) . These expansions can be extended to all n ∈ Z. Thus , the Fourier transform of the measure can be written as the double sum :

X X −2πin( γ m µ(q) = d (q) w(n)e 2 q − ). (4.16) b m π κ m ∈ Z n ∈ Z With our hypothesis on w(n) we know that

X w(n)e−2πinx (4.17)

n ∈ Z \{0} is the derivative , in the distribution sense ( [ 37 ] , Chap . II , §4 ) of the piece - wise continuous periodic function fw(x), of period 1 . From a classical result ( [ 37 ] , Chap . II , §2 ) we have the equality

X −2πin( γ m w(n)e 2 q − ) π κ n ∈ Z σp γ m ) X γ m = w(0) + f 0 ( q − + δ( q − − a ), (4.18) w 2π κ 2π κ p p∈Z

0 where fw has to be taken in a distributional sense . Finally , we are led to the following formula for the Fourier transform of

µ : X γ m ) µ(q) = d (q)(w(0) + f 0 ( q − )+ b m w 2π κ m∈Z

dm(q) 2π X 2π m + σ δ(q − ( + a )) (4.19) γ p γ κ p m,p∈Z within the framework of the distribution theory . With our hypothesis on fw the first term of the right - hand side defines a regular tempered distribution . DIFFRACTION OF BETA hyphen LATTICES .. 2449 DIFFRACTIONCase open parenthesis OF BETA ii− closingLATTICES parenthesis\quad period2449 CaseEquation: ( i i open ) . parenthesis 4 period 1 4 closing parenthesis .. e to the power of minus i qb sub n = sum c sub m open parenthesis q closing parenthesis e to the power of minus 2 pi in open parenthesis gamma\ begin divided{ a l i g n by∗} 2 sub pi q minus m divided by beta closing parenthesis comma m in Z \ tagFor∗{ both$ ( cases 4 we adopt . 1 the generic 4 ) expansion $} e ˆ{ − i qb { n }} = \sum c { m } (Equation: q ) open e ˆ parenthesis{We − infer2 from 4 period\ pi ( 4 . 1 1in 5 9 closing ) that ( if parenthesis\ thef r a Fourier c {\gamma .. transform e to the}{ power2 of}} the of{\ measure minuspi } iµ qbcanq sub be− n =\ f sum r a c { dm sub}{\ m openbeta parenthesisinterpreted} ), q as closing a\\ measurem parenthesis\ tooin , thenZ e to the the first power term of minus of the 2 rhs pi of in ( open 4 . 1 parenthesis 9 ) may gamma\end{ a divided l i g n ∗} by 2 subbe interpretedpi q minus m as divided a “ continuous by kappa part closing ” of parenthesis that measure comma . On m the in other Z hand , where kappa = 1 plusthere beta is inopen ( 4 parenthesis . 1 9 ) a pure case point open part parenthesis which is isupported closing parenthesis by the dense closing set given paren- \ centerline {For both cases we adopt the generic expansion } thesis or kappa = betaby open (4.9) parenthesis.  case open parenthesis ii closing parenthesis closing parenthesis period These expansionsTOME can 5 6 be ( 2 0 0 6 ) , FASCICULE 7 \ beginextended{ a l i tog n all∗} n in Z period Thus comma the Fourier transform of the measure can be \ tagwritten∗{$ (as the 4 double . sum 1 : 5 ) $} e ˆ{ − i qb { n }} = \sum d { m } (Equation: q ) open e ˆparenthesis{ − 2 4 period\ pi 1 6in closing ( parenthesis\ f r a c {\ ..gamma mu-hatwide}{ 2 open}} parenthesis{\ pi } qq closing− parenthesis\ f r a c { m =}{\ sumkappa d sub} m open), parenthesis\\ m q\ closingin Z parenthesis sum w open parenthesis n closing parenthesis\end{ a l i g en ∗} to the power of minus 2 pi in open parenthesis gamma divided by 2 sub pi q minus m divided by kappa closing parenthesis period m in Z n in Z \noindentWith our hypothesiswhere $ on\kappa w open parenthesis= 1 n + closing\beta parenthesis($ we know case(i))or that $ \kappa = Equation:\beta open( $ parenthesis case ( 4 ii period ) ) 1 . 7 These closing parenthesisexpansions .. sumcan w be open parenthesis n closing parenthesisextended e to to the all power $ of n minus\ in 2 pi inxZ n in. $Z backslash Thus , open the brace Fourier 0 closing transform brace of the measure can be writtenis the derivative as the comma double in thesum distribution : sense open parenthesis open square bracket 37 closing square bracket comma Chap period II comma S 4 closing parenthesis of the piece hyphen \ beginwise{ continuousa l i g n ∗} periodic function f sub w open parenthesis x closing parenthesis comma of period 1 period\ tag ∗{ From$ ( a classical 4 . result 1 6 ) $}\widehat{\mu} ( q ) = \sum d { m } (open q parenthesis ) \sum openw square ( bracket n 37 ) closing e ˆ{ square − bracket2 \ pi commain Chap ( period\ f r a II c comma{\gamma S 2 }{ 2 }} {\ pi } closingq − parenthesis \ f r a c { m we}{\ havekappa the equality} ). \\ m \ in Z n \ in Z \endsum{ a w l i open g n ∗} parenthesis n closing parenthesis e to the power of minus 2 pi in open parenthesis gamma divided by 2 sub pi q minus m divided by kappa closing parenthesis n in Z Equation: open parenthesis 4\noindent period 1 8 closingWith parenthesis our hypothesis .. = w open on parenthesis $w ( 0 closing n parenthesis) $ weknow plus f thatsub w to the power of prime open parenthesis gamma divided by 2 pi q minus m divided by kappa to the power of closing parenthesis\ begin { a l plus i g n ∗} sum p in Z to the power of sigma sub p delta open parenthesis gamma divided by 2 pi q\ tag minus∗{$ m (divided 4 by . kappa 1 minus 7 a sub ) $ p}\ closingsum parenthesisw ( comma n ) e ˆ{ − 2 \ pi inx }\\ n \ inwhereZ f sub\setminus w to the power\{ of prime0 has\} to be taken in a distributional sense period \endFinally{ a l i g comma n ∗} we are led to the following formula for the Fourier transform of mu : mu-hatwide open parenthesis q closing parenthesis = sum m in Z d sub m open parenthesis q\noindent closing parenthesisis the open derivative parenthesis , w in open the parenthesis distribution 0 closing sense parenthesis ( [ 37 plus ] f ,sub Chap w to . the II , \S 4 ) of the piece − powerwise of continuous prime open parenthesis periodic gamma function divided $ by f 2 pi{ qw minus} ( m divided x ) by kappa , $ to of the period power of 1 . Froma classical result closing( [ 37 parenthesis ] ,Chap closing . II parenthesis , \S 2 plus ) we Equation: have the open equality parenthesis 4 period 1 9 closing parenthesis .. plus 2 pi divided by gamma sum m comma p in Z to the power of d sub m open parenthesis q closing parenthesis\ begin { a l sigma i g n ∗} sub p delta parenleftbigg q minus 2 pi divided by gamma open parenthesis m divided by\sum kappa plusw a sub( p n closing ) parenthesis e ˆ{ − parenrightbigg2 \ pi in ( \ f r a c {\gamma }{ 2 }} {\ pi } q within− \ thef r a framework c { m }{\ ofkappa the distribution} ) \\ theoryn period\ in WithZ \\\ ourtag hypothesis∗{$ ( on 4 f sub . w 1 8 ) $the} = first termw of ( the 0 right ) hyphen + hand f ˆ side{\ definesprime a} regular{ w } tempered( \ f distribution r a c {\gamma period}{ 2 \ pi } q We− infer \ f r from a c { openm }{\ parenthesiskappa 4}ˆ period{ ) } 1 9+ closing\sum parenthesis{ p that\ in if theZ Fourier}ˆ{\ transformsigma of{ thep }} measure\ delta mu( can\ bef r a c {\gamma }{ 2 \ pi } q − \ f r a c { m }{\kappa } − a { p } ),interpreted as a measure too comma then the first term of the rhs of open parenthesis 4 period 1 9 closing\end{ a parenthesis l i g n ∗} may be interpreted as a quotedblleft continuous part quotedblright of that measure period On the other hand\noindent comma where $ f ˆ{\prime } { w }$ has to be taken in a distributional sense . there is in open parenthesis 4 period 1 9 closing parenthesis a pure point part which is supported by the\ hspace dense∗{\ set f i l l } Finally , we are led to the following formula for the Fourier transform of given by open parenthesis 4 period 9 closing parenthesis period square \ beginTOME{ a 5l i 6g n open∗} parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 \mu : \\\widehat{\mu} ( q ) = \sum { m \ in Z } d { m } (q)(w(0)+fˆ{\prime } { w } ( \ f r a c {\gamma }{ 2 \ pi } q − \ f r a c { m }{\kappa }ˆ{ ) } ) + \\\ tag ∗{$ ( 4 . 1 9 ) $} + \ f r a c { 2 \ pi }{\gamma }\sum { m , p \ in Z }ˆ{ d { m } ( q ) }\sigma { p }\ delta ( q − \ f r a c { 2 \ pi }{\gamma } ( \ f r a c { m }{\kappa } + a { p } )) \end{ a l i g n ∗}

\noindent within the framework of the distribution theory . With our hypothesis on $ f { w }$ the first term of the right − hand side defines a regular tempered distribution . We infer from ( 4 . 1 9 ) that if the Fourier transform of the measure $ \mu $ can be interpreted as a measure too , then the first term of the rhs of ( 4 . 1 9 ) may be interpreted as a ‘‘ continuous part ’’ of that measure . On the other hand , there is in ( 4 . 1 9 ) a pure point part which is supported by the dense set givenby$( 4 . 9 ) . \ square $

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2450 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentNote that the2450 support\quad of theJean pure− pointP i part e r r e mu-hatwide GAZEAU sub $ \ pp& given $ Jean by open− Louis parenthesis VERGER 4 period− GAUGRY 9 closing parenthesis is also \ hspaceequal to∗{\ f i l l }Note that the support of the pure point part $ \widehat{\mu} { pp }$ givenEquation: by ( open 4 . parenthesis 9 ) is also 4 period 20 closing parenthesis .. 2 pi divided by gamma times paren- leftbig sub i in sub N sub beta to the power of union of open parenthesis a sub i plus Z open square bracket\noindent beta closingequal square2450 to bracketJean - P closing i erre GAZEAU parenthesis& Jean parenrightbig - Louis VERGER - GAUGRY where N sub beta is a subsetNote that of open the support brace a of sub the 1 pure comma point a sub part 2µb commapp given period by ( 4 period . 9 ) is period also comma\ begin a{ a sub l i g N n ∗} closingequal brace to such that a sub i comma a sub j in N sub beta comma a sub i minus a sub\ tag j∗{ element-negationslash$ ( 4 . 20 Z open ) square $}\ f r bracket a c { 2 beta\ closingpi }{\ squaregamma bracket}\times ( { i \ in }ˆ{\bigcup } { N {\beta }} ( a { i } + Z [ \beta ])) as long as i negationslash-equal j period There2 isπ noS major problem to consider that open brace a \end{ a l i g n ∗} × ( (a + [β])) (4.20) sub 1 comma a sub 2 comma period period periodγ commai∈N aβ subi NZ closing brace is already composed of discontinuities a sub j .. algebraically independent over \noindent where $ N {\beta }$ is a subset of $ \{ a { 1 } , a { 2 } Z open square bracketwhere betaN closingis a subset square of bracket{a , a , ...,period a } such that a , a ∈ N , a − a element − , . . . , a β{ N }\} $1 such2 thatN $ a { i i j} ,β i a j{ j }\ in If all the singularitiesnegationslash a sub j are in[β Z] open square bracket beta closing square bracket comma then N {\beta } , a { i }Z − a { j } element−negationslash Z [ \beta this support is includedas in long as i 6= j. There is no major problem to consider that {a , a , ..., a } is ] $ 1 2 N 2 pi divided by betaalready to the composedpower of 2 of plusminux discontinuities 1 Z openaj squarealgebraically bracket beta independent closing square over bracket where .... quotedblleft plus quotedblright .... st ands for case open parenthesis i closing parenthesis whereas\noindent .... quotedblleftas long as minus $ quotedblright i \not= is for j case . $open There parenthesis is no i i major closing parenthesis problem to open consider that $ \{ a { 1 } , a { 2 } , . . . , a { N }\} $ parenthesis see Proposition Z[β]. is2 period already 9 closing composed parenthesis of discontinuities period $ a { j }$ \quad algebraically independent over 4 period 2 period .. DiffractionIf all the spectrum singularities of aa setj are of weightedin Z[β], then beta this hyphen support integers is included in \ begin { a l i g n ∗} For the last two decades it has becoming traditional to2 defineπ the Bragg Z[spectrum\ commabeta i period]. e period the more or less bright spots[β] we see in a diffraction experi hyphen β2 ± 1Z \endment{ a l on i g n a∗} long range order material comma as the pure hyphen point component of the measure defined bywhere the Fourier “ + ” transform st ands gamma-hatwide for case ( i ) whereas of the so “ hyphen− ” is called for case autocorrelation ( i i ) ( see \ centerlinegamma of the{ If measure allProposition the mu opensingularities parenthesis for $ the a definition{ j }$ of are gamma in comma $ Z see [ open\beta square bracket] , 23 $ closing then square this2 . 9bracket support ) . closing is parenthesisincluded period in } On the other hand4 comma . 2 . the Diffraction function giving spectrum the intensity of a per set diffracting of weighted site beta - integers \ [ \off r Z a csub{ 2 beta\ ispi defined}{\For asbeta the open lastˆ square{ two2 }\ decades bracketpm it 1 0 has1 closing} becomingZ[ square traditional bracket\beta open to] define square\ ] the bracketBragg 22 closing square bracketspectrum open square, i . bracket e . the 23 more closing or lesssquare bright bracket spots open we see square in a diffraction bracket 25 experi closing square bracket open square- ment bracket on a long 36 closing range ordersquare material bracket :, as the pure - point component of the \noindent where \ h f i l l ‘ ‘ $ + $ ’ ’ \ h f i l l st ands for case ( i ) whereas \ h f i l l ‘‘ Equation: open parenthesismeasure defined 4 period by the2 1 Fourier closing transformparenthesisγb ..of theI sub so w - called open autocorrelationparenthesis q clos-γ ing$ − parenthesis$ ’’ is = for limintof case the l right measure ( i arrow iµ )( for infinity ( thesee definition supremum Proposition of vextendsingle-vextendsingle-vextendsingle-γ, see [ 23 ] ) . vextendsingle-vextendsingleOn 1 the divided other by hand card , theopen function parenthesis giving U thesub intensity l closing perparenthesis diffracting sum site n b of in \noindent 2 . 9 ) . U sub l w open parenthesisZβ is defined n closing as [ parenthesis 1 0 ] [ 22 ] [ e 23 to ] the [ 25 power ] [ 36 ]of : minus iq b sub n vextendsingle- vextendsingle-vextendsingle-vextendsingle-vextendsingle to the power of 2 comma \ centerline {4 . 2 . \quad Diffraction spectrum of a set of weighted beta − i n t e g e r s } where open parenthesis U sub l closing parenthesis l is an1 averaging sequence of finite approximants X −iqbn 2 to the set Z sub beta Iw(q) = lim sup | w(n)e | , (4.21) For the last two decades it has becomingl→∞ traditionalcard(Ul) to define the Bragg open parenthesis Section 2 period 3 closing parenthesis periodn Underb∈Ul some assumptions on spectrumopen parenthesis , i . i e closing . the parenthesis more or .. less the point bright set of spots diffractive we sites see comma in a diffractione period g period experi − modelment sets on comma a long setwhere range Z sub ( betaU orderl)l etcis an period material averaging comma sequence , as the of finite pure approximants− point component to the set Zβ of the measureopen parenthesis defined ii( closing by Section the parenthesis 2 Fourier . 3 ) . Under .. transform the some uniqueness assumptions $ of\ gammawidehat on open{\gamma parenthesis} $ see o f Theorem the so 3− called autocorrelation period$ \gamma 4 in open$ square of the( bracket i ) measure the 23 point closing set $ square\ ofmu diffractive bracket( $ sites closing for , e the parenthesis. g . definitionmodel comma sets , of set Z $β etc\gamma . , ,$open see[23]). parenthesis i ii closing( parenthesis ii ) the uniqueness .. the existence of γ( see of Theorem a limit in 3 . the 4 in sense [ 23 ]of ) Bohr, hyphen Besicovich for the av hyphen( i ii ) the existence of a limit in the sense of Bohr - Besicovich for the av - Oneraged the other Fourier hand transformeraged , the ofFourier finite function transform approximants giving of finite of the the approximants measure intensity mu of the per measure diffractingµ ( see Theorems site o fopen $ parenthesis Z {\beta see5 . Theorems} 1$ and is 5 . defined 54 periodin [ 23 1] as ) and , [ 5 10 period ] 4 [ in 22 open ] [ square 23 ] bracket [ 25 23 ] closing [ 36 ] square : bracket closing parenthesisit can comma be shown that the values of ( 4 . 2 1 ) at the points of the pure - point \ beginit can{ bea l i shown g n ∗} thatspectrum the values of µb ofare open the parenthesis intensities 4 of period the Bragg 2 1 closing peaks parenthesis ; see for instance at the points Def - of the\ tag pure∗{$ hyphen ( 4 pointinition . 2 3 . 1 1 in Strungaru ) $} I { [ 39w ]} .( This statementq ) originates = \lim in the{ sol - called\rightarrow \ inftyspectrum}\ ofsup mu-hatwideANNALES\arrowvert are DE the L ’ I intensities NS T\ ITUTf r a FOURIER c of{ the1 }{ Braggcard peaks semicolon ( U { seel for} instance) }\ Defsum hyphen{ n { b } \ ininitionU 3{ periodl }} 1 inw Strungaru ( n .. open ) esquare ˆ{ − bracketi q 39 bclosing{ n square}}\ bracketarrowvert periodˆ ..{ This2 } statement, originates in the so hyphen called \endANNALES{ a l i g n ∗} DE L quoteright I NS T ITUT FOURIER \noindent where $ ( U { l } ) l $ is an averaging sequence of finite approximants to the set $ Z {\beta }$

