Ito-Sadahiro numbers vs. Parry numbers

We consider positional numeration system with negative base, as introduced by Ito and Sadahiro. In particular, we focus on algebraic properties of negative bases $-\beta$ for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the $(-\beta)$-expansion of $-\frac{\beta}{\beta+1}$ is eventually periodic. We call such numbers $\beta$ Ito-Sadahiro numbers and we compare their properties with Parry numbers, occurring in the same context for R\'enyi positive base numeration system.

Directly from the definition of the transformation T we can derive that the 'digits' x i take values in the set {0, 1, 2, · · · , ⌈β⌉ − 1} for i = 1, 2, 3, · · · . The expression of x in the form (1) is called the β-expansion of x. The number x is thus represented by the infinite word d β (x) = x 1 x 2 x 3 · · · ∈ A N over the alphabet A = {0, 1, 2, . . . , ⌈β⌉ − 1}. From the definition of the transformation β we can derive another important property, namely that the ordering on real numbers is carried over to the ordering of β-expansions. In particular, we have for x, y ∈ [0, 1) that where is the lexicographical order on A N , (ordering on the alphabet A is usual, 0 < 1 < 2 < · · · < ⌈β⌉ − 1). In [10], Parry has provided a criteria which decides whether an infinite word in A N is or not a β-expansion of some real number x. The criteria is formulated using the so-called infinite expansion of 1, denoted by d * β (1), defined as a limit in the space A N equipped with the product topology, by According to Parry, the string x 1 x 2 x 3 · · · ∈ A N represents the β-expansion of a number x ∈ [0, 1) if and only if The condition (2) ensures that the set The notion of β-expansion can be naturally extended to all non-negative real numbers: The expression of a real number y in the form where k ∈ Z and y k y k−1 y k−2 · · · ∈ D β , is called the β-expansion of y.
Real numbers y having in their β-expansion vanishing digits y i for all i < 0 are usually called β-integers and the set of β-integers is denoted by Z β . The notion of β-integers was first considered in [3] as an aperiodic structure modeling non-crystallographic materials with long range order, called quasicrystals.
The choice of the base β > 1 strongly influences the properties of β-expansions. It turns out that important role among bases play such numbers β for which d * β (1) is eventually periodic. Parry, himself, has called these bases beta-numbers; now these numbers are commonly called Parry numbers. One can demonstrate the exceptional properties of Parry numbers on two facts: -The subshift S β is sofic if and only if β is a Parry number [6].
-Distances between consecutive β-integers take finitely many values if and only if β is a Parry number [14].
Recently, Ito and Sadahiro [5] suggested to study positional systems with negative base −β, where β > 1. Representation of real numbers in such a system is defined using the transformation T : Every real x ∈ I β := [l β , r β ) can be written as The above expression is called the (−β)-expansion of x. It can also be written as the infinite word d −β (x) = x 1 x 2 x 3 · · · . One can easily show from (4) that the digits x i , i ≥ 1, take values in the set A = {0, 1, 2, . . . , ⌊β⌋}. In this case, the ordering on the set of infinite words over the alphabet A which would correspond to the ordering of real numbers is the so-called alternate ordering: We say that x 1 x 2 x 3 · · · ≺ alt y 1 y 2 y 3 · · · if for the minimal index j such that x j = y j it holds that x j (−1) j < y j (−1) j . In this notation, we can write for arbitrary x, y ∈ I β that . In their paper, Ito and Sadahiro has provided a criteria to decide whether an infinite word A N belongs to the set of (−β)-expansions, i.e. to the set D −β = {d −β (x) | x ∈ I β }. This time, the criteria is given in terms of two infinite words, namely These two infinite words have close relation: ω . (As usual, the notation w ω stands for infinite repetition of the string w.) In all other cases one has d * −β (r β ) = 0d −β (l β ). Ito and Sadahiro have shown that an infinite word The above condition ensures that the set D −β of infinite words representing (−β)-expansions is shift invariant. In [5] it is shown that the closure of D −β defines a sofic system if and only if d −β (l β ) is eventually periodic. In analogy to the definition of Parry numbers, we suggest that numbers β > 1 such that d −β (l β ) is eventually periodic be called Ito-Sadahiro numbers. The relation of the set of Ito-Sadahiro numbers and the set of Parry numbers is not obvious. We do not know any example of a Parry number which is not an Ito-Sadahiro number or vice-versa. Bassino [2] has shown that quadratic and cubic not totally real numbers β are Parry numbers if and only if they are Pisot numbers. For the same class of numbers, we prove in [9] that β is Ito-Sadahiro if and only if it is Pisot. This means that notions of Parry numbers and Ito-Sadahiro numbers on the mentioned type of irrationals coincide.
In this paper we study numbers with eventually periodic (−β)-expansion. Statements which we show, as well as results of other authors we recall, demonstrate similarities between the behaviour of β-expansions and (−β)-expansions. We mention also phenomena in which the two essentially differ. Nevertheless, we are in favour of the hypothesis that the set of Parry numbers and the set of Ito-Sadahiro numbers coincide.

