Acta Polytechnica Vol. 51 No. 4/2011

Ito-Sadahiro numbers vs. Parry numbers

Z. Mas´akov´a, E. Pelantov´a

Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system.

Keywords: numeration systems, negative base, Pisot number, Parry number.

∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,.... The expres- sion of x in the form (1) is called the β-expansion of k k−1 k−2 x.Thenumberx is thus represented by the infinite y = ykβ + yk−1β + yk−2β + ... , (3) N ∈ Z ∈D word dβ(x)=x1x2x3 ... ∈A over the alphabet where k and ykyk−1yk−2 ... β, A = {0, 1, 2,...,%β&−1}. From the definition of the transformation T we is called the β-expansion of y. can derive another important property, namely that Real numbers y having in the β-expansion of |y| the ordering on real numbers is carried over to the vanishing digits yi for all i<0 are usually called β- ordering of β-expansions. In particular, we have for integers, and the set of β-integers is denoted by Zβ. x, y ∈ [0, 1) that The notion of β-integers was first considered in [3] as an aperiodic structure modeling non-crystallographic x ≤ y ⇐⇒ d (x) ) d (y) , β β materials with long range order, called quasicrystals. N Numbers y with finitely many non-zero digits in the where ) is the lexicographical order on A ,(order- β-expansion of |y| form the set denoted by Fin(β). ing on the alphabet A is usual, 0 < 1 < 2 < ... < The choice of the base β>1 strongly influences %β&−1). the properties of β-expansions. It turns out that an In [11], Parry has provided a criterion which de- N important role among bases is played by such num- cides whether an infinite word in A is or not a β- ∗ bers β for which dβ(1) is eventually periodic. Parry expansion of some real number x. The criterion is himself called these bases beta-numbers; now these formulated using the so-called infinite expansion of ∗ numbers are commonly called Parry numbers. We 1, denoted by dβ(1), defined as a limit in the space can demonstrate the exceptional properties of Parry N A equipped with the product topology, by numbers on two facts: • The subshift S is sofic if and only if β is a Parry ∗ − β dβ(1) := lim dβ(1 ε) . ε→0+ number [6].

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• Distances between consecutive β-integers take defines a sofic system if and only if d−β(lβ)iseven- finitely many values if and only if β is a Parry tually periodic. number [15]. By analogy with the definition of Parry numbers, Recently, Ito and Sadahiro [5] suggested a study we suggest that numbers β>1 such that d−β(lβ)is of positional numeration systems with a negative eventually periodic be called Ito-Sadahiro numbers. base −β,whereβ>1. The representation of real The relation of the set of Ito-Sadahiro numbers and numbers in such a system is defined using the trans- the set of Parry numbers is not obvious. Bassino [2] β has shown that quadratic numbers, as well as cubic formation T :[lβ,rβ ) → [lβ,rβ), where lβ = − , β +1 numbers which are not totally real, are Parry if and 1 only if they are Pisot. For the same class of num- rβ =1+lβ = , 1+β bers, we prove in [10] that β is Ito-Sadahiro if and only if it is Pisot. This means that notions of Parry − −#− − $ T (x):= βx βx lβ . (4) numbers and Ito-Sadahiro numbers on the mentioned type of irrationals do not differ. This would support Every real x ∈ I := [l ,r ) can be written as β β β the hypothesis stated in the first version of this pa- ∞ per, namely that the set of Parry numbers and the x x = i , (5) set of Ito-Sadahiro numbers coincide. However, dur- (−β)i i=1 ing the refereeing process Liao and Steiner [9] found an example of a Parry number which is not an Ito- #− i−1 − $ where xi = βT (x) lβ for i =1, 2, 3,... Sadahiro number, and vice-versa. The above expression is called the (−β)- The main results of this paper are formulated as expansion of x. It can also be written as the infi- Theorems 4 and 7. Theorem 4 gives a bound on the nite word d−β(x)=x1x2x3 ... We can easily show modulus of conjugates of Ito-Sadahiro numbers; The- ≥ from (4) that the digits xi, i 1, take values in the orem 7 shows that periodicity of (−β)-expansion of set A = {0, 1, 2,...,#β$}. In this case, the order- all numbers in the field Q(β)requiresβ to be a Pisot ing on the set of infinite words over the alphabet A or . Statements which we prove, as well which would correspond to the ordering of real num- as results of other authors that we recall, demonstrate bers is the so-called alternate ordering: We say that similarities between the behaviour of β-expansions ≺ x1x2x3 ... alt y1y2y3 ... if for the minimal index j and (−β)-expansions. We mention also phenomena  − j − j such that xj = yj it holds that xj ( 1)