\noindent ( Section 2 . 3 ) . Under some assumptions on

\ centerline {( i ) \quad the point set of diffractive sites , e . g . model sets , set $ Z {\beta }$ e t c . , }

\ centerline {( i i ) \quad the uniqueness of $ \gamma ( $ see Theorem3 . 4 in [ 23 ] ) , }

( i i i ) \quad the existence of a limit in the sense of Bohr − Besicovich for the av − eraged Fourier transform of finite approximants of the measure $ \mu $ ( see Theorems 5 . 1 and 5 . 4 in [ 23 ] ) ,

\noindent it can be shown that the values of ( 4 . 2 1 ) at the points of the pure − point spectrum of $ \widehat{\mu} $ are the intensities of the Bragg peaks ; see for instance Def − inition 3 . 1 in Strungaru \quad [ 39 ] . \quad This statement originates in the so − c a l l e d

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 245 1 \ hspaceBombieri∗{\ hyphenf i l l }DIFFRACTION Taylor conjecture OF open BETA square− LATTICES bracket 6\ commaquad 7245 closing 1 square bracket period Under such assumptions comma the autocorre hyphen \noindentlation is evenBombieri the sum− ofTaylor two pure conjecturepoint measures [ open 6 , parenthesis 7 ] . Under Proposition such 3 assumptions period 3 in open , the autocorre − squarelation bracket is 39even closing the square sum bracket of two closing pure parenthesis point measures comma ( Proposition 3 . 3 in [ 39 ] ) , thethe two two parts parts being being called called strongly strongly and null hyphen and weakly null − almostweakly periodic almost after periodic after ArgabrightArgabright and and Gil Gil de Lamadrid de Lamadrid open square [ 1 ]bracket [ 2 1 1DIFFRACTION closing ] . Though square OF bracket the BETA general - open LATTICES square methodology bracket245 1 2for 1 closing computing square bracket theBombieri diffractionperiod - TaylorThough conjecture the spectrum general[ methodology6 of , 7 ] .$ Under\mu $ such was assumptions provided , the earlier autocorre ( by Meyer infor Proposition computing the 1-diffraction lation in [ is 27 even spectrum ] the p 32 sum of ) mu of , twowas recentpure provided point investigations earlier measures open ( parenthesis Proposition on weighted by 3 Meyer. 3 in Dirac[ 39 combs [in 2 Proposition ] \quad [ 1 in9] open) ] ,\ thequad square two[ parts bracket 23 being ] \ 27quad called closing[ strongly 39 square ] show and bracket null a p -need weakly 32 closing for almost parenthesisintroducing periodic comma after new theories to describe the recent investigations onArgabright weighted Diracand Gil combs de Lamadrid [ 1 ] [ 2 1 ] . Though the general methodology \noindentopen squarestructure bracketfor 2 computing closing of this square the spectrum bracket diffraction .. open . spectrum square of bracketµ was provided 9 closing earliersquare ( bracket by Meyer .. open in square bracket 23 closingProposition square bracket 1 in [ 27 .. ]open p 32 square ) , recent bracket investigations 39 closing on square weighted bracket Dirac show combs a need [ Byfor introducing Section 3 new the theories2 elements ] [ to9 ] describe [ in23 ] the $ [ Z 39 ]{\ showbeta a need}$ for are introducing in one new− theoriesto − one to describe correspondence with elementsstructure of of this $ spectrumthe ( − period \beta , \beta ) ( $ Proposition 3 . 1 ) by the $ ∗By Section − $ 3 operation the elementsstructure . inLet of Z this sub us spectrum beta denote are in . byone hyphen to hyphen one correspondence with elements of open parenthesisBy Section minus 3 beta the comma elements beta in closingZβ are parenthesis in one - to open - one parenthesis correspondence Proposition with 3\noindent period 1 closing$ w parenthesis ˆ{elements ∗ }$ by of the the (−β, * conjugate β hyphen)( Proposition operation weight 3 . period 1 ) defined by Let the us∗− denote onoperation the by dense . Let us subset denote by\ h f i l l $ ( Z ˆ{ + } {\∗beta }ˆ{ ) ˆ{\prime }}\subset ( − 1+)0 , \beta w to the power of *w thethe conjugate conjugate weight weight defined defined on on the the dense dense subset subset .... parenleftbig (Zβ ⊂ Z( sub−1, beta β) or to the) $ power or of plus to the power of parenrightbig to the power of prime subset open parenthesis minus 1 comma beta closing parenthesis or +)0 \ begin { a l i g n ∗} (Zβ ⊂ [0, β)by parenleftbig Z sub beta to the power of plus∗ to0 the power of parenrightbig to the power of prime subset( Z open ˆ{ square+ } {\ bracketbeta 0 comma}ˆ{ ) beta ˆ{\ closingprimew ( parenthesisbn}}\) = subsetw by(n) Equation:, n [∈ N. 0 open parenthesis, \beta 4(4 period.22)) by \\\ tag ∗{$ ( 4 . 22 ) $} w ˆ{ ∗ } ( b ˆ{\prime } { n } ) = 22 closing parenthesisWe .. w assume to the in power the sequel of * thatopen the parenthesis function w b∗ subcan n be to extended the power to aof contin prime - closing uous w ( n ) , n \ in N. parenthesis = w openfunction parenthesis in the n closing interior parenthesis of its support comma supp n (inw∗ N) ⊂ period(−1, β) or ⊂ [0, β), and that \end{ a l i g n ∗} We assume in the sequelit is integrable that the over function the window w to the supp power (w of∗). * can be extended to a contin hyphen uous function in the interiorLet L ofbe its the support space of supp complex open - parenthesis valued functions w to on the power( or equivalently of * closing \noindent We assume in the sequel that the function $ w ˆ{Z ∗β }$ can be extended to a contin − parenthesis subset openon parenthesisthrough n minus→ b ). 1For commaw ∈ L beta, we closingdenote by parenthesisk w k orthe subset pseudo open - norm square ( uous function in theZ interiorn of its support supp $ ( w ˆ{ ∗ } ) \subset bracket 0 comma beta“ closing norm 1 parenthesis ” ) of Marcinkiewicz comma of w defined as ( and− that1 it is integrable , \beta over the) window $ or supp $ \ opensubset parenthesis[ w 0 to the , power\beta of * closing) parenthesis , $ periodand that it is integrable over the windowsup supp1 $ (X w ˆ{ ∗ } ) . $ k w k = lim ∞ | w(n) | Let L be the space of complex hyphen valued functionsl→+ on ZCard sub beta(Ul) open parenthesis or equivalently n∈Z \ hspaceon Z through∗{\ f i ln l } rightLet arrow $ L b $ sub be n closing the space parenthesis of complex period For− wvalued in L comma functions we denote on by $.. Z {\beta } ( $ or equivalentlywhere (Ul)l is an averaging sequence of finite approximants of Zβ. The bar w bar .. the pseudoMarcinkiewicz hyphen norm space M is the quotient space of the subspace open parenthesis quotedblleft norm 1 quotedbl closing parenthesis of Marcinkiewicz of w defined as \noindent on $Z$ through $n \rightarrow b { n } ) . $ For $ w bar w bar = limint l right arrow plus sub infinity{g to∈ the L |k powerg k< of+∞} supremum 1 divided by Card open parenthesis\ in L U ,sub $ l closing we denote parenthesis by \ sumquad n in$ Z\ barparallel w open parenthesisw \ parallel n closing parenthesis$ \quad barthe pseudo − norm (where ‘‘ norm open 1parenthesis ’’of )L of Uby Marcinkiewiczsub the l equivalence closing parenthesis relation of l$R .. w isdefined $an averaging defined by sequence as of finite approximants of Z sub beta period .. The \ [ \ parallel w \ parallel = \lim { l \rightarrow + }ˆ{\sup } {\ infty }\ f r a c { 1 }{ Card Marcinkiewicz space M is the quotient spacehR ofg the⇐⇒ subspace k h − g k = 0 (4.23) (Uopen brace{ l g} in) L bar}\ barsum g bar{ lessn plus\ in infinityZ closing}\mid brace w ( n ) \mid \ ] ( Bertrandias [ 4 ] [ 5 ] , Vo Khac [ 43 ] ) . The class of w is denoted by of L by the equivalence relation R defined by w in M. Though the definition of k · k depends upon the chosen averaging Equation: open parenthesis 4 period 23 closing parenthesis .. h R g arrowdblleft-arrowdblright bar sequence (U )l the space M obviously does not . This equivalence relation is h\noindent minus g barwhere = 0 $ ( U l { l } ) l $ \quad is an averaging sequence of finite approximants of called Marcinkiewicz equivalence relation . The vector space M is normed with $ Zopen{\ parenthesisbeta } Bertrandias. $ \quad openThe square bracket 4 closing square bracket open square bracket 5 k g k = k g k, and is complete ( Bertrandias [ 4 ] [ 5 ] , Vo Khac [ 43 closingMarcinkiewicz square bracket space comma Vo $ M Khac $ open is the square quotient bracket 43 space closing square of the bracket subspace closing parenthesis ] ) . By L∞ we will mean the subspace of L of bounded weights endowed with period The class of w is denoted by overbar w in M period the M− topology in the sequel . \ [ Though\{ g the definition\ in L of bar\mid times bar\ parallel depends upong the chosen\ parallel averaging< sequence+ \ infty \}\ ] open parenthesis U sub l closing parenthesis l the space M obviously does not period This equivalence relation is called \noindent of $ L $ by the equivalence− − − relation $ R $ defined by Marcinkiewicz equivalenceProposition relation period4.2 The vectorLet h,space g ∈ LMsuch is normed that withh, g ∈ M. Then , bar to the power of hline g bar = bar g bar. comma and is complete open parenthesis Bertrandias for q ∈ , open\ begin square{ a l i bracket g n ∗} 4 closing squareR bracket .. open square bracket 5 closing square bracket comma Vo\ tag Khac∗{$ open ( square 4 bracket . 23 43 closing ) $} squareh R bracket g closing\ i f f parenthesis\ parallel period Byh L to− the powerg \ ofparallel = 0 2 infinity Ih(q) 6 k h k , (4.24) \endwe{ willa l i g mean n ∗} the subspace of L of bounded weights endowed with the M hyphen topologyTOME in the 5 6 ( sequel 2 0 0 6 ) period , FASCICULE 7 \noindenthline ( Bertrandias [ 4 ] [ 5 ] , VoKhac [ 43 ] ) . The class of $w$ isProposition denoted by 4 period $\ overline Row 1 emdash{\}{ Roww 2} period$ in . h $ comma M g. in $ L such that h comma to the power ofThough hline g in the M period definition Then comma of for $ \ qparallel in R comma \cdot \ parallel $ depends upon the chosen averaging sequence $Equation: ( U open{ l parenthesis} ) l 4 $ period the 24 space closing $M$ parenthesis obviously .. I sub h does open parenthesis not . This q closing equivalence relation is called parenthesisMarcinkiewicz leqslant bar equivalence h bar to the powerrelation of 2 comma . The vector space $ M $ is normed with $TOME\ parallel 5 6 openˆ parenthesis{\ r u l e {3em 2 0}{ 0 60.4 closing pt }} parenthesisg \ parallel comma FASCICULE= \ parallel 7 g \ parallel , $ and is complete ( Bertrandias [ 4 ] \quad [ 5 ] ,VoKhac [ 43 ] ) . By $ L ˆ{\ infty }$ we will mean the subspace of $ L $ of bounded weights endowed with the $ M − $ topology in the sequel .