Preliminaries
Let us first recall some number theoretical notions. A complex number β is called an algebraic number, if it is root of a monic polynomial x n + a n−1 x n−1 + · · ·+a 1 x+a 0 , with rational coefficients a 0 , . . . , a n−1 ∈ Q. Monic polynomial with rational coefficients and root β of the minimal degree among all polynomials with the same properties is called the minimal polynomial of β, its degree is called the degree of β. The roots of the minimal polynomial are algebraic conjugates.
If the minimal polynomial of β has integer coefficients, β is called an algebraic integer. An algebraic integer β > 1 is called a Perron number, if all its conjugates are in modulus strictly smaller than β. An algebraic integer β > 1 is called a Pisot number, if all its conjugates are in modulus strictly smaller than 1. An algebraic integer β > 1 is called a Salem number, if all its conjugates are in modulus smaller than or equal to 1 and β is not a Pisot number.
If β is an algebraic number of degree n, then the minimal subfield of the field of complex numbers containing β is denoted by Q(β) and is of the form If γ is a conjugate of an algebraic number β, then the fields Q(β) and Q(γ) are isomorphic. The corresponding isomorphism is given by the prescription In particular, it means that β is a root of some polynomial f with rational coefficients if and only if γ is a root of the same polynomial f .

Ito-Sadahiro polynomial
From now on, we shall consider for bases of the numeration system only Ito-Sadahiro numbers, i.e. numbers β such that Without loss of generality we shall assume that m ≥ 0, p ≥ 1 are minimal values so that d −β (l β ) can be written in the above form. Recall that l β = − β β+1 . Therefore (7) can be rewritten as Multiplying by (−β) m (−β) p − 1 , we obtain the following lemma.
Lemma 1 Let β be an Ito-Sadahiro number and let d −β (l β ) be of the form (7). Then β is a root of the polynomial Corollary 2 An Ito-Sadahiro number is an algebraic integer of degree smaller than or equal to m + p, where m, p are given by (7).
It is useful to mention that the Ito-Sadahiro polynomial is not necessarily irreducible over Q. As an example one can take the minimal Pisot number. For such β, we have d −β (l β ) = 1001 ω , and thus the Ito-Sadahiro polynomial is equal to Remark 3 Note that for p = 1 and d m+1 = 0, we have d −β (l β ) = d 1 · · · d m 0 ω , and the Ito-Sadahiro polynomial of β is of the form and thus β is an algebraic integer of degree at most m + 1.
Proof. Since β is a root of the Ito-Sadahiro polynomial P , there must exist a polynomial Q such that P (x) = (x − β)Q(x). Let us first determine Q and show that it is a monic polynomial with coefficients in modulus not exceeding 1. The coefficients d i in the polynomial P in the form (8) are the digits of the (−β)-expansion of l β , and thus, by (5), they satisfy For simplicity of notation in this proof, denote T i = T i (l β ), for i = 0, 1, . . . , m+p.
First realize that T m − T m+p = 0, since d −β (l β ) is eventually periodic with preperiod of length m and period of length p. As T 0 = T 0 (l β ) = − β β+1 , we can derive that Putting back to (10), we obtain that the desired polynomial Q defined by which can be rewritten in another form, namely, Note that the coefficients at individual powers of −x are of two types, namely In order to complete the proof, realize that every root γ, γ = β, of the polynomial P satisfies Q(γ) = 0. We thus have and hence From this, one easily derives that |γ| < 2.

Periodic expansions in the Ito-Sadahiro system
Representations of numbers in the numeration system with negative base from the point of view of dynamical systems has been studied by Frougny and Lai [7]. They have shown the following statement.

