DOCSLIB.ORG

Cyclotomic Matrices Over Quadratic Integer Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cyclotomic Matrices Over Quadratic Integer Rings Cyclotomic Matrices over Quadratic Integer Rings Gary Greaves Thesis submitted to the University of London for the degree of Doctor of Philosophy 2012 Declaration of Authorship I, Gary Greaves, hereby declare that this thesis and the work presented in it is entirely my own. Where I have consulted the work of others, this is always clearly stated. Signed: (Gary Greaves) Date: Summary This thesis has been motivated largely by Lehmer’s problem [20], which was stated in 1933 and it is still a problem that mathematicians have not completely solved. The Mersenne n sequence, (2 1)n N, has properties that make it useful for finding large primes but its terms become− very2 large very fast. Lehmer’s problem is related to finding large primes in sequences that are analogous to the Mersenne sequence but that grow as slowly as possible and Lehmer’s conjecture implies a lower bound on the growth rate of any such sequence. Lehmer’s problem is usually stated in terms of a geometric constraint on the zeros of polynomials having integer coefficients and top coefficient 1. Breusch [4] and Smyth [38] have reduced the problem to one where it is only necessary to consider a much smaller class of polynomials. Some recent progress has been made on this restricted version of Lehmer’s problem by associating some of the polynomials of this smaller class to combinatorial objects. The characteristic polynomial χA (x) of a matrix A is taken to be det(x I A). For an n n integer symmetric matrix A we define its associated polynomial as − × n RA (z ) := z χA (z + 1=z ). A Hermitian matrix A is called cyclotomic if all of the zeros of RA lie on the unit circle. McKee and Smyth [26] showed that Lehmer’s conjecture holds for the polynomials RA for all integer symmetric matrices A. Their method involved first classifying all cyclotomic integer symmetric matrices. McKee [23] used the classification of cyclotomic integer symmetric matrices to classify certain polynomials (which he called small-span polynomials) that are also characteristic polynomials of integer symmetric matrices. A large part of my research has involved developing the method of McKee and Smyth of associating algebraic numbers to combinatorial objects. The main results are the following. 1. The classification of cyclotomic matrices over the Eisenstein and Gaussian integers. 2. The classification of cyclotomic matrices over real quadratic integer rings. 3. Reducing to a finite search the proof that Lehmer’s conjecture holds for polynomials RA for all Hermitian matrices A over the Eisenstein and Gaussian integers. 4. Confirmation that Lehmer’s conjecture holds for polynomials RA for all real sym- metric matrices A over real quadratic integer rings. 5. The classification of small-span polynomials that are also characteristic polynomials of Hermitian matrices over quadratic integer rings. 3 Acknowledgments To my supervisor James McKee from whom I have learnt so much and whose patience, advice, and support have been invaluable to me. To the EPSRC and the Heilbronn insti- tute for their funding which enabled me to go to various conferences and meetings. In particular, I found it very useful to discuss my research in Edinburgh with Chris Smyth and Graeme Taylor. To the mathematics department whose inhabitants, past and present, have created an open and friendly atmosphere. To Laurence for finding typos and to Anastasia, Ciaran, Kenny, Max, and Millie for musical interludes. To my friends for keeping me aware of life outside of the mathematics department. And finally, to my family for their love. I extend my deepest gratitude. 4 Contents 1 Introduction 9 1.1 Boyd and Lehmer . 9 1.2 Simple graphs . 11 1.3 Integer symmetric matrices . 13 1.4 Recurrent themes . 15 2 Radical Integer Trees 19 2.1 Cyclotomics . 19 2.2 Minimal non-cyclotomics . 22 2.3 Coxeter systems . 28 2.4 Unfolding trees . 30 2.5 Boyd’s conjecture . 34 3 Hermitian Matrices over Imaginary Quadratic Integer Rings 40 3.1 Cyclotomic matrices over Z[i ] and Z[!] ......................... 