\ [ \ r u l e {3em}{0.4 pt }\ ]

\ hspace ∗{\ f i l l } Proposition $\ l e f t . 4 . 2\ begin { array }{ c} −−− \\ . \end{ array } Let \ right . h , g \ in L$ suchthat $h ,ˆ{\ r u l e {3em}{0.4 pt }} g \ in M .$ Then,for $q \ in R , $

\ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 24 ) $} I { h } ( q ) \ l e q s l a n t \ parallel h \ parallel ˆ{ 2 } , \end{ a l i g n ∗}

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2452 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentEquation:2452 open parenthesis\quad Jean 4 period− P 25 i closing e r r e parenthesisGAZEAU $ ..\ h& R $g equal-arrowdblright Jean − Louis VERGER I sub h open− GAUGRY parenthesis q closing parenthesis = I sub g open parenthesis q closing parenthesis period \ beginProof{ perioda l i g n ∗} emdash Immediate period square \ tagThe∗{ relation$ ( 4open parenthesis. 25 4 ) period $} h 25 closingR g parenthesis\Longrightarrow means that the setI of{ weightedh } ( Dirac q combs) = on beta I { hypheng } ( q ) . \endintegers{ a l i g isn ∗} classified2452 by theJean Marcinkiewicz - P i erre GAZEAU relation& Jean period - Louis The VERGER intensity - GAUGRY function I sub w is a class function on M period We now express the diffracting intensity I sub w open parenthesis\ hspace ∗{\ q closingf i l l } Proof parenthesis . −−− Immediate $ . \ square $ by taking the sequence .. open parenthesish openRg parenthesis=⇒ Ih(q) minus = BIg sub(q). N closing parenthesis(4.25) cup The relation ( 4 . 25 ) means that the set of weighted Dirac combs on beta − B sub N closing parenthesis N .. as averaging sequence of finite Proof . — Immediate . integersapproximants is to classified Z sub beta comma by the inorder Marcinkiewicz to compute it relation period The . following The intensity formulae are function $ I { w }$ is aThe class relation function ( 4 . 25 ) onmeans $M that the . set $ of Wenow weighted express Dirac combs the on diffracting beta - intensity important in the senseintegers that is I classified sub w open by the parenthesis Marcinkiewicz q closing relation parenthesis . The intensity is what function we observeI is in diffraction$ I { w experi} ( hyphen q ) $ w by taking the sequencea class function\quad on M$. We ( now ( express− B the diffracting{ N } ) intensity\cupIw(qB) by{ takingN } ment period the sequence ((−B ) ∪ B )N as averaging sequence of finite approximants )Theorem N $ \ 4quad periodas Row averaging 1 emdash Row sequence 2N periodN of . w finitein L to the power of infinity .. be a weight on Z approximants toto $Zβ Z, in{\ orderbeta to compute} , $ it . Thein order following to formulae compute are it . The following formulae are sub beta period Assumeimportant that the in the sense that I (q) is what we observe in diffraction experi - singularities a sub j are all in Z open square bracketw beta closing square bracket period .... Then the \noindent importantment in. the sense that $ I { w } ( q ) $ is what we observe in diffraction experi − pure point part open parenthesis Bragg part closing− − − parenthesis of Theorem 4.3 Let w ∈ L∞ be a weight on . Assume that the intensity I sub w open parenthesis q closing. parenthesis is equal to Zβ \noindent ment . I sub w open parenthesisthe q closing parenthesis = Equation: open parenthesis 4 period 26 closing parenthesis .. vextendsingle-vextendsingle-vextendsingle-vextendsingle-vextendsinglesingularities a are all in [β]. Then the pure point part ( Bragg 1 divided part )by of bar \ hspace ∗{\ f i l l }Theorem $\ l ej f t . 4 .Z 3\ begin { array }{ c} −−− \\ . \end{ array } Let \ right . supp open parenthesisthe w to intensity the powerIw of(q) bigis equalstar closing to parenthesis bar minus to the power of beta 1 wopen square\ in bracketL ˆ{\ winfty to the power}$ \ ofquad big starbe open a weight parenthesis on x closing$ Z {\ parenthesisbeta } e to. the $ power Assume of i that the q to the power of prime x plus w to the power of big star open parenthesis minus x closing parenthesis e \noindent singularities $ a { j }$ areallin $Z [ \beta ] . $ to the power of minus i q to the power of prime x closing square bracket dx vextendsingle-vextendsingle-Iw(q) = \ h f i l l Then the pure point part ( Bragg part ) of vextendsingle-vextendsingle-vextendsingle1 to the power0 of 2 case open0 parenthesis i closing parenthesis | −β 1[w?(x)eiq x + w?(−x)e−iq x] dx|2 case(i), (4.26) comma Equation: open parenthesis? 4 period 27 closing parenthesis .. vextendsingle-vextendsingle- \noindent the intensity| supp( $w I) | { w } ( q )$ isequalto vextendsingle-vextendsingle-vextendsingle 1β divided by bar supp open parenthesis w to the power of big 1 Z 0 0 star closing parenthesis bar integral sub 0 to the? poweriq x of? beta open−iq x square2 bracket w to the power | ? [w (x)e + w (−x)e ] dx| case(ii). (4.27) of\ begin big star{ a openl i g n ∗} parenthesis| xsupp( closingw ) parenthesis| 0 e to the power of iq to the power of prime x plus w I { w } ( q ) = \\\ tag ∗{$ ( 4 . 26 ) $}\arrowvert \ f r a c { 1 }{\mid to the power of big star open parenthesis minus x closing2 parenthesis e to the power of minus i q to supp ( w ˆ{\wherestar q} = ) 2π κ\midand }q0 = −2πβˆ{\κ0,betawith }κ ∈1[β], [and wwhere ˆ{\ star “ } + ”( the power of prime x closing squareβ bracket2±1 dx vextendsingle-vextendsingle-vextendsingle-vextendsingle-β2±1 Z vextendsinglex ) e ˆ to{ thei powerstands q ˆ{\ of 2prime case open} parenthesisx } + ii w closing ˆ{\ parenthesisstar } ( period− x ) e ˆ{ − iwhere q ˆ{\ q =prime 2 pi divided}for casex by} beta (] i ) towhereas dx the power\arrowvert of “ 2− plusminux” isˆ for{ case2 1} kappa (case ii and ) . q to( the i power ) of prime , \\\ tag ∗{$ ( 4 . 27 ) $}\arrowvert− − − \ f r a c { 1 }{\mid supp ( w ˆ{\ star } ) = 2 pi beta to the powerP roof of 2 dividedT he byintensity beta to the function power is of equal 2 plusminux to : ( 4 . 1 28 kappa ) to the power of prime\mid comma}\ int withˆ{\ kappabeta in Z} open.{ 0 square} [ bracket w ˆ{\ betastar closing} square( x bracket ) comma e ˆ{ andi q where ˆ{\prime .... } quotedblleftx } + w plus ˆ{\ quotedblrightstar } stands( − x ) e ˆ{ − i q ˆ{\prime } x } ] 1 dx \arrowvert ˆ{ 2 } case ( i i )X . −iqbn iqbn 2 for case open parenthesisIw i( closingq) = lim parenthesissup | whereas .. quotedblleft[w(n)e + minusw(−n quotedblright)e − w(o)] is| for. case \end{ a l i g n ∗} N→∞ 2cN − 1 open parenthesis ii closing parenthesis period nb∈BN ProofRow 1 emdash Row 2 period Proof intensity function is equal to : \noindentopen parenthesiswhere 4Note period $ q that 28 = closing the\ f term r a parenthesis c {w(2o) can\ pi be}{\ droppedbeta forˆ{ obvious2 }\ reasonspm .1 The}\ Galoiskappa $ and $ q ˆ{\prime } = \ f r a c { 2 \0 pi \beta1 ˆ{ 2 }}{\beta ˆ{ 2 }\pm I sub w open parenthesisconjugation q closingκ = parenthesisr + sβ → =κ limint= r ∓ Ns β rightr, s arrow∈ Z( case infinity ( i supremum ) or case vextendsingle- ( ii ) ) in 1 }\kappa ˆ{\prime } , $ with $ \kappa, \ in Z[ \beta ] , $ vextendsingle-vextendsingle-vextendsingle-vextendsinglethe ring Z[β] leads to : 1 divided by 2 c sub N minus 1 sum n b in B suband N where bracketleftbig\ h f i l lw open‘‘ parenthesis $+$ ’’ n closing stands parenthesis e to the power of minus i qb sub n plus w open parenthesis minus n closing parenthesis e to the power of i qb2 sub0 n minus w open parenthesis o clos- ing\noindent parenthesisfor bracketrightbig case ( i ) vextendsingle-vextendsingle-vextendsingle-vextendsingle-vextendsingle whereas \quad κ‘ ‘ $ −β κ$ ’’ is for case ( ii ) . 2 ± 2 ∈ Z. (4.29) to the power of 2 period β ± 1 β ± 1 \noindentNote that the$\ l term e f t . w Proof open\ parenthesisbegin { array o closing}{ c} −−−parenthesis \\ . can\end be{ droppedarray } forThe obvious\ right reasons.$ intensity function is equal to : So we can write , for any q ∈ 2π [β], period( 4 The. 28 Galois ) β2±1 Z conjugation kappa = r plus s beta right arrow kappa to the power of prime = r minusplus s 1 divided \ [I { w } ( q ) = \lim { N \rightarrow \ infty }\sup \arrowvert by beta sub comma r comma s in Z open parenthesis case open0 0 parenthesis i closing parenthesis or case qbn ≡ −q bn mod (2πZ), (4.30) open\ f r a cparenthesis{ 1 }{ 2 ii closing c { parenthesisN } − closing1 }\ parenthesissum { inn { b }\ in B { N }} [ w ( n ) e ˆ{ − i qb { n }} + w ( − 2 n ) e ˆ{ i qb { n }} the ring Z open square bracket beta closing square0 bracket leads2πβ to :0 2π − w ( owhere ) ] we have\arrowvert denoted byˆq{ the2 } number. \ ]± β2±1 κ if q = β2±1 κ for κ ∈ Z[β] Equation: open parenthesis( “ + ” 4 period stands 29 for closingcase ( parenthesis i ) whereas .. kappa “ − ” dividedis for case by ( beta i i ) to) the . Hence power of 2 plusminux 1 plusminux beta to the power of 2 kappa to the power of prime divided by beta to the , the intensity ANNALES DE L ’ I NS T ITUT FOURIER power of 2 plusminux 1 in Z period \noindentSo we canNote write commathat the for any term q in $ 2 w pi divided ( oby beta ) $ to the can power be dropped of 2 plusminux for obvious 1 Z open reasons . The Galois squareconjugation bracket beta $ closing\kappa square= bracket r comma + s \beta \rightarrow \kappa ˆ{\prime } =Equation: r \mp opens parenthesis\ f r a c { 41 period}{\ 30beta closing} { parenthesis, } r .. qb , sub s n equiv\ in minusZ q to ($ the power case( i )orcase( ii ))in of prime b sub n to the power of prime modulo open parenthesis 2 pi Z closing parenthesis comma \noindentwhere we havethe denoted r i n g by $ Z q to the [ power\beta of prime] the$ number l e a d s plusminux to : 2 pi beta to the power of 2 divided by beta to the power of 2 plusminux 1 kappa to the power of prime if q = 2 pi divided by beta to\ begin the power{ a l i gof n 2∗} plusminux 1 kappa for kappa in Z open square bracket beta closing square bracket \ tagopen∗{$ parenthesis ( 4 .. . quotedblleft 29 ) $ plus}\ f rquotedblright a c {\kappa ..}{\ standsbeta forˆ case{ 2 open}\ parenthesispm 1 }\ i closingpm parenthesis\ f r a c {\beta whereasˆ{ ..2 quotedblleft}\kappa minusˆ{\ quotedblrightprime }}{\ is forbeta case openˆ{ 2 parenthesis}\pm i i closing1 }\ parenthesisin Z closing. parenthesis period Hence comma the intensity \endANNALES{ a l i g n ∗} DE L quoteright I NS T ITUT FOURIER \noindent So we can write , for any $ q \ in \ f r a c { 2 \ pi }{\beta ˆ{ 2 } \pm 1 } Z[ \beta ] , $

\ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 30 ) $} qb { n }\equiv − q ˆ{\prime } b ˆ{\prime } { n } \bmod ( 2 \ pi Z), \end{ a l i g n ∗}

\noindent where we have denoted by $ q ˆ{\prime }$ the number $ \pm \ f r a c { 2 \ pi \beta ˆ{ 2 }}{\beta ˆ{ 2 }\pm 1 }\kappa ˆ{\prime }$ i f $ q = \ f r a c { 2 \ pi }{\beta ˆ{ 2 }\pm 1 }\kappa $ f o r $ \kappa \ in Z[ \beta ] $