Theorem 6
If β is a Pisot number, then d −β (x) is eventually periodic for any x ∈ I β ∩ Q(β).
In particular, their result implies that every Pisot number is an Ito-Sadahiro number. Here, we show a 'reversed' statement.
Proof. First realize that since l −β ∈ Q(β), by assumption, d −β (l β ) is eventually periodic, and thus β is an Ito-Sadahiro number. Therefore using Corollary 2, β is an algebraic integer. It remains to show that all conjugates of β are in modulus smaller than or equal to 1.
Consider a real number x whose (−β)-expansion is of the form d −β (x) = x 1 x 2 x 3 · · · . We now show that . (12) In order to see this, we estimate the series is a (−β)-expansion of some y ∈ I β . Therefore we can write .
As β > 1, there exists L ∈ N such that Let M ∈ N satisfy M > 2L + 1. Choose a rational number r such that According to the auxiliary statement (12), the (−β)-expansion of r must be of the form As r is rational, by assumption, the infinite word r M+1 r M+2 · · · is eventually periodic and by summing a geometric series, where n is the degree of β.
In order to prove the theorem by contradiction, assume that a conjugate γ = β is in modulus greater than 1. By application of the isomorphism between Q(β) and Q(γ), we get Subtracting (15) from (14), we obtain where η = max{|β| −1 , |γ| −1 } < 1. Obviously, for any M > 2L + 1, we can find a rational r satisfying (13) and thus derive the inequality (16). However, the left-hand side of (16) is a fixed positive number, whereas the right-hand side decreases to zero with increasing M , which is a contradiction.
In order to stress the analogy of the Ito-Sadahiro numeration system with Rényi β-expansions of numbers, recall that already Schmidt in [12] has shown that for a Pisot number β, any x ∈ [0, 1) ∩ Q(β) has an eventually periodic βexpansion and also the converse, that every x ∈ [0, 1)∩Q(β) having an eventually periodic β-expansion forces β to be either Pisot or Salem number. In fact, the proof of Theorem 6 given by Frougny and Lai, as well as our proof of Theorem 7 are using the ideas presented in [12].
A special case of numbers with periodic (−β)-expansion is given by those numbers x for which the infinite word d −β (x) has suffix 0 ω . We then say that the expansion d −β (x) is finite. An example of such a number is x = 0 with (−β)expansion d −β (x) = 0 ω . As it is shown in [9], if β < 1 2 (1 + √ 5), then x = 0 is the only number with finite (−β)-expansion. This property of the Ito-Sadahiro numeration system has no analogue in Rényi β-expansions; for positive base, the set of finite β-expansions is always dense in [0, 1).
Just as in the numeration system with positive base, we can extend the definition of (−β)-expansions of x to all real numbers x, and define the notion of a (−β)-integer as a real number y such that where y k · · · y 1 y 0 0 ω is the (−β)-expansion of some number in I β . The set of (−β)-integers is denote by Z −β . In this notation, we can write for the set of all numbers with finite (−β)-expansions It is not surprising that arithmetical properties of β-expansions and (−β)-expansions depend on the choice of the base β. It can be shown that both Z β and Z −β is closed under addition and multiplication if and only if β ∈ N. On the other hand, Fin(β) and Fin(−β) can have a ring structure even if β is not an integer. Frougny and Solomyak [8] have shown that if Fin(β) is a ring, then β is a Pisot number. Similar result is given in [9] for negative base: Fin(−β) being a ring implies that β is either a Pisot or a Salem number. In [9] we also prove the conjecture of Ito and Sadahiro that in case of quadratic Pisot base β the set Fin(−β) is a ring if and only if the conjugate of β is negative.

Comments and open questions
-Every Pisot number is a Parry number and every Parry number is a Perron number, and neither of the statements can be reversed. The former is a consequence of the mentioned result of Schmidt, the latter statement follows for example from the fact that every Perron number has an associated canonical substitution ϕ β , see [4]. The substitution is primitive, and its incidence matrix has β as its eigenvalue. The fixed point of ϕ β is an infinite word which codes the sequence of distances between consecutive β-integers.
-For the negative base numberation system, we can derive from Theorem 6 that every Pisot number is an Ito-Sadahiro number. From Corollary 5 we know that an Ito-Sadahiro number β ≥ 2 is a Perron number. Based on our investigation, we conjecture that for any Ito-Sadahiro number β ≥ 1 2 (1+ √ 5), the sequence of distances between consecutive (−β)-integers can be coded by a fixed point of a 'canonical' substitution which is primitive and its incidence matrix has β 2 for its dominant eigenvalue. Thus we expect that every Ito-Sadahiro number β ≥ 1 2 (1 + √ 5) is also a Perron number. In case that β < 1 2 (1 + √ 5), we have Z −β = {0} and so the situation is not at all obvious.
-In [13], Solomyak has explicitely described the set of conjugates of all Parry numbers. In particular, he has shown that this set is included in the complex disc of radius 1 2 (1 + √ 5), and that this radius cannot be diminished. For his proof it was important that all conjugates of a Parry number are roots of a polynomial with real coefficients in the interval [0, 1). In the proof of Theorem 4 we show that conjugates of an Ito-Sadahiro number are roots of a polynomial (11) with coefficients in [−1, 1]. From this, we derive that conjugates of Ito-Sadahiro numbers lie in the complex disc of radius ≤ 2. We do not know whether this value can be diminished.