40 3.2 Excluded subgraphs and Gram matrices . 45 3.3 Proof of Theorem 3.1 . 48 3.4 Proof of Theorem 3.2 . 57 3.5 Proof of Theorem 3.3 . 62 3.6 The Eisenstein integers . 69 3.7 Lehmer’s problem . 71 4 Hermitian Matrices over Real Quadratic Integer Rings 78 4.1 Integral characteristic polynomials . 78 4.2 Classification of cyclotomic R-matrices . 79 4.3 Proof of Theorem 4.3 . 83 4.4 Applying Perron-Frobenius theory . 89 4.5 Lehmer’s problem . 94 5 Small-Span Hermitian Matrices over Quadratic Integer Rings 102 5.1 Orientation . 102 5.2 Computation of small-span matrices of up to 8 rows . 103 5.3 Maximal small-span infinite families . 113 5.4 Missing small-span polynomials . 121 Bibliography 123 5 List of Figures 1.1 Extended simply-laced Coxeter graphs . 12 ˜ ˜ ˜ ˜ 2.1 The maximal cyclotomic radical integer trees A1, Bn (n ¾ 3), Cn (n ¾ 2), Dn (n ¾ ˜ ˜ ˜ ˜ ˜ 4), E6, E7, E8, F4, and G2. The numbers on the vertices correspond to eigen- vectors with largest eigenvalue 2. The number of vertices is one more than the subscript. 21 2.2 The 9 minimal non-cyclotomic simple trees. 22 2.3 The minimal non-cyclotomic radical integer trees on more than 2 vertices with at least one irrational edge-weight. 23 2.4 Positive semidefinite Coxeter graphs. The number of vertices is one more than the subscript. 29 (x) 3.1 The families T2k and T2k (respectively) of 2k -vertex maximal connected cyclo- tomic Z[x]-graphs, for k ¾ 3 and x i ,! . (The two copies of vertices A and B should be identified to give a toral2 ftessellation.)g . 42 3.2 The family of 2k -vertex maximal connected cyclotomic Z[i ]-graphs C2k for k ¾ 2...................................................... 42 ++ 3.3 The families of 2k -vertex maximal connected cyclotomic Z-graphs C2k and + C2k− for k ¾ 2................................................ 42 3.4 The family of (2k +1)-vertex maximal connected cyclotomic Z[i ]-graphs C2k +1 for k ¾ 1.................................................... 42 3.5 The sporadic maximal connected cyclotomic Z[!]-graphs S10, S12, and S14 of orders 10, 12, and 14 respectively. The Z-graph S14 is also a Z[i ]-graph. 43 3.6 The sporadic maximal connected cyclotomic Z-hypercube S16. 43 3.7 The sporadic maximal connected cyclotomic Z[!]-graphs of orders 1, 2, 4, 5, and 6. The Z-graphs S1 and S2 are also Z[i ]-graphs. 43 3.8 The sporadic maximal connected cyclotomic Z[i ]-graphs of orders 4, 7, and 8. The Z-graphs S7, S8, and S08 are also Z[!]-graphs. 43 3.9 some non-cyclotomic uncharged Z-graphs. 48 3.10 some cyclotomic Z[i ]-graphs that are contained as subgraphs of fixed maximal connected cyclotomic Z[i ]-graphs. 48 3.11 some non-cyclotomic uncharged Z[i ]-graphs. 58 3.12 some cyclotomic Z[i ]-graphs that are contained as subgraphs of fixed maximal connected cyclotomic Z[i ]-graphs. 58 3.13 some non-cyclotomic charged Z[i ]-graphs. 63 3.14 some charged cyclotomic Z[i ]-graphs that are contained as subgraphs of fixed maximal connected cyclotomic Z[i ]-graphs. 63 3.15 some cyclotomic Z[!]-graphs that are contained as subgraphs of fixed maxi- mal connected cyclotomic Z[!]-graphs. 69 6 List of Figures 3.16 some non-cyclotomic charged Z[!]-graphs. 71 3.17 some charged cyclotomic Z[!]-graphs that are contained as subgraphs of fixed maximal connected cyclotomic Z[!]-graphs. 71 3.18 Some Z[i ]-graphs that are not subgraphs of any non-supersporadic graph having at least 5 vertices. 72 3.19 Some Z[i ]-graphs that are not subgraphs of any non-supersporadic graph on at least 10 vertices. 75 4.1 The family T2k of 2k -vertex maximal connected cyclotomic Z-graphs, for k ¾ 3. (The two copies of vertices A and B should be identified to give a toral tessellation.) . 80 4.2 The family of 2k -vertex maximal connected cyclotomic Z[p2]-graphs C2k for k ¾ 2...................................................... 80 ++ 4.3 The families of 2k -vertex maximal connected cyclotomic Z-graphs C2k and + C2k− for k ¾ 2................................................ 80 4.4 The family of (2k + 1)-vertex maximal connected cyclotomic Z[p2]-graphs C2k +1 for k ¾ 1............................................... 