\noindent ( \quad ‘ ‘ $ + $ ’ ’ \quad stands for case ( i ) whereas \quad ‘‘ $ − $ ’’ is for case ( i i ) ) . Hence , the intensity ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2453 DIFFRACTIONopen parenthesis OF BETA 4 period− 28LATTICES closing parenthesis\quad 2453 can be rewritten as (Equation: 4 . 28 ) open can parenthesis be rewritten 4 period as 3 1 closing parenthesis .. I sub w open parenthesis q clos- ing parenthesis = limint N right arrow infinity supremum vextendsingle-vextendsingle-vextendsingle- vextendsingle-vextendsingle\ begin { a l i g n ∗} 1 divided by 2 c sub N minus 1 sum n b in B sub N open square bracket w open\ tag parenthesis∗{$ ( 4 n closing . parenthesis 3 1 )e to $} theI power{ w of} i q( to the q power ) of = prime\\\ blim sub n to{ N the power\rightarrow of\ infty prime plus}\ wsup open parenthesis\arrowvertDIFFRACTION minus OF\ nf BETAr closing a c { - LATTICES1 parenthesis}{ 2 c2453 e to{ the(N 4 . power} 28 − ) can of minus1 be}\ rewritten iqsum to the as{ powern { b } of\ in primeB b sub{ N n to}} the[ power w of prime ( n minus ) w open e ˆ{ parenthesisi q ˆ{\ o closingprime parenthesis} b ˆ{\ closingprime square} { n }} +bracket w vextendsingle-vextendsingle-vextendsingle-vextendsingle-vextendsingle ( − n ) e ˆ{ − i q ˆ{\prime } b ˆ{\prime to the} power{ n }} of 2 period − w (Let o us now ) use ] the\arrowvert quotedblleft algebraicˆ{ 2 } cut. hyphen and hyphen project quotedblrightIw(q) = properties (4.31) \end{ a l i g n ∗} open parenthesis 3 period 1 1 closing parenthesis1 X and openiq parenthesis0b0 3 period−iq0b0 1 2 closing2 parenthesis lim sup | [w(n)e n + w(−n)e n − w(o)]| . of the positive beta hyphenN→∞ integers2c andN − the1 fact that the sets Z sub beta to the power of plus arise n ∈B from\noindent model Let us now use the ‘‘ algebraicb N cut − and − project ’’ properties ( 3 . 1 1 ) and ( 3 . 1 2 ) sets period The conjugateLet us now sets use parenleftbig the “ algebraic Z sub cut beta - and to the - project power” of properties plus to the ( 3 power . 1 1 of ) and paren- ( \noindent of the positive beta − integers and the fact that the sets $ Z ˆ{ + } {\beta }$ rightbig to the power of3 . prime 1 2 ) = open brace x in Z open square bracket beta closing square bracket cap arise from model open parenthesis minusof 1 the comma positive beta beta closing - integers parenthesis and barthe xfact to thethat power the sets of prime+ arise geqslant from 0 model closing sets . The conjugate sets $ ( Z ˆ{ + } {\beta }ˆ{ ) ˆ{\Zβ prime }} = \{ brace open parenthesis case open parenthesis i closing+)0 parenthesis closing parenthesis0 sets . The conjugate sets (Zβ = {x ∈ Z[β] ∩ (−1, β) | x > 0}( case ( i ) ) x and\ in open parenthesisZ[ \ Zbeta sub beta] to the\cap power( of plus− closing1 parenthesis , \beta to the) power\mid of primex = ˆ{\prime } and ( +)0 = {x ∈ [β] ∩ [0, β) | x0 0}( case ( ii ) ) densely fill the inter - open\ g e q brace s l a n t x in 0Z open\} squareZ(β $ bracket case betaZ ( closing i ) ) square> bracket cap open square bracket 0 comma vals (−1, β) ( of length 1 + β) and [0, β) ( of length β) respectively with betaand closing $ ( parenthesis Z ˆ{ + bar} x{\ to thebeta power} of) prime ˆ{\prime geqslant} 0 closing= \{ brace openx parenthesis\ in Z[ case open\beta uniform distribution modulo 1 . Hence we can assert that the finite sum parenthesis] \cap ii closing[ 0 parenthesis , \beta closing parenthesis) \mid denselyx ˆ fill{\ theprime inter hyphen}\ g e q s l a n t 0 \} ( $ case ( ii ) ) densely fill the inter − vals open parenthesis minus 1 comma beta1 closingX parenthesisiq0b0 open parenthesis−iq0b0 of length 1 plus beta closingv a l s parenthesis $ ( − .. and1 .. open , square\beta bracket) 0 ($ comma[w(n oflength)e betan closing+ w(−n parenthesis) $1e n ] + open\beta parenthesis) $ of \quad and \quad $ [ 0 , \2betacN ) ($ oflength $ \beta ) $ respectively with length beta closing parenthesis respectively withnb∈BN uniform distributionis moduloj ust a Riemann 1 period Hence sum approximating we can assert ,that forthe large finiteN and sum for “ reasonable ” \noindent1 divideduniform by 2 c subweight distribution N sumw, the n integralb in B modulo sub : N open 1 . square Hence bracket we can w assertopen parenthesis that the n closing finite sum parenthesis e to the power of i q to the power of prime b sub n to the power of prime plus w open parenthesis\ [ \ f r a c { minus1 }{ n2 closing c parenthesis{ N }}\ esum to the{ powern { of minusb }\ iqin to theB power{ N of}} prime[ b sub w n to ( the n ) e ˆ{ i q ˆ{\prime } b ˆ{\prime1 X} { n }}iq0b0 + w− (iq0b0 − n ) power of prime closing square bracket [w(n)e n + w(−n)e n ] e ˆ{ − i q ˆ{\prime } b ˆ{\prime }2c{N n }} ] \ ] is j ust a Riemann sum approximating comma for largenb∈ NBN and for .. quotedblleft reasonable quoted- blright 1 0 0 −β 1[w?(x)eiq x + w?(−x)e−iq x] dx, case(i) (4.32) weight w comma the integral≈ | supp( : w?) | \noindent1 divided byis 2 j c ust sub aN Riemannsum n b in sum B sub approximating N open square bracket, for large w open parenthesis $ N $ and n closing for \quad ‘‘ reasonable ’’ weight $w , $ the integral : 0 0 parenthesis e to the power of i q to1 the powerR β ? of primeiq x b sub? n to−iq thex power of prime plus w open (4.33) ≈ |supp(w?)| 0 [w (x)e + w (−x)e ] dx, case ( i i ) . parenthesis minus n closing parenthesis e to the power of minus i q to the power of prime b sub n to At the limit , using Proposition 2 . 9 , we obtain ( 4 . 26 ) and (4.27).  the\ begin power{ a of l i primeg n ∗} closing square bracket Equation: open parenthesis 4 period 32 closing parenthesis .. \ f r a c { 1 }{ 2 c The{ N proof}}\ ofsum Theorem{ n 4 . 3{ showsb }\ thatin the limsupB { (N 4 .}} 2 1 )[ becomes w a ( true n thickapprox 1 dividedlimit by bar because supp open we have parenthesis used the w formalism to the power of cut of big - and star - closing project parenthesis schemes and bar minus) e to ˆ the{ i power q of ˆ{\ betaprime 1 open} squareb ˆ bracket{\prime w to} the{ powern }} of+ big star w open ( parenthesis− n x )closing e ˆ{ − i q ˆ{\prime }thatb the ˆ{\ positiveprime and} negative{ n }} parts] of\\\ thetag beta∗{$ - integers ( 4 can . be interpreted 32 ) $ in}\ thickapprox parenthesis e to the powerterms of of i q model to the sets power . of prime x plus w to the power of big star open parenthesis minus\ f r a c x{ closing1 }{\ parenthesismid supp e to the ( power w ofˆ{\ minusstar iq to} the) power\mid of prime} − xˆ closing{\beta square} bracket1 [ w ˆ{\ star } ( x )Let us e now ˆ{ consideri q ˆ discontinuities{\prime } aj xnot} in+Z[β]. wTheorem ˆ{\ star 4 . 3} gives( dx comma case open parenthesis i closing parenthesis 0 − x ) e ˆ{an − expressioni q ˆ{\ ofprimeIw(q) as} a continuousx } ] function dx of , the case variable (q . Since i ) open parenthesis 4 period0 33 closing2π parenthesis2 thickapprox 1 divided by bar supp open parenthesis \end{ a l i g n ∗} q ∈ line − betaβ2±1Z[β] and that Z[β] is dense in R, we can prolongate by w to the power of big starcontinuity closing parenthesis bar integral sub 0 to the power of beta open square bracket w to the power of big star open parenthesis x closing parenthesis e to the power of i q to2π theSN power of the intensity function Iw(q) given by ( 4 . 26 ) or ( 4 . 27 ) : if q ∈ ( (aj+ prime\noindent x plus w$ to ( the power 4 .of big 33 star open ) parenthesis\ thickapprox minus x\ closingf r a c { parenthesis1 }{\mid e toγ thesuppj=1 power of ( [β])), there exist j ∈ {1, 2, ..., N} and κ ∈ [β] such that wminus ˆ{\ i qstar to the} power) Z of\mid prime}\ x closingint squareˆ{\beta bracket} { dx0 comma} Z [ .. case w ˆ open{\ star parenthesis} ( i i closing x ) e ˆ{ i q ˆ{\prime } x } + w ˆ{\ star } ( − x ) e ˆ{ − i parenthesis period 2π q ˆ{\prime } x } ] dx , $ q\=quad case(a + (κ) i, i κ )∈ . [β]. At the limit comma using Proposition 2 periodβ2 9± comma1 j we obtainZ open parenthesis 4 period 26 closing parenthesis andTOME open 5 6 parenthesis ( 2 0 0 6 ) , FASCICULE 4 period 7 27 closing parenthesis period square \noindentThe proofAt of Theorem the limit 4 period , using 3 shows Proposition that the limsup 2 . open 9 , parenthesiswe obtain 4 ( period 4 . 226 1 )closing and parenthesis$ ( 4 becomes . 27 a true ) . \ square $ limit because we have used the formalism of cut hyphen and hyphen project schemes and Thethat proof the positive of Theorem and negative 4 . parts3 shows of the that beta hyphen the limsup integers ( can 4 be . interpreted2 1 ) becomes a true limitin terms because of model we sets have period used the formalism of cut − and − project schemes and thatLet us the now positive consider discontinuities and negative a sub parts j not in of Z open the square beta bracket− integers beta closing can square be interpreted bracket periodin terms Theorem of 4 model period 3 sets gives . an expression of I sub w open parenthesis q closing parenthesis as a continuous function of the variable\ hspace q∗{\ to thef i l power l } Let of us prime now period consider Since discontinuities $ a { j }$ not in $ Z [ q to\beta the power] of prime . $ in Theorem line-beta beta 4 . to 3 the gives power of 2 plusminux 1 to the power of 2 pi 2 Z open square bracket beta closing square bracket and that Z open square bracket beta closing square bracket is\noindent dense in R commaan expression we can prolongate of $ by I continuity{ w } ( q ) $ as a continuous function of the variable $ qthe ˆ{\ intensityprime function} . $I sub Since w open parenthesis q closing parenthesis given by open parenthesis 4 period 26 closing parenthesis or open parenthesis 4 period 27 closing parenthesis : if q in 2 pi divided by\noindent gamma parenleftbig$ q ˆ{\ unionprime of sub}\ j =in 1 to thel i n power e −beta of N open{\beta parenthesisˆ{ 2 a}\ subpm j plus 1 }ˆ{ 2 \ piZ open2 } squareZ[ bracket\ betabeta closing]$ square andthat bracket closing $Z parenthesis [ \beta parenrightbig] $ is comma dense there in exist$ R j in , open $ we brace can 1 comma prolongate 2 comma by period continuity period period comma N closing brace and kappa in Z open square bracket beta closing square bracket such that \noindentq = 2 pi dividedthe intensity by beta to the function power of 2 $plusminux I { w 1} open( parenthesis q )$ a sub givenby(4.26)or(4.27): j plus kappa closing if parenthesis$ q \ in comma\ f r a kappa c { 2 in Z\ openpi }{\ squaregamma bracket} beta( closing\bigcup squareˆ{ bracketN } { periodj = 1 } ( a TOME{ j } 5 6+ open $ parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 \noindent $ Z [ \beta ] ) ) ,$ thereexist $j \ in \{ 1 , 2 , . . . , N \} $ and $ \kappa \ in Z[ \beta ] $ such that

\ [ q = \ f r a c { 2 \ pi }{\beta ˆ{ 2 }\pm 1 } ( a { j } + \kappa ), \kappa \ in Z[ \beta ]. \ ]