≤ ⇐⇒ ) 2 Preliminaries x y d−β(x) alt d−β(y) . Letusfirstrecallsomenumber theoretical notions. A In their paper, Ito and Sadahiro have provided a N complex number β is called an , if it A n n−1 criterion to decide whether an infinite word be- is a root of a monic polynomial x +a − x +...+ − n 1 longs to the set of ( β)-expansions, i.e. to the set a x + a , with rational coefficients a ,...,a − ∈ Q. D { | ∈ } 1 0 0 n 1 −β = d−β(x) x Iβ . This time, the criterion is A monic polynomial with rational coefficients and given in terms of two infinite words, namely root β of the minimal degree among all polynomials ∗ − with the same properties is called the minimal poly- d−β(lβ)andd−β(rβ) := lim d−β(rβ ε) . ε→0+ nomial of β, and its degree is called the degree of β. The roots of the minimal polynomial are algebraic These two infinite words have a close relation: If conjugates. d−β(lβ) is purely periodic with odd period length, i.e. If the minimal polynomial of β has integer coeffi- ω ∗ d−β(lβ)=(d1d2 ...d2k+1) ,thenwehaved−β(rβ )= cients, β is called an . An algebraic ω ω 0d1d2 ...(d2k+1 − 1) . (As usual, the notation w integer β>1 is called a Perron number, if all its con- stands for infinite repetition of the string w.) In all jugates are in modulus strictly smaller than β.An ∗ other cases we have d−β(rβ )=0d−β(lβ). algebraic integer β>1 is called a Pisot number, if all Ito and Sadahiro have shown that an infinite its conjugates are in modulus strictly smaller than 1. word x1x2x3 ...represents a (−β)-expansion of some An algebraic integer β>1 is called a Salem number, x ∈ [lβ,rβ) if and only if for every i ≥ 1 it holds that if all its conjugates are in modulus smaller than or equal to 1 and β is not a Pisot number. ) ≺ ∗ d−β(lβ) alt xixi+1xi+2 ... alt d−β(rβ) . (6) If β is an algebraic number of degree n, then the minimal subfield of the field of complex numbers con- D The above condition ensures that the set −β of in- taining β is denoted by Q(β) and is of the form finite words representing (−β)-expansions is shift in- n−1 variant. In [5] it is shown that the closure of D−β Q(β)={c0 + c1β + ...+ cn−1β | ci ∈ Q} .

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If γ is a conjugate of an algebraic number β,then Remark 3 Note that for p =1and dm+1 =0,we ω the fields Q(β)andQ(γ) are isomorphic. The corre- have d−β(lβ)=d1 ...dm0 , and the Ito-Sadahiro sponding isomorphism is given by polynomial of β is of the form

n−1 → n−1 m+1 m m−1 c0 +c1β+...+cn−1β c0 +c1γ +...+cn−1γ . P (x)=(−x) + d1(−x) +(d2 − d1)(−x) +