80 4.5 The sporadic maximal connected cyclotomic Z-graph S14 of order 14. 80 4.6 The sporadic maximal connected cyclotomic Z-hypercube S16. 81 4.7 The sporadic maximal connected cyclotomic R-graphs of orders 1, 2, 3 and 4. 81 4.8 The sporadic maximal connected cyclotomic R-graphs of orders 6, 7, and 8. 81 4.9 some non-cyclotomic Z[p2]-graphs. 83 4.10 some cyclotomic Z[p2]-graphs that are contained as subgraphs of fixed maxi- mal connected cyclotomic Z[p2]-graphs. 84 4.11 Four infinite families of nonnegative cyclotomic Z[p2]-graphs each having spectral radius 2. The numbers on the vertices correspond to an eigenvector with largest eigenvalue 2. The subscript is the number of vertices. 90 5.1 The infinite family T2k of 2k -vertex maximal connected cyclotomic templates. (The two copies of vertices A and B should be identified to give a toral tessella- tion.) . 114 5.2 The infinite family of 2k -vertex maximal connected cyclotomic templates C2k C and 20 k (respectively) for k ¾ 2. 115 5.3 The infinite family of (2k +1)-vertex maximal connected cyclotomic templates C2k +1 for k ¾ 1. .............................................. 115 5.4 Cyclotomic templates having span equal to 4. In the first three templates, the subscript denotes the number of vertices.
Load more

Recommended publications
  • Quartic Salem Numbers Which Are Mahler Measures of Non-Reciprocal 2-Pisot Numbers Tome 32, No 3 (2020), P
    Toufik ZAÏMI Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers Tome 32, no 3 (2020), p. 877-889. <http://jtnb.centre-mersenne.org/item?id=JTNB_2020__32_3_877_0> © Société Arithmétique de Bordeaux, 2020, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est con- stitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 877–889 Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers par Toufik ZAÏMI Résumé. Motivé par une question de M. J. Bertin, on obtient des paramé- trisations des polynômes minimaux des nombres de Salem quartiques, disons α, qui sont des mesures de Mahler des 2 -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres α, de trace donnée, et de déduire que pour tout entier naturel t (resp. t ≥ 2), il y a un nombre de Salem quartique, de trace t, qui est (resp. qui n’est pas) une mesure de Mahler d’un 2 -nombre de Pisot non-réciproque.
    [Show full text]
  • The Smallest Perron Numbers 1
    MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers.
    [Show full text]
  • Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems
    Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems Dissertation zur Erlangung des akademischen Grades eines Doktors der Mathematik (Dr. math.) vorgelegt von Chrizaldy Neil Ma~nibo Fakult¨atf¨urMathematik Universit¨atBielefeld April 2019 Gedruckt auf alterungsbest¨andigem Papier ◦◦ISO 9706 1. Berichterstatter: Prof. Dr. Michael Baake Universit¨atBielefeld, Germany 2. Berichterstatter: A/Prof. Dr. Michael Coons University of Newcastle, Australia 3. Berichterstatter: Prof. Dr. Uwe Grimm The Open University, Milton Keynes, UK Datum der m¨undlichen Pr¨ufung:04 Juni 2019 i Contents Acknowledgementsv Introduction viii 1. Prerequisites 1 d 1.1. Point sets in R .....................................1 1.2. Symbolic dynamics and inflation rules........................1 1.2.1. Substitutions..................................1 1.2.2. Perron{Frobenius theory............................3 1.2.3. The symbolic hull...............................4 1.2.4. Inflation systems and the geometric hull...................5 1.3. Harmonic analysis and diffraction...........................7 1.3.1. Fourier transformation of functions......................7 1.3.2. Measures....................................7 1.3.3. Decomposition of positive measures.....................9 1.3.4. Autocorrelation and diffraction measure...................9 1.4. Lyapunov exponents.................................. 11 1.4.1. Lyapunov exponents for sequences of matrices................ 11 1.4.2. Matrix cocycles................................. 12 1.4.3. Ergodic theorems...............................