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2454 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentIt suffices to2454 take\ anyquad sequenceJean open− P parenthesis i e r r e GAZEAU a sub j comma $ \& l $ closing Jean parenthesis− Louis l VERGERgeqslant 0− ofGAUGRY elementsIt suffices of Z open to square take bracket any sequence beta closing square $ ( bracket a { whichj , converges l } ) l \ g e q s l a n t 0$to the of discontinuity elements ofa sub $Z j : [ \beta ] $ which converges a sub j = limint l right arrow infinity a sub j comma l \noindentand to computeto the the discontinuityassociated numbers $ a { j } : $ q l = 2 pi divided2454 by betaJean to - the P i errepower GAZEAU of 2 plusminux& Jean - Louis 1 open VERGER parenthesis - GAUGRY a subIt suffices j comma to takel plus \ [ a { j } = \lim { l \rightarrow \ infty } a { j , l }\ ] kappa closing parenthesisany comma sequence q (subaj,l) ll to> the0 of power elements of prime of Z[β =] which plusminux converges 2 pi beta to the power of 2 divided by beta to theto powerthe discontinuity of 2 plusminuxaj : 1 open parenthesis a to the power of prime j sub comma l plus kappa to the power of prime closing parenthesis period \noindent and to compute the associated numbers Then the intensity I sub w open parenthesis q closingaj parenthesis= lim aj,l is given by l→∞ I sub w open parenthesis q closing parenthesis = I sub w open parenthesis sub l limint right arrow and to compute the associated numbers infinity\ [ q q l l closing = \ parenthesisf r a c { 2 =\ limintpi }{\ l rightbeta arrowˆ{ 2 infinity}\pm I sub w1 open} ( parenthesis a { j q l closing , l } + \kappa ) , q ˆ{\prime } { l } = \pm \ f r a c { 2 \ pi \beta ˆ{ 2 }}{\beta ˆ{ 2 } parenthesis 2π 2πβ2 \pm 1 } ( a ˆ{\primeql }{= j } { (,a } + κl), q+0 =\kappa± ˆ{\(a0jprimel + κ0).} ). \ ] where I sub w open parenthesis q l closingβ2 ± 1 parenthesisj,l isl computedβ by2 ± open1 parenthesis, 4 period 26 closing parenthesis or open parenthesis 4 period 27 closing parenthesis using q sub l to the power of prime and passing to theThen the intensity Iw(q) is given by \noindentlimit q to theThen power the of intensityprime = limint $ sub I l right{ w arrow} ( infinity q q )$sub l to isgivenby the power of period to the Iw(q) = Iw(l lim ql) = lim Iw(ql) power of prime →∞ l→∞ \ [ILet us{ noww } give( simple q or ) standard = examples I { w period} ( For{ al uniform}\lim distribution{\rightarrow \ infty } where I (ql) is computed by ( 4 . 26 ) or ( 4 . 27 ) using q0 and passing to the qw lopen ) parenthesis = \lim n closingw{ parenthesisl \rightarrow = 1 comma Formula\ infty open} I parenthesis{ w }l 4( period q 26 closingl ) \ ] parenthesis reduces to 0 0 Equation: open parenthesis 4 period 34 closing parenthesislimitq = lim .. Iql sub. w open parenthesis q closing parenthesis\noindent =where 4 parenleftbigg $ I { cosinew } q( to the q power l of )$ primel is→∞ open computedby(4 parenthesis beta minus . 26) 1 closing or (4 . 27) using $ q ˆ{\prime } { l }$ and passing to the parenthesis divided by 2 parenrightbiggLet us now give 2 simpleparenleftbigg or standard sinc q examples to the power . For of a prime uniform open distribution parenthesis beta plus 1 closing parenthesisw(n) = 1, dividedFormula by ( 2 4 parenrightbigg . 26 ) reduces to2 comma \ beginwhere{ a sinc l i g open n ∗} parenthesis x closing parenthesis = sine x divided by x designates the sinus cardinal periodl i m i tOn the q other ˆ{\prime hand comma} = \lim { l \rightarrow \ infty } q ˆ{\prime } { l ˆ{ . }} \end{ a l i g n ∗} q0(β − 1) q0(β + 1) Formula open parenthesis 4 period 27Iw closing(q) = 4(cos parenthesis becomes)2 (sinc )2, (4.34) Equation: open parenthesis 4 period 35 closing parenthesis2 .. I sub w2 open parenthesis q closing Let us now give simple or standard examples . For a uniform distribution parenthesis = 4 parenleftbiggwhere sinc sinc (x q) to = thesin powerx designates of prime betathe sinus divided cardinal by 2 parenrightbigg . On the other 2 commahand , $w ( n ) = 1 ,$x Formula(4.26)reducesto In Case open parenthesisFormula i closing ( 4 . 27 parenthesis ) becomes comma for a model set Capital Sigma to the power of Capital Omega comma determined algebraically by a window \ beginCapital{ a l Omega i g n ∗} subset open square bracket minus 1 comma 1 closing square bracket through the \ tag ∗{$ ( 4 . 34 ) $} I { w } ( qq )0β = 4 ( \cos \ f r a c { q ˆ{\prime } sieving procedure Iw(q) = 4(sinc )2, (4.35) ( Equation:\beta open− parenthesis1 ) }{ 4 period2 } 36) closing 2 parenthesis ( s i n c .. Capital\2f r a c { Sigmaq ˆ{\ to theprime power} of Capital( \beta + 1 ) }{ 2 } ) 2 , Omega = open brace x inIn Z openCase square ( i ) bracket, for a modelbeta closing set ΣΩ square, determined bracket algebraically bar x to the power by a window of prime \end{ a l i g n ∗} in Capital Omega closingΩ ⊂ brace[−1, 1] = through open brace the sieving x in Z proceduresub beta bar x to the power of prime in Capital Omega closing brace comma \noindentwhere thewheresinc corresponding weight $( is x given ) by w = to\ thef r a power c {\ sin of big starx }{ openx parenthesis}$ designates x closing the sinus cardinal . On the other hand , Formula ( 4 . 27 ) becomes Ω 0 0 parenthesis = chi Capital Omega openΣ parenthesis= {x ∈ Z x[β closing] | x ∈ parenthesisΩ} = {x ∈ Z theβ | x characteris∈ Ω}, hyphen(4.36) tic function of Capital Omega : then the diffraction intensity of such a model set is given where the corresponding weight is given by w?(x) = χΩ(x) the characteris - tic \ beginby { a l i g n ∗} function of Ω : then the diffraction intensity of such a model set is given by \ tagEquation:∗{$ ( open 4 parenthesis . 35 4 ) period $} I 37{ closingw } parenthesis( q ..) I sub= w 4 open ( parenthesis s i n c \ qfclos- r a c { q ˆ{\prime } ing\beta parenthesis}{ 2 } = 4) vextendsingle-vextendsingle-vextendsingle-vextendsingle 2 , 1 divided by bar Capital \end{ a l i g n ∗} Z Omega bar integral sub Capital Omega e to the power of i q1 to theiq power0x 2 of prime x dx vextendsingle- Iw(q) = 4| e dx| . (4.37) vextendsingle-vextendsingle-vextendsingle to the power of| 2Ω period| Ω InIn Case Case ( open i ) parenthesis , for a ii model closing setparenthesis $ \Sigma commaˆ dealing{\Omega with} model, $ sets determined for the computation algebraically by a window In Case ( ii ) , dealing with model sets for the computation of the intensity of the$ \ intensityOmega \subset [ − 1 , 1 ] $ through the sieving procedure function forces us to consider the decorated version ( Section 3 ) of . A slight function forces us to consider the decorated version open parenthesis Section 3 closing parenthesisZβ of adaptation of the above formalism becomes then necessary . Z\ begin sub beta{ a lperiod i g n ∗} A Our aim is to reconstruct to a certain extent ( the phase problem ! ) the \ tagslight∗{$ adaptation ( 4 of . the 36above formalism ) $}\Sigma becomesˆ{\ thenOmega necessary} = period\{ x \ in Z[ weight function n → w(n) through its “ Galois conjugate ” (−1, β) or \betaOur aim] is to\ reconstructmid x toˆ{\ a certainprime extent}\ openin parenthesis\Omega the\} phase= problem\{ ! closingx parenthesis\ in Z {\beta } [0, β) 3 x → w?(x). This reconstruction should take into account the partitioning the\mid x ˆ{\prime }\ in \Omega \} , of the Marcinkiewicz classes , and what we may expect is probably the \endweight{ a l i gfunction n ∗} n right arrow w open parenthesis n closing parenthesis through its .. quotedblleft reconstruction of a peculiar representant for each class . This reconstruction Galois conjugate quotedblright .. open parenthesis minus 1 comma beta closing parenthesis or open square\noindent bracketwhere 0 commaANNALES the beta correspondingDE closing L ’ I NS parenthesis T ITUT FOURIER weight ni is given by $ w ˆ{\ star } ( x )x =right\ arrowchi w to\Omega the power( of big x star ) open $parenthesis the characteris x closing− parenthesis period This recon- structiontic function should take of into $ account\Omega the partitioning: $ then the diffraction intensity of such a model set is given byof the Marcinkiewicz .. classes comma .. and what we may expect is probably the reconstruction of a peculiar representant for each class period This reconstruction \ beginANNALES{ a l i g n DE∗} L quoteright I NS T ITUT FOURIER \ tag ∗{$ ( 4 . 37 ) $} I { w } ( q ) = 4 \arrowvert \ f r a c { 1 }{\mid \Omega \mid }\ int {\Omega } e ˆ{ i q ˆ{\prime } x } dx \arrowvert ˆ{ 2 } . \end{ a l i g n ∗}

\noindent In Case ( ii ) , dealing with model sets for the computation of the intensity function forces us to consider the decorated version ( Section 3 ) of $ Z {\beta } . $ A slight adaptation of the above formalism becomes then necessary .

Our aim is to reconstruct to a certain extent ( the phase problem ! ) the weight function $ n \rightarrow w ( n )$ throughits \quad ‘‘ Galois conjugate ’’ \quad $ ( − 1 , \beta ) $ or $ [ 0 , \beta ) \ ni $ $ x \rightarrow w ˆ{\ star } ( x ) . $ This reconstruction should take into account the partitioning of the Marcinkiewicz \quad c l a s s e s , \quad and what we may expect is probably the

\noindent reconstruction of a peculiar representant for each class . This reconstruction

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2455 \ hspacecan be∗{\ partiallyf i l l } implementedDIFFRACTION through OF BETA a sort− ofLATTICES multiresolution\quad analysis2455 of the intensity function I sub w open parenthesis q closing parenthesis comma a method developed in\noindent open squarecan bracket be partially 1 2 closing square implemented bracket and through open square a sort bracket of 1 multiresolution 1 closing square bracket analysis of periodthe Thereintensity function $ I { w } ( q ) , $ amethod developed in [ 12 ] and [ 11 ] . There are alternative expressions for I sub w open parenthesis q closing parenthesis which could reveal themselves\noindent usefulare alternative expressions for $DIFFRACTION I { w } OF( BETA q - LATTICES ) $ which2455 could reveal themselves useful f ofor r this t h i s so so hyphen− calledcan called be homometry partially homometry implemented problem problem period through . a sort of multiresolution analysis of the 4 period 3 period ..intensity Diffraction function spectrumIw(q) of, a a method weighted developed beta hyphen in [ 1 latt 2 ] ice and [ 1 1 ] . There \ centerline {4 . 3 . \quad Diffraction spectrum of a weighted beta − l a t t i c e } Theorem 4 period 1are and alternative Theorem expressions4 period 3 give for theIw(q pure) which point could part reveal of the themselvesdiffraction useful for spectra of weightedthis beta so hyphen - called integers homometry period problem We are . now ready to deduce the Bragg Theorempart of diffraction4 . 1 and spectra Theorem4 . of 3 weighted . 4 Diffraction . 3 Delone give sets the spectrum on pure beta hyphen ofpoint a weighted lattices part inof beta general the - latt diffraction ice spectraopen parenthesis of weighted see FigureTheorem beta 4 period 4− . 1integers and3 for Theorem an i llustration . We 4 . are 3 give closing now thepure ready parenthesis point to part deduceperiod of the the diffraction Bragg partLet us of briefly diffraction recallspectra what is spectra of a weightedbeta hyphen of beta weighted lattice - integers period Delone . We We are report now sets to ready open on to beta square deduce− bracket thelattices Bragg 1 3 closing part in general square( see bracket Figure for the 4of . full diffraction3 for an spectra i llustration of weighted Delone ) . sets on beta - lattices in general ( see description of symmetryFigure groups 4 . 3 for on an beta i llustration hyphen lattices ) . period LetBeta us hyphen briefly lattices recall areLet aimed us what briefly to is replace recall a beta whatlattices− is ainlattice beta the context - lattice . We of . We quasicrystals report report to to [ period [ 1 3 1 ] 3for ] the for full the full descriptionThey are based of on symmetrydescription beta hyphen of groups integers symmetry on comma groups beta like on− lattices betal a t t - i arelatticesc e s based . . on integers : d Equation: open parenthesisBeta - lattices 4 period are 38 aimed closing to parenthesis replace lattices .. Capital in the Gamma context of= quasicrystalssum Z sub beta . eBeta sub i− commalattices i = 1 They are aimed are based to on replace beta - integers lattices , like lattices in the are context based on integers of quasicrystals : . Theywith areopen based parenthesis on ebeta sub i− closingintegers parenthesis , like 1 leqslant lattices i leqslant are da based base of on R to integers the power : of d comma d geqslant 1 period Figure 4 period 1 gives the example of the \ begintau hyphen{ a l i g n lattice∗} open parenthesis or t au hyphen grid closingd parenthesis Capital Gamma sub 1 d \\\ tag ∗{$ ( 4 . 38 ) $}\Gamma X= \sum Z {\beta } e { i } open parenthesis tau closing parenthesis = Z sub tau plusΓ = e i piZ dividedβei, by 5 sub Z sub tau in R(4 to.38) the power, \\ ofi 2 period = 1 \endFigure{ a l i g 4 n period∗} 1 period tau hyphen lattice Capital Gammai = sub1 1 open parenthesis tau closing parenthesis = Z sub tau plus e i pi divided by 5 subd Z sub tau in R to the power of 2 period \noindent with $with ( (ei)1 e 6{i 6i d}a base) of 1R , d\>l e q s1 l. aFigure n t 4 i . 1 gives\ leqslant the example d of $ the a base of TOME 5 6 open parenthesis 2 0 0 6 closing parenthesis commaiπ FASCICULE2 7 $ R ˆ{ d } , dtau -\ latticegeqslant ( or t au -1 grid . )Γ $1(τ) Figure = τ + e 4 . 1in gives. the example of the Z 5 Zτ R iπ 2 Figure 4.1. τ− lattice Γ1(τ) = τ + e in . Z 5 Zτ R \noindent tau − TOMElattice 5 6 ( 2 0 ( 0 6 or ) , FASCICULE t au − 7 g r i d $ ) \Gamma { 1 } ( \tau ) = Z {\tau } + e \ f r a c { i \ pi }{ 5 } { Z {\tau }}$ in $ R ˆ{ 2 } . $

\ centerline { Figure $4 . 1 . \tau − $ l a t t i c e $ \Gamma { 1 } ( \tau ) = Z {\tau } + e \ f r a c { i \ pi }{ 5 } { Z {\tau }}$ in $ R ˆ{ 2 } . $ }