In particular, this means that β is a root of some ...+(dm − dm−1)(−x) − dm , (9) polynomial f with rational coefficients if and only if and thus β is an algebraic integer of degree at most γ is a root of the same polynomial f. m +1. 3 Ito-Sadahiro polynomial Theorem 4 Let β be an Ito-Sadahiro number. All roots γ, γ = β, of the Ito-Sadahiro polynomial (in From now on, we shall consider for bases of the particular all conjugates of β) satisfy |γ| < 2. numeration system only Ito-Sadahiro numbers, i.e. numbers β such that Proof. Since β is a root of its Ito-Sadahiro poly- ω nomial P , there must exist a polynomial Q such that d−β(lβ)=d1 ...dm(dm+1 ...dm+p) . (7) P (x)=(x − β)Q(x).LetusfirstdetermineQ and Without loss of generality we shall assume that m ≥ show that it is a monic polynomial with coefficients 0, p ≥ 1 are minimal values so that d−β(lβ)canbe in modulus not exceeding 1. The coefficients di in the polynomial P in the form (8) are the digits of the − β written in the above form. Recall that lβ = . − β +1 ( β)-expansion of lβ, and thus, by (5), they satisfy i−1 Therefore (7) can be rewritten as di = #−βT (lβ) − lβ$. Relation (4) then implies T i(l )=−βT i−1(l )−#−βT i−1(l )−l $,wherefrom β d d β β β β − = 1 + ...+ m + we have − − m β +1 β ( β) d = −T i(l ) − βT i−1(l ) . ∞ i β β dm+1 dm+p 1 + ...+ , For simplicity of notation in this proof, denote Ti = (−β)m+1 (−β)m+p (−β)pi i i=0 T (lβ), for i =0, 1,...,m + p. Substituting di = − − and after arrangement Ti βTi−1 into (8), we obtain −β d d (−β)p p−1 0= + 1 + ...+ m + · − m+1 − i − p − · −β − 1 −β (−β)m (−β)p − 1 P (x)=( x) ( x) + ( x) 1 i=0 d d m+1 + ...+ m+p . m − m+1 − m+p m−i ( β) ( β) (−Ti − βTi−1)(−x) + i=1 Multiplying by (−β)m (−β)p −1 , we obtain the fol- m+p lowing lemma. m+p−i (−T − βT − )(−x) = Lemma 1 Let β be an Ito-Sadahiro number and let i i 1 i=m+1 d− (l ) be of the form (7). Then β is a root of the β β p−1 polynomial (−x)m+1 (−x)i + (−x)p − 1 (x − β) · − p1 i=0 P (x)=(−x)m+1 (−x)i + (−x)p − 1 · (8) m m−i i=0 Ti−1(−x) + (10) m m+p i=2 m−i m+p−i p di(−x) + di(−x) . p−i i=1 i=m+1 (x − β) Tm+i−1(−x) − i=1 Such a polynomial is called the Ito-Sadahiro polyno- − p − − m−1 − mial of β. ( x) 1 βT0( x) + Tm Tm+p . − Corollary 2 An Ito-Sadahiro number is an alge- First realize that Tm Tm+p =0,sinced−β(lβ)is braic integer of degree smaller than or equal to m+p, eventually periodic with a preperiod of length m and β where m, p are given by (7). a period of length p.AsT = T 0(l )=− ,we 0 β β +1 It is useful to mention that the Ito-Sadahiro poly- can derive that nomial is not necessarily irreducible over Q.As p−1 an example one can take the minimal Pisot num- − m+1 − i − − p − − m−1 ω ( x) ( x) ( x) 1 βT0( x) = − ber. For such β,wehaved β(lβ) = 1 001 ,and i=0 thus the Ito-Sadahiro polynomial is equal to P (x)= p−1 x4 −x3 −x2 +1 = (x−1)(x3 −x−1), where x3 −x−1 m−1 i (−x) (x − β)(x − T0) (−x) . is the minimal polynomial of β. i=0