    [Show full text]
  • A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected]
    The Waterloo Mathematics Review 35 A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected] Abstract: Motivated by the search for “almost integers”, we describe the algebraic integers known as Pisot numbers, and explain how they can be used to easily find irrational values that can be arbitrarily close to whole numbers. Some properties of the set of Pisot numbers are briefly discussed, as well as some applications of these numbers to other areas of mathematics. 1 Introduction It is a curious occurrence when an expression that is known to be a non-integer ends up having a value surprisingly close to a whole number. Some examples of this phenomenon include: eπ π = 19.9990999791 ... − 23 5 = 109.0000338701 ... 9 88 ln 89 = 395.0000005364 ... These peculiar numbers are often referred to as “almost integers”, and there are many known examples. Almost integers have attracted considerable interest among recreational mathematicians, who not only try to generate elegant examples, but also try to justify the unusual behaviour of these numbers. In most cases, almost integers exist merely as numerical coincidences, where the value of some expression just happens to be very close to an integer. However, sometimes there actually is a clear, mathematical reason why certain irrational numbers should be very close to whole numbers. In this paper, we’ll look at the a set of numbers called the Pisot numbers, and how they can be used to systematically construct infinitely many examples of almost integers. In Section 2, we will prove a result about powers of roots of polynomials, and use this as motivation to define the Pisot numbers.
    [Show full text]
  • Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds
    COUNTING SALEM NUMBERS OF ARITHMETIC HYPERBOLIC 3{ORBIFOLDS MIKHAIL BELOLIPETSKY, MATILDE LAL´IN, PLINIO G. P. MURILLO, AND LOLA THOMPSON Abstract. It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines cQ1=2 +O(Q1=4) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, 4 3=2 which is shown to be asymptotic to 3 Q + O(Q). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3. 1. Introduction A Salem number is a real algebraic integer λ > 1 such that all of its Galois conjugates except λ−1 have absolute value equal to 1. Salem numbers appear in many areas of mathematics including algebra, geometry, dynamical systems, and number theory. They are closely related to the celebrated Lehmer's problem about the smallest Mahler measure of a non-cyclotomic polynomial. We refer to [Smy15] for a survey of research on Salem numbers. It has been known for some time that the exponential lengths of the closed geodesics of an arithmetic hyperbolic n-dimensional manifold or orbifold are given by Salem numbers.
    [Show full text]
  • Small Salem Graphs
    Department of Mathematics, Royal Holloway College, University of London July 2016 Small Salem Graphs PhD thesis by Jonathan Charles James Cooley Supervised by Professor James F McKee 1 Declaration of Authorship I Jonathan Charles James Cooley hereby declare that this thesis and the work presented in it is entirely my own. Where I have consulted the work of others, this is always clearly stated. July 2016 2 Abstract The aim for this thesis was to produce the first systematic catalogue of small Salem graphs, and to illustrate and enumerate those with interesting properties. This is done in the central section, chapters 3 to 5. That is preceded by two introductory chapters, the first dealing with definitions and motivation, the second concerning the computational methods used to construct the catalogue. A new isomorphism testing algorithm is presented which has proved highly successful in practice, but an example is constructed in which it is useless. A complete classification of circulant Salem graphs is conjectured. For Salem graphs, their Salem number and their Mahler measure are precisely the same thing, and some observations were made which enabled the completion of the classification of all graphs with Mahler measure below , the very well-known ‘golden number’. The final chapter is an exposition of the paper which reports this completion of the classification. 3 Contents Chapter 1 Definitions and motivation 1.1 Salem numbers 1.2 Totally real algebraic integers, and graphs 1.3 Linking the previous two sections 1.4 Combinatorial objects and polynomials
    [Show full text]
  • Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks