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2456 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentBeta hyphen2456 lattices\quad are eligibleJean − framesP i in e r which r e GAZEAU one could $ think\& $ of the Jean properties− Louis VERGER − GAUGRY of quasiperiodic point hyphen sets and tilings comma thus generalizing the notion of lattice Betain periodic− lattices cases period are eligible frames in which one could think of the properties ofAs quasiperiodic a matter of fact comma point it− hassets become and like tilings a paradigm , thusthat geometrical generalizing sup hyphen the notion of lattice inports periodic of quasi hyphen cases crystalline . structures should be Delone sets obtained through quotedblleft cut and2456 projectionJean - P quotedblright i erre GAZEAU from& Jean higher - Louis hyphen VERGER dimensional - GAUGRY lattices period Now commaAs a matter it is easily of un fact hyphenBeta , it - lattices has become are eligible like frames a paradigm in which one that could geometrical think of the properties sup − portsderstood of from quasi Section−of quasiperiodiccrystalline 3 that most pointof suchstructures - setscut hyphen and tilings should and , hyphen thus be generalizing Delone project sets sets the are notion obtained subsets of lattice through ‘‘of cut suitably and rescaled projectionin beta periodic hyphen ’’ cases from lattices . higher period− Wedimensional show in Figure lattices 4 period 2 . a Now cut hyphen , it is and easily un − hyphenderstood project from SectionAs a matter 3 that of mostfact , it of has such become cut like− aand paradigm− project that geometrical sets are sup subsets - of2 D suitably decagonal rescaledset andports it of s embeddingbeta quasi -− crystallinelattices into the structures tau . hyphenWe show should lattice in be of DeloneFigure Figure sets 4 4 period obtained . 2 a 1 is cut through− and “ − p r o j e c t 2shown D decagonal in Figure 4 setcut period and 3 projection period it s embedding ” from higher into - dimensional the tau − latticeslattice . Now of , it Figure is easilyun 4 .- 1 is shownFigure in 4 period Figure 2 periodderstood 4 ... 3 A from . decagonal Section cut 3 that hyphen most and of such hyphen cut project- and - project set sets are subsets of It is well known thatsuitably the condition rescaled 2 betacos open - lattices parenthesis . We show 2 pi slash in Figure N closing 4 . 2 parenthesis a cut - and in - Z project comma i\ periodcenterline e period{ Figure N =2 1D comma 4 decagonal . 2 2 comma . \ setquad and 3 commaA it sdecagonal embedding 4 intocut the− and tau -− latticep r o j of e c Figuret s e t 4} . 1 is and 6 comma .... characterizesshown in Figure N hyphen 4 . 3 fold. Bravais lattices in R to the power of 2 open parenthesis and\ hspace in R to∗{\ thef i power l l } It of is 3 closing wellFigure parenthesis known 4 . that 2 .open the parenthesis A decagonal condition see cut also - 2 and cos - project $ ( set 2 \ pi / N)open square\ in bracketZ 3 ,$i.e$.N=1closingIt is well square known bracket that for the Pisot condition hyphen 2 Cyclotomic cos , (2π/N 2) numbers∈ ,Z, i 3 . eclosing.N ,= parenthesis 4$ 1, 2, 3, 4 period Now comma whatand can 6 , we characterizes do when NN is− quafold hyphen Bravais lattices in R2 ( and in R3) ( see also \noindentsicrystallographicand 6 i[, period 3\ ]h for f ie Pisot l period l characterizes - Cyclotomic N = 5 comma numbers 1 0 $ comma N ) . Now− 8 and$ , what 1 fold 2 comma can Bravais we respectively do when latticesN associatedis qua in - with$ R one ˆ{ 2 } ( $sicrystallographic and in $ R ˆi{ . e3 }.N =) 5, 10 (, 8 $ and see 1 2 , a respectively l s o associated with one of the cyclotomic Pisotof the units cyclotomic tau = 2 Pisot cos open units parenthesisτ = 2 cos (2 2π/ pi10) slash, δ = 1 10 + closing 2 cos parenthesis(2π/8) and comma delta\noindent = 1 plus[ 2 3 cos ] openθ for= parenthesis 2 Pisot + 2 cos− (2 2Cyclotomicπ/ pi12)? slash Possible 8 closing numbers answers parenthesis are ) .provided andNow , by what beta can - lattices we do in the when $ Ntheta $ = i s 2 qua plus− 2 cosplane open . These parenthesis point sets 2 pi are slash defined 1 2 as closing parenthesis ? Possible answers are providedsicrystallographic by beta hyphen lattices i . e in the $ . N = 5 , 1 0 , 8$ and12 , respectively associated with one of the cyclotomic Pisot units $ \tau = 2 $ cos $ ( 2 \ pi / 1 plane period These point sets are defined as q 0Equation: ) , open\ delta parenthesis= 4 period 1 + 39 closing 2$cos$( parenthesisΓq(β) = Z ..β + CapitalZ 2βζ , Gamma\ pi sub/ q 8open parenthesis) $(4.39) and beta closing parenthesis = Z sub beta plus Z sub beta zeta to the power of q comma i 2π \noindentwith zeta =$ e to\theta thewith powerζ == ofe N i 2, 2for pi divided 1 +6 q 6 2$cos$( byN N− 1 comma. Besides for the 1 leqslant example 2 \ qpi of leqslant t au/ - Nlattice 1 minus shown 2 1 period ) in Besides? $ Possible the example answers ofFigure t au hyphen 4 are . 1 , provided lattice beta - shown lattices by for betaβ =−τ, δ,latticesand θ are in given the in [ 1 3 ] . Note the planein Figure . These 4 period pointfollowing 1 comma sets important beta are hyphen defined features lattices : as for beta = tau comma delta comma and theta are given in open square bracketANNALES 1 DE 3 L closing ’ I NS T ITUT square FOURIER bracket period Note the \ beginfollowing{ a l i important g n ∗} features : \ tagANNALES∗{$ ( DE 4 L quoteright . 39 I NS ) $ T}\ ITUTGamma FOURIER{ q } ( \beta ) = Z {\beta } + Z {\beta }\zeta ˆ{ q } , \end{ a l i g n ∗}

\noindent with $ \zeta = e ˆ{ i \ f r a c { 2 \ pi }{ N }} , $ f o r $ 1 \ l e q s l a n t q \ l e q s l a n t N − 1 . $ Besides the example of t au − lattice shown in Figure 4 . 1 , beta − lattices for $ \beta = \tau , \ delta , $ and $ \theta $ are given in [ 1 3 ] . Note the following important features :

\noindent ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2457 \ hspaceFigure∗{\ 4 periodf i l l } 3DIFFRACTION period .. Embedding OF BETA of the− cuLATTICES t hyphen and\quad hyphen2457 project set ofFigure 4 period 2 into the \ begintau hyphen{ c e n t lattice e r } of Figure 4 period 1 period The tau hyphen lattice is here the set of the intersection Figurepoints 4 of .the 3 lines . \quad period Embedding of the cu t − and − project set ofFigure 4 . 2 into the $bullet\tau .. they− are$ lattices lattice for a of new Figure internal 4 law . oplus 1 . : The Capital $ Gamma\tau sub− q$ open lattice parenthesis is beta here the set of the intersection closingpoints parenthesis of the oplus lines Capital . Gamma sub q open parenthesisDIFFRACTION beta closing OF BETA parenthesis - LATTICES = Capital2457 Gamma\end{ c esub n t e q r open} parenthesis beta closing parenthesis comma bullet .. they are selfFigure hyphen 4 . 3 similar . Embedding : beta Capital of the Gamma cu t - and sub - q project open parenthesisset ofFigure beta 4 . 2 closing into parenthesis\ hspace ∗{\ subsetf i l l Capital} $ \ bulletthe Gammaτ− lattice sub$ \ qquad open of Figure parenthesisthey 4 . are 1 . beta The lattices closingτ− lattice parenthesis for is a here newcomma the internal set of the law $ \bulletoplus .. they: satisfy\Gamma a more{ generalq } ( quotedblleftintersection\beta quasi) points quotedblright\oplus of the lines self\Gamma . hyphen{ similarityq } ( open\beta parenthesis) = \ withGamma new exter{ q } hyphen( \beta ) , $ • they are lattices for a new internal law ⊕ :Γ (β) ⊕ Γ (β) = Γ (β), nal law closing parenthesis : Z sub beta oslash Capital Gamma subset Capitalq Gammaq commaq • they are self - similar : βΓ (β) ⊂ Γ (β), \ centerlinebullet .. however{ $ \ commabullet they$ are\quad neitherthey rotationally are self invariant− s inor mq i translationallyl a r $q : \beta \Gamma { q } • they satisfy a more general “ quasi ” self - similarity ( with new exter - ( invariant\beta comma) \subset \Gamma { q } ( \beta ) , $ } bullet .. as already pointed out open square bracket 8 closing square bracket .. open square bracket nallaw): ⊗ Γ ⊂ Γ, 1\ hspace 3 closing∗{\ squaref i l l bracket} $ \ bullet comma a$ large\quad class ofthey aperiodic satisfy setsZβ like a more general ‘‘ quasi ’’ self − similarity ( with new exter − model sets open parenthesis• however or quotedblleft , they are cut neither hyphen rotationally and hyphen invariant project quotedblright nor translationally sets closing in- parenthesis\ [ nal comma law currently )variant : used , Z by{\ physicistsbeta }\otimes \Gamma \subset \Gamma , \ ] as geometric models supporting• as already atomic pointed sites in out quasicrystals [ 8 ] [ 1 3 comma ] , a large can class of aperiodic sets like be embedded in thesemodel beta sets hyphen ( or “ cut lattices - and Capital - project Gamma ” sets sub ) , currently q open parenthesis used by physicists beta closing as parenthesis$ \ bullet or in$ some\quadgeometric .. quotedbllefthowever models decorated , supporting they are quotedblright atomic neither sites in rotationally quasicrystals , can invariant be embedded nor in translationally i n v a r i a n t , version of them periodthese beta - lattices Γq(β) or in some “ decorated ” version of them . Let us extend to higherLet dimensional us extend tocases higher the formulaedimensional open cases parenthesis the formulae 4 period ( 4 . 26 26 closing ) and paren- ( 4 . thesis$ \ bullet and open$ parenthesis\quad27 ) foras 4 theperiod already Bragg 27 closing part pointed of parenthesis the diffraction out [ 8 intensity ] \quad . It[ is 1more 3 ]or less, a straightfor large class of aperiodic sets like modelfor the sets Bragg ( part or- of ward‘‘ the cut diffraction , depending− and intensity− onproject the chosenperiod ’’ It weight is sets more and ) or geometry , less currently straightfor . For the hyphenused sake by of physicistssim - asward geometric comma depending modelsplicity on supporting, the the general chosen caseweight atomic being and similar sitesgeometry , inwe period consider quasicrystals For here the sake the plane of , sim can (d hyphen= 2) case beplicity embedded comma in theβ thesegeneral= τ only casebeta and being− itsl asimilar corresponding t t i c e comma s $ \ tauweGamma consider - grid{ , actually hereq } the( the plane simplest\beta open parenthesis one) among $ or d in some \quad ‘‘ decorated ’’ =version 2 closing parenthesis of themseveral . possible ones : case beta = tau only and its corresponding tau hyphen grid comma actually the simplest one Let us extend to higher dimensional cases the formulae ( 4 . 26 ) and ( 4 . 27 ) among several possible ones : π for the Bragg part of the diffraction intensity .i 5 It is more or less straightfor − Equation: open parenthesis 4 periodΓ1 = 40{z closing∈ C | z parenthesism,n = bm + ..bn Capitale , b Gammam, bn ∈ Z subτ }. 1 = braceleftbig(4.40) zward in C bar , zdepending sub m comma on n the = b subchosen m plus weight b sub n and e to thegeometry power of . i pi For divided the by sake 5 comma of sim b sub− mplicity comma b sub , the n in general ZTOME sub 5 tau 6 ( 2 casebracerightbig 0 0 6 ) , FASCICULEbeing period similar 7 , we consider here the plane $ ( dTOME = 25 6 open ) $ parenthesis 2 0 0 6 closing parenthesis comma FASCICULE 7 case $ \beta = \tau $ only and its corresponding tau − grid , actually the simplest one among several possible ones :

\ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 40 ) $}\Gamma { 1 } = \{ z \ in C \mid z { m , n } = b { m } + b { n } e ˆ{ i \ f r a c {\ pi }{ 5 }} , b { m } , b { n }\ in Z {\tau }\} . \end{ a l i g n ∗}

\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2458 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentThis discrete2458 set\ isquad a subsetJean of the− P dense i e cyclotomic r r e GAZEAU ring Z $ open\& $ square Jean bracket− Louis e to the VERGER power of− iGAUGRY pi divided by 5 closing square bracket = Z open square bracket tau closing square bracket plus \ centerlineZ open square{ This bracket discrete tau closing set square is bracket a subset e to theof power the dense of i pi divided cyclotomic by 5 period ring $ Z [We e now ˆ{ consideri \ f r a the c {\ purepi point}{ 5 measure}} ] supported = Z by Capital [ \ Gammatau sub] 1 + : $ } Equation: open parenthesis 4 period 4 1 closing parenthesis .. mu = sum w open parenthesis m comma\ begin n{ closinga l i g n ∗} parenthesis2458 deltaJean - sub P i erre z sub GAZEAU m comma& Jean n m- Louis comma VERGER n in - Z GAUGRY i π Z[where open\tau parenthesis]This ex comma ˆdiscrete{ i y\ set closingf r ais c a{\ subset parenthesispi }{ of the5 right dense}} arrow. cyclotomic w open ring parenthesisZ[e 5 ] = Z x[τ comma]+ y closing\end{ a parenthesis l i g n ∗} is a generally complex hyphen valued weight comma assumed split