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Putting back to (10), we obtain that the desired poly- 4 Periodic expansions in the nomial Q defined by P (x)=(x − β)Q(x)isofthe form Ito-Sadahiro system

p−1 Representations of numbers in the numeration sys- Q(x)=(−x)m−1(x − T ) (−x)i + 0 tem with a negative base from the point of view of i=0 dynamical systems have been studied by Frougny and m p m−i Lai [7]. They have shown the following statement. (−x) − 1 Ti−1(−x) + − i=2 Theorem 6 If β is a Pisot number, then d β(x) is ∈ ∩ Q p eventually periodic for any x Iβ (β). p−i T − (−x) , m+i 1 In particular, their result implies that every Pisot i=1 number is an Ito-Sadahiro number. Here, we show a which can be rewritten in another form, namely, ‘reversed’ statement.

− Theorem 7 If any x ∈ I ∩ Q(β) has eventually pe- Q(x)=−(−x)m+p 1 + β riodic (−β)-expansion, then β is either a Pisot num- m+p−2 i ber or a Salem number. (Tm+p−1−i − T0 − 1)(−x) + (11) i=m Proof. First realize that since l−β ∈ Q(β), by as- m−1 sumption, d−β(lβ) is eventually periodic, and thus β − − i (Tm+p−1−i Tm−1−i)( x) . is an Ito-Sadahiro number. Therefore, using Corol- i=0 lary 2, β is an algebraic integer. It remains to show that all conjugates of β are in modulus smaller than Note that the coefficients at individual powers of −x or equal to 1. are of two types, namely Consider a real number x whose (−β)-expansion is of the form d−β(x)=x1x2x3 ...We now show that Tm+p−1−i − T0 − 1 ∈ [−1, 0) ,

x1 = x2 = ...= xk−1 =0 and xk =0 and T − − − T − − ∈ (−1, 1) . implies m+p 1 i m 1 i 1 |x|≥ . (12) In order to complete the proof, realize that every root βk(β +1)  γ, γ = β, of the polynomial P satisfies Q(γ)=0.We In order to see this, we estimate the series thus have ' ∞ ' ' ∞ ' ' xk xk+i ' 1 1 ' xk+i ' m+p−2 | | ' ' ≥ − ' ' x = k + k+i k k i . m+p−1 i (−β) (−β) β β (−β) (−γ) = (Tm+p−1−i − T0 − 1)(−γ) + i=1 i=1 i=m Since the set D−β of all (−β)-expansions is shift in- m−1 ∞ − − i xk+i (Tm+p−1−i Tm−1−i)( γ) , variant, the sum is a (−β)-expansion of (−β)i i=0 i=1 some y ∈ Iβ. Therefore we can write and hence 1 1 1 1 β 1 m+p−2 |x|≥ − |y|≥ − = . βk βk βk βk β +1 βk(β +1) |γ|m+p−1 ≤ |γ|i = i=0 As β>1, there exists L ∈ N such that | |m+p−1 − | |m+p−1 γ 1 γ β 1 | |− < | |− . − < . γ 1 γ 1 β +1 (−β)2L+1 From this, we easily derive that |γ| < 2. Let M ∈ N satisfy M>2L + 1. Choose a rational As a consequence, we can easily deduce the rela- number r such that tion between Ito-Sadahiro numbers greater or equal 1 1 1