    Dynamics on K3 surfaces: Salem numbers and Siegel disks Curtis T. McMullen 19 January, 2001 Abstract This paper presents the first examples of K3 surface automorphisms f : X X with Siegel disks (domains on which f acts by an irrational rotation).→ The set of such examples is countable, and the surface X must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of de- gree 22 and trace 1, which play the role of the leading eigenvalue for f ∗ H2(X). The construction− uses the Torelli theorem, the Atiyah-Bott fixed-point| theorem and results from transcendence theory. Contents 1 Introduction............................ 1 2 K3surfaces ............................ 7 3 AutomorphismsofK3surfaces . 11 4 ErgodicdynamicsonKummersurfaces. 14 5 Siegel disks and transcendence theory . 18 6 HolomorphicLefschetznumbers. 20 7 SiegeldisksonK3surfaces. 22 8 Latticesinnumberfields. 24 9 From Salem numbers to automorphisms . 29 10 ExamplesofSiegeldisks . 30 11 Limits of K¨ahler-Einstein metrics . 34 Research supported in part by the NSF. 2000 Mathematics Subject Classification: 37F50 (11R06, 14J50, 32H50). 1 Introduction The first dynamically interesting automorphisms of compact complex man- ifolds arise on K3 surfaces. Indeed, automorphisms of curves are linear (genus 0 or 1) or of finite order (genus 2 or more). Similarly, automorphisms of most surfaces (includ- ing P2, surfaces of general type and ruled surfaces) are either linear, finite order or skew-products over automorphisms of curves. Only K3 surfaces, Enriques surfaces, complex tori and certain non-minimal rational surfaces admit automorphisms of positive topological entropy [Ca2]. The automor- phisms of tori are linear, and the Enriques examples are double-covered by K3 examples.
    [Show full text]
  • Special Classes of Algebraic Integers in Low-Dimensional Topology
    Special classes of algebraic integers in low-dimensional topology Eriko Hironaka April 15, 2005 Abstract This note describes some open problems concerning distributions of special classes of real algebraic integers such as algebraic units, and Salem, P-V and Perron numbers. These special algebraic integers appear naturally as geometric invariants in low-dimensional topology. We relate properties of Salem, P-V and Perron to minimization problems in various geometric settings. 1 Introduction A complex number α ∈ C is an algebraic integer if it is a root of a monic integer polynomial. Two algebraic integers α and β are algebraically conjugate, written α ∼ β, if α and β satisfy the same irreducible monic integer polynomial. An algebraic integer α is an algebraic unit if α ∼ α−1. Let α be a real algebraic integer with α > 1. Consider all β ∼ α such that β 6= α: (i) if |β| < |α|, then α is a Perron number; (ii) if |β| < 1, then α is a P-V number; and (iii) if |β| ≤ 1 with at least one |β| = 1, then α is a Salem number. In this short note, we review definitions and known results concerning distributions of P-V, Salem and Perron numbers (Section 2), and relate them to geometric invariants in low-dimensional topology, including lengths of geodesics, growth rates of automatic groups, and homological and geometric dilatations of surface homeomorphisms (Section 3). 2 Distributions of algebraic integers and Lehmer’s problem Let P be the set of monic integer polynomials. Given f ∈ P, let Sf be the set of complex roots of + f counted with multiplicity, and let S (f) ⊂ Sf be the subset of points outside the unit circle C.
    [Show full text]
  • A Proof of the Conjecture of Lehmer
    A PROOF OF THE CONJECTURE OF LEHMER AND OF THE CONJECTURE OF SCHINZEL-ZASSENHAUS JEAN-LOUIS VERGER-GAUGRY September 13, 2017 Abstract. The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f α (z) associated with the dynamical zeta functions ζ α (z) of the Rényi–Parry arithmetical dynamical systems, for α an algebraic integer α of house α greater than 1, (ii) the discovery of lenticuli of poles of ζ α (z) which uniformly equidistribute at the limit on a limit “lenticular" arc of the unit circle, when α tends to 1+, giving rise to a contin- uous lenticular minorant Mr( α ) of the Mahler measure M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant Mr( α ) as a function of the dynamical degree. With the same arguments the conjecture of Schinzel- Zassenhaus is proved to be true. An inequality improving those of Dobrowolski and Voutier ones is obtained. The set of Salem numbers is shown to be bounded −1 from below by the Perron number θ31 =1.08545 ..., dominant root of the trinomial 1 z30 + z31. Whether Lehmer’s number is the smallest Salem number remains open.− − A lower bound for the Weil height of nonzero totally real algebraic num- bers, = 1, is obtained (Bogomolov property). For sequences of algebraic integers of Mahler6 ± measure smaller than the smallest Pisot number, whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on z =1 (limit equidistribution).