as follows : i π [τ]e 5 . \ centerlinew open parenthesis{We now m comma consider n closing the parenthesis pure point = wZ submeasure 1 open supportedparenthesis m by closing $ parenthesis\Gamma { 1 } w: sub $ 2} open parenthesis n closing parenthesis comma with w sub 1 comma w sub 2 in L to the power We now consider the pure point measure supported by Γ1 : of infinity period \ beginThe Fourier{ a l i g n transform∗} of the measure open parenthesis 4 period 4 1 closing parenthesis is defined open\ tag parenthesis∗{$ ( 4 in a distributional . 4 1 ) $}\mu = X\sum w ( m , n ) \ delta { z { m µ = w(m, n)δzm,n (4.41) ,sense n }}\\ closingm parenthesis , n comma\ in by Z \endEquation:{ a l i g n ∗} open parenthesis 4 period 42 closing parenthesis .. mu-hatwidem, n ∈ Z open parenthesis k closing parenthesis = mu parenleftbig e to the power of minus i k times z sub m comma n parenrightbig = sum \noindent wherewhere $( (x, x y) → ,w(x, y) is a ) generally\rightarrow complex - valuedw weight ( x, assumed , split y as ) $ w open parenthesis mfollows comma : n closing parenthesis e to the power of minus i k times z sub m comma n commais a generally m comma n in complex Z − valued weight , assumed split as f o l l o w s : where z sub m comma n denotes the vector defined by the complex z sub m comma∞ n period w(m, n) = w1(m)w2(n), withw1, w2 ∈ L . We choose to study the diffraction pattern in the plane with quotedblleft oblique quotedblright \ [components w ( m open , parenthesisThe n Fourier ) k transform sub = 1 comma w of{ the k1 sub measure} 2( closing ( m 4 parenthesis. 4 ) 1 ) is w defined of{ the2 (} wavevector in a( distributional n k comma ) , namelywith components w { 1 } definedsense, ) through w , by{ 2 }\ in L ˆ{\ infty } . \ ] the Euclidean scalar product .. k times z sub m comma n comma .. with k .. = parenleftbig k sub 1 plus k sub 2 cos open parenthesis pi divided by 5 closing parenthesis parenrightbig e sub 1 plus −ik·zm,n X −ik·zm,n \noindentk sub 2 sinThe open Fourier parenthesis transform pi dividedµb(k) =by ofµ 5(e closingthe measure parenthesis) = ( 4ew sub( .m, 4 2 n comma) 1e ) is e sub, defined i times(4 (e. sub42) in j a distributional sense ) , by = delta sub ij in the plane reads as : m, n ∈ Z open parenthesis 4 period 43 closing parenthesis .. k times z sub m comma n = b sub m open \ begin { a l i g n ∗} parenthesis k sub 1 pluswhere k subzm,n 2 cosdenotes open parenthesisthe vector defined pi divided by the by 5complex closingz parenthesism,n. closing paren- thesis\ tag ∗{ plus$ (b sub 4 n openWe . parenthesis choose 42 to ) study$ k}\ sub thewidehat 2 plus diffraction k{\ submu 1pattern} cos open( in the parenthesis k plane ) with = pi “ divided oblique\mu by ”( compo- 5 closing e ˆ{ − i k \cdot z { m , n }} ) = \sum w ( m , n ) e ˆ{ − parenthesis closing parenthesisnents (k1, period k2) of the wavevector k , namely components defined through i k \cdot z { m , n }} , \\ m , n \ in Z π Hence comma the Fourierthe Euclidean transform scalar open product parenthesisk ·z 4m,n period, with 42 closingk = parenthesis (k1 + k2 ofcos the ( 5 measure))e1+ \end{ a l i g n ∗} π factorizes as the productk2 sin ( 5 )e2, ei · ej = δij in the plane reads as : π π of two one hyphen( dimensional 4 . 43 ) k Fourier·zm,n = transformbm(k1 + k of2 cos the ( type5 )) + openbn(k2 parenthesis+ k1 cos ( 5 4)) period. 2 closing parenthesis\noindent : whereHence $ z ,{ them Fourier , transform n }$ (denotes 4 . 42 ) of the the measure vector factorizes defined as by the theproduct complex $ zEquation:{ m open , parenthesis nof} two. one $ 4 -period dimensional 44 closing Fourier parenthesis transform .. mu-hatwide of the type open ( 4 . parenthesis2 ) : k closing parenthesis = Row 1 sum w sub 1 open parenthesis m closing parenthesis e to the power of minus iq 1 \noindent We choose to study the diffraction pattern in the plane with ‘‘ oblique ’’ to the power of b m Row 2 m in Z . Row 1 sum w sub 2 openb parenthesis n closingb parenthesis e to the  P w (m)e−iq1 m   P w (n)e−iq2 n  powercomponents of minus iq $ 2 to ( the k power{ 1 ofµ(} bk) n =, Row k 2 n1{ in2 Z .} comma) $ of the2 wavevector, k ,(4 namely.44) components defined through b m ∈ n ∈ with q 1 = parenleftbig k sub 1 plus k sub 2 cos openZ parenthesis pi dividedZ by 5 closing parenthesis \noindent the Euclidean scalar product \quad k $ \cdot z { m , n } parenrightbig comma q 2 = parenleftbig k sub 2π plus k sub 1 cos open parenthesisπ pi divided by 5 closing with q1 = (k1 + k2 cos ( )), q2 = (k2 + k1 cos ( )), i . e . parenthesis, $ \quad parenrightbigwith k \quad comma i$ period = e ( period5 k { 1 } + k 5 { 2 }$ cos $ ( \ f r a c {\ pi }{ 5 } ) ) e { 1 } + $ π Line 1 k sub 1 = q 1 minus q 2 cosine openq1 − parenthesisq2 cos( 5 ) pi divided1 by2 5 closing3 parenthesis divided k1 = = (4τ q1 − 2τ q2), by sine to the power of 2 open parenthesis pi dividedsin2( π by) 5 closingτ 2 + parenthesis 1 = 1 divided by tau to the power\noindent of 2 plus$ 1 parenleftbig k { 2 }$ 4 tau s i nto the $ power ( \ f of r a 2 c q5{\ 1 minuspi }{ 2 tau5 } to the) power e of{ 32 q 2} parenrightbig, e { i } \cdot e { j } = \ delta { iq j2 −}$q1 cos( inπ the) plane1 reads as : comma Line 2 k sub 2 = q 2 minus qk 1= cosine open parenthesis5 = pi(4 dividedτ 2q2 − 2 byτ 3q 51) closing parenthesis 2 2 π τ 2 + 1 divided by sine to the power of 2 open parenthesissin pi( 5 ) divided by 5 closing parenthesis = 1 divided by \noindent ( 4 . 43 ) \quad k $ \cdot z { m , n } = b { m } ( tau to the power of 2 plusSince 1 we parenleftbig know from 4 Section tau to the 4 . 1 power thatthe of 2 pure q 2 minus - point 2 support tau to the of each power factor of 3 in q 1 k { 1 } + k { 2 }$ cos $ ( \ f r a c {\ pi }{ 5 } ) ) + b { n } parenrightbig the rhs of ( 4 . 44 ) is 2π [τ], we conclude that the pure - point part of the ( k { 2 } + k { 1 }$ cosτ 2 $+1 (Z \ f r a c {\ pi }{ 5 } ) ) . $ Since we know fromdiffraction Section 4 is period supported 1 that by the the pure cyclotomic hyphen ring point up support to a scale of each factor factor : in the rhs of open parenthesis 4 period 44 closing parenthesis is 2 pi divided by tau to the power of 2 plus\noindent 1 Z openHence square bracket , the tau Fourier closing square transform bracket ( comma 4 . 42 we conclude ) of the that measure the pure hyphen factorizes point as the product o f two one − dimensional Fourier transformi π of the2π typei π ( 4 . 2 ) : part of 5 5 k = k1 + k2e ∈ 2 2 Z[e ], (4.45) the diffraction is supported by the cyclotomic ring up to a scale(τ + factor 1) : \ beginEquation:{ a l i g open n ∗} parenthesis 4 period 45 closing parenthesis .. k = k sub 1 plus k sub 2 e to the power in ( abusive ) complex notations . ANNALES DE L ’ I NS T ITUT FOURIER of\ tag i pi∗{ divided$ ( by 4 5 in .2 pi divided 44 ) by $ open}\widehat parenthesis{\mu tau} to the( power k of ) 2 plus = 1\ closingl e f t (\ parenthesisbegin { array }{ c}\sum wto the{ power1 } of( 2 Z m open square ) e bracket ˆ{ − e toi q the power 1 ˆ{ ofb i pi} dividedm }\\ by 5m closing\ in squareZ bracket\end{ commaarray }\ right ) \ l e f t (\ begin { array }{ c}\sum w in{ open2 } parenthesis( n abusive ) e closing ˆ{ − parenthesisi q 2 complex ˆ{ b } notationsn }\\ periodn \ in Z \end{ array }\ right ) , ANNALES DE L quoteright I NS T ITUT FOURIER \end{ a l i g n ∗}

\noindent with $ q 1 = ( k { 1 } + k { 2 }$ cos $ ( \ f r a c {\ pi }{ 5 } ) ) , q 2 = ( k { 2 } + k { 1 }$ cos $ ( \ f r a c {\ pi }{ 5 } ) ) , $ i . e .

\ [ \ begin { a l i g n e d } k { 1 } = \ f r a c { q 1 − q 2 \cos ( \ f r a c {\ pi }{ 5 } ) }{\ sin ˆ{ 2 } ( \ f r a c {\ pi }{ 5 } ) } = \ f r a c { 1 }{\tau ˆ{ 2 } + 1 } ( 4 \tau ˆ{ 2 } q 1 − 2 \tau ˆ{ 3 } q 2 ) , \\ k { 2 } = \ f r a c { q 2 − q 1 \cos ( \ f r a c {\ pi }{ 5 } ) }{\ sin ˆ{ 2 } ( \ f r a c {\ pi }{ 5 } ) } = \ f r a c { 1 }{\tau ˆ{ 2 } + 1 } ( 4 \tau ˆ{ 2 } q 2 − 2 \tau ˆ{ 3 } q 1 ) \end{ a l i g n e d }\ ]

\noindent Since we know from Section 4 . 1 that the pure − point support of each factor in the rhs of ( 4 . 44 ) is $\ f r a c { 2 \ pi }{\tau ˆ{ 2 } + 1 } Z[ \tau ] , $ we conclude that the pure − point part of the diffraction is supported by the cyclotomic ring up to a scale factor :

\ begin { a l i g n ∗} \ tag ∗{$ ( 4 . 45 ) $} k = k { 1 } + k { 2 } e ˆ{ i \ f r a c {\ pi }{ 5 }} \ in \ f r a c { 2 \ pi }{ ( \tau ˆ{ 2 } + 1 ) ˆ{ 2 }} Z [ e ˆ{ i \ f r a c {\ pi }{ 5 }} ], \end{ a l i g n ∗}