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As r is rational, by assumption, the infinite word Just as in the numeration system with a positive rM+1rM+2 ... is eventually periodic and by sum- base, we can extend the definition of (−β)-expansions ∞ r of x to all real numbers x, and define the notion of a ming a geometric series, the sum i can (−β)i (−β)-integer as a real number y such that i=M+1 be rewritten as k y = yk(−β) + ...+ y1(−β)+y0 , ∞ ri n−1 = c + c β + ...+ c − β ∈ Q(β) , where y ...y y 0ω is the (−β)-expansion of some (−β)i 0 1 n 1 k 1 0 i=M+1 number in Iβ.Thesetof(−β)-integers is denoted by Z− . With this notation, we can write the set of where n isthedegreeofβ. β all numbers with finite (−β)-expansions as In order to prove the theorem by contradiction, assume that a conjugate γ = β is in modulus greater ∞ 1 than 1. By application of the isomorphism between Fin(−β)= Z− . (−β)k β Q(β)andQ(γ), we get k=0 ∞ It is not surprising that the arithmetical properties n−1 ri c + c γ + ...+ c − γ = , − 0 1 n 1 (−γ)i of β-expansions and ( β)-expansions depend on the i=M+1 choice of the base β. It can be shown that both Zβ and Z− is closed under addition and multiplication and thus β if and only if β ∈ N. On the other hand, Fin(β)and ∞ 1 r Fin(−β)canhavearingstructureevenifβ is not r = + i . (15) (−γ)2L+1 (−γ)i an integer. Frougny and Solomyak [8] have shown i=M+1 that if Fin(β) is a ring, then β is a Pisot number. A similar result is given in [10] for a negative base: Subtracting (15) from (14), we obtain Fin(−β) being a ring implies that β is either a Pisot ' ' ' 1 1 ' number or a Salem number. In [10] we also prove 0 < ' − ' ≤ (16) (−β)2L+1 (−γ)2L+1 the conjecture of Ito and Sadahiro that in the case ∞ of quadratic Pisot base β the set Fin(−β)isaringif ' ' ηM+1 r '(−β)−i − (−γ)−i' ≤ 2#β$ , and only if the conjugate of β is negative. i 1 − η i=M+1 where η =max{|β|−1, |γ|−1} < 1. Obviously, for any M>2L + 1, we can find a rational r satisfy- 5 Comments and open ing (13) and thus derive the inequality (16). However, questions the left-hand side of (16) is a fixed positive number, whereas the right-hand side decreases to zero with • Every Pisot number is a Parry number and every increasing M, which is a contradiction. Parry number is a Perron number, and neither In order to stress the analogy of the Ito-Sadahiro of these statements can be reversed. The for- numeration system with R´enyi β-expansions of num- mer is a consequence of the mentioned result of bers, recall that already Schmidt in [13] has shown Schmidt, the latter statement follows for exam- that for a Pisot number β,anyx ∈ [0, 1) ∩ Q(β) ple from the fact that every Perron number has has an eventually periodic β-expansion and also, con- an associated canonical substitution ϕβ, see [4]. versely, that every x ∈ [0, 1) ∩ Q(β)havinganeven- The substitution is primitive, and its incidence tually periodic β-expansion force β is either a Pisot has β as its eigenvalue. The fixed point of number or a Salem number. In fact, the proof of The- ϕβ is an infinite word which codes the sequence orem 6 given by Frougny and Lai, as well as our proof of distances between consecutive β-integers. of Theorem 7 are using the ideas presented in [13]. • For the negative base numeration system, we can A special case of numbers with periodic (−β)- derive from Theorem 6 that every Pisot number expansion is given by those numbers x for which the is an Ito-Sadahiro number. From Corollary 5 ω infinite word d−β(x)hassuffix0 .Wethensaythat we know that an Ito-Sadahiro number β ≥ 2is the expansion d−β(x) is finite. An example of such a a Perron number. Based on our investigation, ω number is x =0with(−β)-expansion d−β(x)=0 . we conjecture that for any Ito-Sadahiro number 1 √ 1 √ As is shown in [10], if β< (1 + 5), then x =0 β ≥ (1 + 5), the sequence of distances be- 2 2 is the only number with finite (−β)-expansion. This tween consecutive (−β)-integers can be coded by property of the Ito-Sadahiro numeration system has a fixed point of a ‘canonical’ substitution which no analogue in R´enyi β-expansions; for positive base, is primitive and its incidence matrix has β2 for the set of finite β-expansions is always dense in [0, 1). its dominant eigenvalue. Thus we expect that