    [Show full text]
  • On Beta Expansions for Pisot Numbers
    MATHEMATICS OF COMPUTATION Volume 65, Number 214 April 1996, Pages 841–860 ON BETA EXPANSIONS FOR PISOT NUMBERS DAVID W. BOYD Abstract. Given a number β>1, the beta-transformation T = Tβ is defined for x [0, 1] by Tx := βx (mod 1). The number β is said to be a beta- number∈ if the orbit T n(1) is finite, hence eventually periodic. In this case β is the root of a monic{ polynomial} R(x) with integer coefficients called the characteristic polynomial of β.IfP(x) is the minimal polynomial of β,then R(x)=P(x)Q(x) for some polynomial Q(x). It is the factor Q(x)which concerns us here in case β is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether Q(x)mustbe cyclotomic in this case, particularly if 1 <β<2. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in [1, 1.9324] [1.9333, 1.96] (an infinite set), byasearchuptodegree50in[1.9,2],todegree60in[1∪ .96, 2], and to degree 20 in [2, 2.2]. We find the smallest counterexample, the counterexample of smallest degree, examples where Q(x) is nonreciprocal, and examples where Q(x) is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to 2 from above, and infinite sequences of β with Q(x) nonreciprocal which converge to 2 from below and to the 6th smallest limit point of the Pisot numbers from both sides.
    [Show full text]
  • On the Complexity of Algebraic Number I. Expansions in Integer Bases. Boris Adamczewski, Yann Bugeaud
    On the complexity of algebraic number I. Expansions in integer bases. Boris Adamczewski, Yann Bugeaud To cite this version: Boris Adamczewski, Yann Bugeaud. On the complexity of algebraic number I. Expansions in integer bases.. 2005. hal-00014567 HAL Id: hal-00014567 https://hal.archives-ouvertes.fr/hal-00014567 Preprint submitted on 28 Nov 2005 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the complexity of algebraic numbers I. Expansions in integer bases Boris ADAMCZEWSKI (Lyon) & Yann BUGEAUD (Strasbourg) Abstract. Let b 2 be an integer. We prove that the b-adic expan- sion of every irrational≥ algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcen- dental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial tran- scendence criterion. 1. Introduction Let b 2 be an integer. The b-adic expansion of every rational number is eventually periodic, but≥ what can be said on the b-adic expansion of an irrational algebraic number? This question was addressed for the first time by Emile´ Borel [11], who made the conjecture that such an expansion should satisfies the same laws as do almost all real numbers.
    [Show full text]
  • An Arithmetical Property of Powers of Salem Numbers ✩
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Number Theory 120 (2006) 179–191 www.elsevier.com/locate/jnt An arithmetical property of powers of Salem numbers ✩ Toufik Zaïmi Department of Mathematics, College of Sciences, PO Box 2455, King Saud University, Riyadh 11451, Saudi Arabia Received 11 November 2004; revised 4 September 2005 Available online 4 January 2006 Communicated by David Goss Abstract Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζαn,whenn runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α − 1 is a unit. © 2005 Elsevier Inc. All rights reserved. MSC: 11J71; 11R04; 11R06 Keywords: Salem numbers; Fractional parts 1. Introduction The problem of studying the distribution mod 1 of the powers of a fixed real number α greater than 1, has been of interest for some time. In his monograph [4], R. Salem considered the case of certain special real numbers α. For instance, he showed that if α is a Pisot number then αn mod 1 tends to zero, whereas if α is a Salem number then the sequence αn mod 1 is dense in the unit interval. Recall that a Pisot (respectively a Salem) number is a real algebraic integer greater than 1 whose other conjugates are of modulus less than 1 (respectively are of modulus at most 1 and with a conjugate of modulus 1).
    [Show full text]