\noindent in ( abusive ) complex notations . ANNALES DE L ’ I NS T ITUT FOURIER DIFFRACTION OF BETA hyphen LATTICES .. 2459 DIFFRACTIONThe computation OF BETA of the− correspondingLATTICES \ intensityquad 2459 function is j ust a repe hyphen Thetition computation of the computation of the for thecorresponding one hyphen dimensional intensity case function : is j ust a repe − I sub w open parenthesis k closing parenthesis = vextendsingle-vextendsingle-vextendsingle-vextendsingle bracketleftbigg\noindent tition 1 divided of by the barsupp computation open parenthesis for the w sub one 1 to− thedimensional power of big star case closing : paren- thesis bar minus-integraldisplay to the power of tau sub 1 open square bracket w big star sub 1 open parenthesis\ begin { a l x i g closing n ∗} parenthesisDIFFRACTION e to the OF power BETA - of LATTICES i q sub 1 to2459 the The power computation of prime x of plus the w correspond- big star sub 1I open{ parenthesisw } ( minus kingx intensity ) closing = parenthesis function\arrowvert is j e ust to the a repe[ power -\ f rof a cminus{ 1 iq}{\ submid 1 to thesupp power of ( prime w x ˆ{\ star } { 1 } closing) \mid square} bracketminustition dx−integraldisplay bracketrightbigg of the computation times ˆ{\ for Equation: thetau one} open-{ dimensional1 parenthesis} [ wcase{\ 4 : periodstar 46} { closing1 } paren-( thesisx ) .. times e ˆ bracketleftbigg{ i q ˆ{\ 1 dividedprime by} bar{ 1 supp} openx } parenthesis+ w{\ wstar big star} { sub1 2} closing( parenthesis− x bar) minus e ˆ{ to − the poweri q ˆ{\ of tauprime 1 open} { square1 } bracketx } w] to the dx power ] of\ bigtimes star 2\\\ opentag parenthesis∗{$ ( 4 x 1 τ iq0 x −iq0 x . 46 ) $}\timesI (k) = |[[ \ f r a c { minus1 }{\−midintegraldisplaysupp[w? (( wx{\)e 1star+ w?}(−{ x)2e } 1 ]) dx]× closing parenthesis e tow the power| supp( of i qw? sub) | 2 to the power of prime1 x plus1 w to the power1 of big star 2\mid open} parenthesis − ˆ{\tau minus} x closing1 [ parenthesis w1 ˆ{\ estar to the}{ power2 } of minus( x iq sub ) 2 to e the ˆ{ poweri qofprime ˆ{\prime } { 2 } x } + w ˆ{\ star }{ 2 } ( − x )1 e ˆ{ −τ ?i q ˆ{\iq0 x prime? } { −2iq0 x} x }2 x closing square bracket dx bracketrightbigg vextendsingle-vextendsingle-vextendsingle-vextendsingle× [ − 1[w 2(x)e 2 + w 2(−x)e 2 ] todx]| . the] power dx of ] 2 period\arrowvert ˆ{ 2 } . | supp(w?2) | \endFor{ a instance l i g n ∗} comma for the tau hyphen grid itself with a uniform distribution : (4.46) open parenthesis 4 period 47 closing parenthesis I open parenthesis k closing parenthesis = 1 6 \ centerline {For instanceFor instance , for , for the the tau tau− - gridgrid itself itself with a uniformwith a distribution uniform distribution : : } parenleftbigg cos q sub 1 to the power of primeq0 dividedq0 by 2 tau cosq0 τ 2 q subq 20 τ 2 to the power of prime (4.47) I( k ) = 16( cos 1 cos 2 )2 ( sinc 1 sinc 2 )2. divided by 2 tau parenrightbigg 2 parenleftbigg2 sincτ q sub2τ 1 to the power2 of prime2 tau to the power of 2\noindent divided by 2$( sinc q sub 4For 2 to . a the non 47 power - split ) of weight prime Iw tau( ($k$)m, to n) the6= w power1(m)w of2( =2n) divided, it 1 is clear by 6 2 that parenrightbigg ($cos$ we have to 2\ f r a c { q ˆ{\prime } { 1 }}{ 2 period\tau }$ cos $\ resumef r a c { theq ˆ approach{\prime we} have{ 2 followed}}{ 2 in the\tau one -} dimensional) 2 case ( $ within s i n the c $\ f r a c { q ˆ{\prime } { 1 } \tauForˆ a{ non2 hyphen}}{ 2 split}theory$ weight s iof n cdistributions w open $\ f rparenthesis a c { . Howeverq ˆ{\ mprime comma , we expect n} closing{ that2 }\ theparenthesis resulttau concerningˆ negationslash-equal{ 2 }}{ the2 pure} ) w2 sub . 1 $ open parenthesis- point m closing support parenthesis of the Fourier w sub transform 2 open parenthesis will not appear n closing different parenthesis of ( 4 . 45comma ) . it is clear that we have to Acknowledgements Forresume a non the− approachsplitweight weThe have authors followed $w are in ( indebtedthe onem hyphen to , L . n Balkova dimensional ) for\not valuable case= within comments w { 1 } and( dis - m )the w theory{ 2 of} distributions(cussions n )period . ,$ However it comma is clear we expect thatwehaveto that the result concerning resumethe pure the hyphen approach point support we have of the followed Fourier transformBIBLIOGRAPHY in the will one not− appeardimensional different case within theof open theory parenthesis of distributions 4 period[ 145 ] L closing . Argabright . However parenthesis& J , period . we Gil expectde Lamadrid that, Fourier the Analysis result of Unboundedconcerning theAcknowledgements pure − pointMeasures support of the Fourier transform will not appear different o fThe ( authors4 . 45 are ) indebted . on to Locally L period Compact Balkova Abelian for Groups valuable, commentsMemoirs of the and American dis hyphen Mathematical So - cussions period ciety , vol . 145 , American Mathematical Society , Providence , RI , 1 974 . \ centerlineBIBLIOGRAPHY{Acknowledgements[ 2 ] M .} Baake & R . V . Moody , “ Weighted Dirac combs with pure point open square bracketdiffraction 1 closing ” , square bracket L period Argabright ampersand J period Gil de Lamadrid\ hspace ∗{\ commaf i l l Fourier}The Analysisauthors ofJ are Unbounded. Reine indebted Angew Measures . Math to . L573 . Balkova( 2004 ) , p for . 6 1 - valuable 94 . comments and dis − on Locally Compact Abelian Groups[ 3 ] J . P comma . Bell Memoirs& K . G of . 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\noindent TOME 5 6 ( 2 0 0 6 ) , FASCICULE 7 2460 Jean - P i erre GAZEAU & Jean - Louis VERGER - GAUGRY [8] C.ˇ Burd´ık , C . Frougny , J . - P . Gazeau & R . Krejcarˇ , “ Beta - integers as natural counting systems for quasicrystals ” , J . of Physics A : Math . Gen . 31 ( 1 998 ) , p . 6449 - 6472 . [ 9 ] A . Cordoba , “ Dirac combs ” , Lett . Math . Phys . 1 7 ( 1 989 ) , p . 1 9 1 - 1 96 . [ 10 ] J . - M . Cowley , Diffraction Physics , North - Holland , Amsterdam , 1 986 , 2 nd edition . [ 1 1 ] F . Denoyer , A . Elkharrat & J . - P . Gazeau , “ Beta - lattice multiresolution of qua - sicrystalline Bragg peaks ” , submitted , 2006 . [ 1 2 ] A . Elkharrat , “ Scale dependent partitioning of one - dimensional aperiodic set diffraction ” , Europ . Phys . J . B 39 ( 2004 ) , p . 287 - 294 , and Th e` se de l ’ Universit´e Paris 7 - Denis Diderot ( 2004 ) . [ 13 ] A . Elkharrat , C . Frougny , J . - P . Gazeau & J . - L . 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Moody , ed . ) , NATO advances Science Institutes , Series C : Mathematical and Physical Sciences 489 , Kluwer Academic Publishers , Dordrecht , 1 997 , p . 175 - 1 98 . [ 20 ] J . - P . Gazeau & J . - L . Verger - Gaugry , “ Geometric study of the beta - integers for a Perron number and mathematical quasicrystals ” , J . Th´eorie Nombres Bordeaux 1 6 ( 2004 ) , p . 1 25 - 149 . [ 2 1 ] J . Gil de Lamadrid & L . Argabright , “ Almost Periodic Measures ” , Memoirs of the American Mathematical Society , American Mathematical Society , Providence , RI 85 ( 1 990 ) , no . 428 , p . vi +219. [ 22 ] A . Guinier , Theory and Techniques for X - Ray Crystallography , Dunod , Paris , 1 964 . [ 23 ] A . Hof , “ On diffraction by aperiodic structures ” , Commun . Math . Phys . 1 69 ( 1 995 ) , p . 25 - 43 . [ 24 ] J . C . Lagarias , “ Geometric Models for Quasicrystals I . Delone Sets of Finite Type ” , Discr . Comput . Geom . 2 1 ( 1 999 ) , p . 1 6 1 - 1 9 1 . 2460 .. Jean hyphen P i erre GAZEAU ampersand Jean hyphen Louis VERGER hyphen GAUGRY \noindentopen square2460 bracket\quad 8 closingJean square− P bracket i e r r e Ccaron GAZEAU period $ Burdiacutek\& $ Jean comma− Louis C period VERGER Frougny− GAUGRY comma J period hyphen P period Gazeau ampersand R period Krejccaronar comma quotedblleft Beta hyphen\ hspace integers∗{\ f i as l l natural} $ [ 8 ] \check{C} . $ Burd \ ’{\ i }k ,C . Frougny , J . − P . Gazeau $ \counting& $ R systems . Krej for\v{ quasicrystalsc} ar , ‘ ‘ quotedblright Beta − integers comma J as period natural of Physics A : Math period Gen period 31 open parenthesis 1 998 closing parenthesis comma p period 6449 hyphen counting6472 period systems[ 25 for ] — quasicrystals — , “ Mathematical Quasicrystals ’’ , J . and of the Physics problem of A diffraction : Math ” , . in GenDirections . 31 ( 1 998 ) , p . 6449 − 6472open . square bracket 9in closing Mathematical square Quasicrystals bracket A period( M . Baake Cordoba& R comma. V . Moody .. quotedblleft , eds . ) , CRM Dirac Monograph combs quotedblright comma LettSeries period , Amer Math . Math period . Soc . Phys , Providence period ,1 RI 7 open , 2000 parenthesis , p . 6 1 - 93 1 . 989 closing parenthesis comma\ centerline p period{ [ 1 9 9 1[ ] hyphen 26 A ] M . . Cordoba 1 Lothaire 96 period, , Algebraic\quad Combinatorics‘‘ Diracon combs Words ’’, Cambridge , Lett University . Math Press . Phys , . 1 7 ( 1 989 ) , p . 1 9 1 − 1 96 . } open square bracket2002 10 . closing square bracket J period hyphen M period Cowley comma Diffraction Physics\noindent comma[ North 10 ][ hyphen J27 ] . Y− . Holland MeyerM . Cowley, comma “ Nombres ,Amsterdam Diffraction de Pisot , Nombres comma Physics de1 986 Salem comma et , Analyse North 2 nd Harmonique edition− Holland period ” , in , Amsterdam , 1 986 , 2 nd edition . open square bracketLect 1 1 . closingNotes Math square . , bracket vol . 1 1 F 7 ,period Springer Denoyer , 1 969 , comma p . 63 . A period Elkharrat amper- sand\noindent J period[ hyphen 1 1[ ]P 28 Fperiod ] — . — Denoyer Gazeau , Algebraic comma , Numbers A . quotedblleft Elkharrat and Harmonic Beta Analysis $ hyphen\& $, North lattice J . - Holland− multiresolutionP . , 1 Gazeau 972 . of, qua ‘‘ Beta − lattice multiresolution of qua − hyphensicrystalline Bragg[ 29 ] — peaks — , “ Quasicrystals ’’ , submitted , Diophantine , approximation 2006 . and algebraic numbers ” , in Be - sicrystalline Bragg peaksyond quotedblright Quasicrystals comma( F . Axel submitted& D . 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Moody and algebraic , \ h f inumbers l l ‘‘ Meyer quotedblright sets comma and their in Be hyphen duals ’’ , in The Mathematics of Long − Range yond Quasicrystals open parenthesis F period Axel ampersand D period Gratias comma eds period closing\ centerline parenthesis{ Aperiodic comma Springer Order hyphen ( R .Verlag V . ampersand Moody , Lesed Editions . ) , Kluwer de , 1 997 , p . 403 − 442 . } Physique comma 1 995 comma p period 3 hyphen 1 6 period \noindentopen squareANNALES bracket DE 30 closing L ’ I square NS T bracket ITUT FOURIER R period V period Moody comma .... quotedblleft Meyer sets and their duals quotedblright comma in The Mathematics of Long hyphen Range Aperiodic Order open parenthesis R period V period Moody comma ed period closing parenthesis comma Kluwer comma 1 997 comma p period 403 hyphen 442 period ANNALES DE L quoteright I NS T ITUT FOURIER DIFFRACTION OF BETA - LATTICES 246 1 [ 31 ] — — , “ From quasicrystals to more complex systems ” , in Model Sets : A Survey ( F . Axel , F . Denoyer & J . - P . Gazeau , eds . ) , Springer & Les Editions de Physique , 2000 , p . 145 - 1 66 . [ 32 ] G . Muraz & J . - L . Verger - Gaugry , “ On lower bounds of the density of Delone sets and holes in sequences of sphere packings ” , Exp . Math . 1 4 ( 2005 ) , no . 1 , p . 47 - 57 . [ 33 ] W . Parry , “ On the β− expansions of real numbers ” , Acta Math . Acad . Sci . Hungar . 1 1 ( 1 960 ) , p . 40 1 - 41 6 . [ 34 ] N . Pytheas´ Fogg , “ Substitutions in dynamics , arithmetics and combinatorics ” , in Lecture Notes in Math . , vol . 1 794 , Springer , 2003 . [ 35 ] A.Renyi´ , “ Representations for real numbers and their ergodic properties ” , Acta Math . Acad . Sci . Hung . 8 ( 1 957 ) , p . 477 - 493 . [ 36 ] M . Schlottmann , “ Cut - and - Proj ect sets in locally compact Abelian groups ” , in Quasicrystals and Discrete Geometry ( J . Patera , ed . ) , Fields Institute Monograph Series , vol . 10 , Amer . Math . Soc . , Providence , RI , 1 998 , p . 247 - 264 . [ 37 ] L . S chwartz , Th´eoriedes distributions , Hermann , Paris , 1 973 . [ 38 ] D. Shechtman , I . Blech , D . Gratias & J . Cahn , “ Metallic phase with long - range orientational order and no translational symmetry ” , Phys . Rev . Lett . 5 3 ( 1 984 ), p . 1 951 - 1 953 , 1 95 1 . [ 39 ] N . Strungaru , “ Almost Periodic Measures and Long - Range Order in Meyer Sets ”, Discr . Comput . Geom . 33 ( 2005 ) , p . 483 - 505 . [ 40 ] W . P . Thurston ,“ Groups , tilings , and finite state automata ” , A . M . S . Colloquium Lectures , Boulder , Summer 1 989 . [ 41 ] J . - L . Verger - Gaugry , “ On gaps in R´enyi β− expansions of unity for β > 1 an algebraic number ” , Annales Institut Fourier , 2006 . [ 42 ] — — , “ On self - similar finitely generated uniformly discrete ( SFU - ) sets and sphere packings ” , in Number Theory and Physics ( L . Nyssen , ed . ) , IRMA Lectures in Mathematics and Theoretical Physics , E . M . S . Publishing House , 2006 . [ 43 ] K. Vo Khac , “ Fonctions et distributions stationnaires . Application a` l ’ ´etude des solutions stationnaires d ’ ´equations aux d´eriv´eespartielles ” , in Espaces de Marcinkiewicz , corr´elations, mesures , syst e` mes dynamiques , Masson , Paris , 1 987 , p . 1 1 - 57 . Jean - Pierre GAZEAU Universit´eParis 7 - Denis Diderot APC - UMR CNRS 7164 Boite 7020 7525 1 Paris cedex 5 ( France ) gazeau @ ccr . j ussieu . fr Jean - Louis VERGER - GAUGRY Universit´eGrenoble I Institut Fourier - UMR CNRS 5582 DIFFRACTION OF BETA hyphen LATTICES .. 246 1 \ hspaceopen square∗{\ f ibracket l l }DIFFRACTION 31 closing square OF bracket BETA − emdashLATTICES emdash\quad comma246 quotedblleft 1 From quasicrystals to more complex systems quotedblright comma in Model Sets : A Survey \noindentopen parenthesis[ 31 ] F−−− period −−− Axel, comma ‘‘ From F period quasicrystals Denoyer ampersand to more J complexperiod hyphen systems P period ’’ , in Model Sets : A Survey Gazeau comma eds period closing parenthesis comma Springer ampersand Les Editions de Physique comma\ hspace ∗{\ f i l l }( F . Axel , F . Denoyer $ \& $ J . − P . Gazeau , eds . ) , Springer $ \2000& $ comma Les pEditions period 145 de hyphen Physique 1 66 period , BP 74 open square bracket 32 closing square38402 bracket Saint .... - Martin G period d ’ H Muraze` res ( ampersand France ) J period hyphen L period\ centerline Verger hyphen{2000 Gaugry , p . comma 145 − quotedblleft1 66 . j lverger} On lower@ ujbounds f - grenoble of .the fr density of Delone sets and holes in sequencesTOME of 5 6 sphere( 2 0 0 6 ) packings , FASCICULE quotedblright 7 comma Exp period Math period 1 4 open parenthesis\noindent 2005[ 32 closing ] \ parenthesish f i l l G . comma Muraz no period $ \& 1$ comma J . − p periodL . Verger 47 hyphen− 57Gaugry period , ‘‘ On lower bounds of the density of Delone sets open square bracket 33 closing square bracket W period Parry comma .. quotedblleft On the beta hyphenand holes expansions in sequences of real numbers of sphere quotedblright packings comma ’’ Acta , Exp Math . period Math Acad . 1 4 period ( 2005 Sci period ) , no . 1 , p . 47 − 57 . Hungar[ 33 period ]W. Parry , \quad ‘ ‘ On the $ \beta − $ expansions of real numbers ’’ , Acta Math . Acad . Sci . Hungar . 1 1 open parenthesis 1 960 closing parenthesis comma p period 40 1 hyphen 41 6 period \ centerlineopen square{11(1960) bracket 34 closing ,p.401 square bracket− N41 period 6 . Pytheacuteas} Fogg comma quotedblleft Substitutions in dynamics comma arithmetics and combinatorics quotedblright comma in \noindentLecture Notes[ 34 in Math] N . period Pyth comma\ ’{ e} as vol Fogg period ,1 794 ‘‘ comma Substitutions Springer comma in dynamics 2003 period , arithmetics and combinatorics ’’ , in open square bracket 35 closing square bracket A period Reacutenyi comma quotedblleft Representa- tions\ centerline for real numbers{ Lecture and their Notes ergodic in Math properties . , quotedblright vol . 1 794 comma , Springer Acta , 2003 . } Math period Acad period Sci period Hung period 8 open parenthesis 1 957 closing parenthesis comma p\noindent period 477 hyphen[ 35 ] 493 A period . R\ ’{ e} nyi , ‘‘ Representations for real numbers and their ergodic properties ’’ , Acta open square bracket 36 closing square bracket M period Schlottmann comma quotedblleft Cut hyphen and\noindent hyphen ProjMath ect . sets Acad in locally . Sci compact . 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Publishing Fonctions House , 2006 . et[ distributions 43 ] K . stationnaires\quad Vo \ periodquad ..Khac Application , \quad agrave‘‘ l Fonctions quoteright eacutetude et distributions stationnaires . \quad Application $ \desgrave solutions{a} $ stationnaires l ’ \ ’{ e d} tude quoteright eacutequations aux deacuteriveacutees partielles quoted- blright comma in Espaces de \ hspaceMarcinkiewicz∗{\ f i l l comma} des correacutelations solutions stationnaires comma mesures d comma ’ \ ’{ syste} quations egrave mes dynamiques aux d\ ’{ e comma} r i v \ ’{ e} es partielles ’’ , in Espaces de Masson comma Paris comma 1 987 comma \ hspacep period∗{\ 1f 1 i hyphen l l } Marcinkiewicz 57 period , corr \ ’{ e} lations , mesures , syst $ \grave{e} $ mesJean dynamiques hyphen Pierre , MassonGAZEAU , Paris , 1 987 , Universiteacute Paris 7 hyphen Denis Diderot \ centerlineAPC hyphen{p UMR . 1 CNRS 1 − 57 7164 . } Boite 7020 \ centerline7525 1 Paris{ cedexJean 5− openP i e parenthesis r r e GAZEAU France} closing parenthesis gazeau at ccr period j ussieu period fr \ centerlineJean hyphen{ U Louis n i v eVERGER r s i t \ ’{ e hyphen} Paris GAUGRY 7 − Denis Diderot } Universiteacute Grenoble I \ centerlineInstitut Fourier{APC hyphen− UMR UMR CNRS CNRS 7164 5582} BP 74 \ centerline38402 Saint{ hyphenBoite Martin 7020 d} quoteright H egrave res open parenthesis France closing parenthesis j lverger at uj f hyphen grenoble period fr \ centerlineTOME 5 6 open{7525 parenthesis 1 Paris 2 0cedex 0 6 closing 5 ( parenthesis France ) comma} FASCICULE 7 \ centerline { gazeau $@$ ccr . j ussieu . fr }

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