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√ 1 − every Ito-Sadahiro number β ≥ (1 + 5) is [5] Ito, S., Sadahiro, T.: ( β)-expansions of real 2 numbers, Integers 9 (2009), 239–259. also a Perron number. In the case that β< 1 √ (1 + 5), we have Z− = {0} and so the situ- [6] Ito, S., Takahashi, Y.: Markov subshifts and re- 2 β ation is not at all obvious. alization of β-expansions, J. Math. Soc. Japan • In [14], Solomyak has explicitly described the set 26 (1974), 33–55. of conjugates of all Parry numbers. In particu- [7] Frougny, Ch., Lai, A. C.: Negative bases and lar, he has shown that this set is included in the automata, Discr. Math. Theor. Comp. Sci. 13, 1 √ complex disc of radius (1 + 5), and that this No 1 (2011), 75–94. 2 radius cannot be diminished. For his proof it was [8] Frougny, Ch., Solomyak, B.: Finite β-expan- important that all conjugates of a Parry number sions, Ergodic Theory Dynamical Systems 12 are roots of a polynomial with real coefficients in (1994), 713–723. the interval [0, 1). In the proof of Theorem 4 we show that conjugates of an Ito-Sadahiro number [9] Liao, L., Steiner, W.: Dynamical properties of are roots of a polynomial (11) with coefficients the negative beta transformation, preprint 2011, in [−1, 1]. From this, we derive that conjugates 18pp. http://arxiv.org/abs/1101.2366 of Ito-Sadahiro numbers lie in the complex disc of radius ≤ 2. We do not know whether this [10] Mas´akov´a, Z., Pelantov´a, E., V´avra, T.: Arith- value can be diminished. metics in number systems with negative base, Theor. Comp. Sci. 412 (2011), 835–845.

[11] Parry, W.: On the β-expansions of real num- Acknowledgement bers, Acta Math. Acad. Sci. Hung. 11 (1960), We acknowledge financial support from Czech Sci- 401–416. ence Foundation grant 201/09/0584 and from grants [12] R´enyi, A.: Representations for real numbers and MSM6840770039 and LC06002 of the Ministry of Ed- their ergodic properties, Acta Math. Acad. Sci. ucation, Youth, and Sports of the Czech Republic. Hung. 8 (1957), 477–493. References [13] Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers, Bull. London [1] Ambroˇz,P.,Dombek,D.,Mas´akov´a, Z., Pelan- Math. Soc. 12 (1980), 269–278. tov´a, E.: Numbers with integer expansion in the [14] Solomyak, B.: Conjugates of beta-numbers and numeration system with negative base,preprint the zero-free domain foraclassofanalytic 2009, 13pp. http://arxiv.org/abs/0912.4597 functions, Proc. London Math. Soc. 68 (1994), [2] Bassino, F.: β-expansions for cubic Pisot num- 477–498. bers, 5th Latin American Theoretical INformat- [15] Thurston, W. P.: Groups, tilings, and fi- ics Symposium (LATIN’02), 2286 LNCS. Can- nite state automata, AMS Colloquium Lecture cun, Mexico. April, 2002. pp. 141–152. Springer- Notes, American Mathematical Society, Boul- Verlag. der, 1989. [3] Burd´ık, C.,ˇ Frougny, Ch., Gazeau, J. P., Krej- car, R.: Beta-Integers as Natural Counting Sys- Zuzana Mas´akov´a tems for Quasicrystals, J. Phys. A: Math. Gen. E-mail: zuzana.masakova@fjfi.cvut.cz 31 (1998), 6 449–6 472. Edita Pelantov´a E-mail: edita.pelantova@fjfi.cvut.cz [4] Fabre, S.: Substitutions et β-syst`emes de Department of Mathematics FNSPE num´eration, Theoret. Comput. Sci. 137 (1995), Czech Technical University in Prague 219–236. Trojanova 13, 120 00 Praha 2, Czech Republic

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