# A NON-TRIVIAL MINORATION FOR THE SET OF SALEM NUMBERS

## JEAN-LOUIS VERGER-GAUGRY

**ABSTRACT.** The set of Salem numbers is proved to be bounded from below by $\theta_{31}^{-1} = 1.08544...$ where $\theta_n, n \ge 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280...$ belongs to the interval $(\theta_{12}^{-1}, \theta_{11}^{-1})$. We conjecture that there is no Salem number in $(\theta_{31}^{-1}, \theta_{12}^{-1}) = (1.08544..., 1.17295...)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the Rényi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1.

**Keywords:** Lehmer conjecture, minoration, Salem number, numeration system, asymptotic expansion, dynamical zeta function, $\beta$-transformation, Parry Upper function, Perron number, Pisot number, Parry number.

2020 Mathematics Subject Classification: 11K16, 11M41, 11R06, 11R09, 30B10, 30B40, 37C30, 37N99, 03D45.

- Dedicated to the 75th birthday of Christiane Frougny -

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Dynamical zeta function of the Rényi-Parry numeration dynamical system</td><td>5</td></tr><tr><td>3. Dynamics of the small Salem numbers known</td><td>9</td></tr><tr><td>4. Proof of the Main Theorem 1.1</td><td>11</td></tr><tr><td>4.1. A non-real lenticular pole of ζ<sub>β</sub>(z) in a small angular sector</td><td>11</td></tr><tr><td>4.2. Rewriting trails</td><td>19</td></tr><tr><td>4.3. Kala-Vavra Theorem and Galois identification</td><td>23</td></tr><tr><td>References</td><td>25</td></tr></table>
# 1. INTRODUCTION

A Salem number is an algebraic integer $\beta > 1$ such that its Galois conjugates $\beta^{(i)}$ satisfy: $|\beta^{(i)}| \le 1$ for all $i = 1, 2, \dots, m-1$, with $m = \deg(\beta) \ge 4$, $\beta^{(0)} = \beta$ and at least one conjugate $\beta^{(i)}$, $i \ne 0$, on the unit circle [13] [15] [16]. All the Galois conjugates of a Salem number $\beta$ lie on the unit circle, by pairs of complex conjugates, except $1/\beta$ which lies in the open interval $(0, 1)$. Salem numbers are of even degree $\ge 4$. The minimal polynomial $P_{\beta}(X) \in \mathbb{Z}[X]$ of $\beta$ is reciprocal. Recall that an integer reciprocal $P(X) \in \mathbb{Z}[X]$ is reciprocal if it is equal to its reciprocal polynomial $P^*(X)$, where $P^*(X)$ is defined by $P^*(X) := X^{\deg(P)}P(1/X)$. Let us denote by $T$ the set of Salem numbers.

**Q.** A long-standing basic question [16] [68] [99] [100] is about the existence of a non-trivial minorant of the set $T$. More precisely, does there exist a constant $c > 0$ such that: $\beta \in T \implies \beta \ge 1+c$? This question asks the more general question of the topology of $T$ and of its adherence $\bar{T}$: in this context, is $1$ a limit point of $T$ or not?

Salem numbers appear in many domains of mathematics, not only in numeration systems or number theory. For studying the localization of Salem numbers in $(1, +\infty)$, the set $T$ has been compared with the set $S$ of Pisot numbers by the “Construction of Salem”, introduced in [92], and also using association equations, and the theory of interlacing roots. Let us recall the definitions. A Perron number is either 1 or a real algebraic integer $\theta > 1$ such that the Galois conjugates $\theta^{(i)}, i \ne 0$, of $\theta^{(0)} := \theta$ satisfy: $|\theta^{(i)}| < \theta$. Denote by $\mathbb{P}$ the set of Perron numbers. A Pisot number is a Perron number $> 1$ for which $|\theta^{(i)}| < 1$ for all $i \ne 0$. The set of Pisot numbers admits the minorant $\Theta = 1.3247\dots$, unique root $> 1$ of $X^3 - X - 1$ by a result of Siegel [97].

Salem (1944) [92] (Samet [94]) proved that every Salem number is the quotient of two Pisot numbers. Apart from this direct result, few relations are known between Salem numbers and Pisot numbers. The set of Pisot numbers is better known than the set of Salem numbers.

To study Salem numbers association equations between Pisot numbers and Salem numbers have been introduced by Boyd ([20] Theorem 4.1), Bertin and Pathiaux-Delefosse ([14], [15] pp 37-46, [16] chapter 6). Association equations are generically written

$$ (1.1) \qquad (X^2 + 1)P_{\text{Salem}} = XP_{\text{Pisot}}(X) + P_{\text{Pisot}}^*(X), $$

to investigate the links between infinite collections of Pisot numbers and a given Salem number, of respective minimal polynomials $P_{\text{Pisot}}$ and $P_{\text{Salem}}$.

The theory of interlacing of roots on the unit circle is a powerful tool for studying classes of polynomials having special geometry of zeroes of modulus one [57] [59] [60] [61] [62], in particular Salem polynomials [68] [69] [70]. In [14] [15] Bertin and Boyd obtained two interlacing theorems, namely Theorem A and Theorem B, turned out to be fruitful with their limit-interlacing versions. McKee and Smyth in [70] obtained new interlacing theorems. Theorem 5.3 in [70] shows that all Pisot numbers are produced by a suitable interlacing condition, supporting the second Conjecture of Boyd, i.e. (1.4). Similarly Theorem 7.3 in [70], using Boyd's association Theorems, shows that all Salem numbers are produced by interlacing and that a classification of Salem numbers can be made.

In [44] is reconsidered the interest of the interlacing Theorems of [14], as potential tools for the study of families of algebraic integers in neighbourhoods of Salem numbers, as
analogues of those of McKee and Smyth. Focusing on Theorem A of [14] association equations are obtained between Salem polynomials (and/or cyclotomic polynomials) and expansive polynomials, generically

$$ (1.2) \qquad (z-1)P_{\text{Salem}}(z) = zP_{\text{expansive}}(z) - P_{\text{expansive}}^{*}(z), $$

which allow to deduce rational $n$-dimensional representations of the neighbourhoods of a Salem number of degree $n$, using the formalism of Stieltjès continued fractions. These representations are tools to study the limit points of sequences of algebraic numbers in the neighbourhood of a given Salem number, the existence of Salem numbers in small neighbourhoods of Salem numbers.

In the same direction association equations between Salem numbers and (generalized) Garsia numbers are obtained by Hare and Panju [45] using the theory of interlacing on the unit circle.

As a counterpart, in number fields, Salem numbers are linked to units: they are given by closed formulas from Stark units in Chinburg [27] [28], exceptional units in Silverman [98]. From [28] they are related to relative regulators of number fields [29] [31] [43].

In some domains negative Salem numbers naturally occur. A negative Salem number is by definition the opposite of a Salem number, a negative Pisot number is by definition the opposite of a Pisot number. Negative Salem numbers occur, e.g. for graphs or integer symmetric matrices in [68] [69] [70], and in other domains, like Alexander polynomials of links of the variable "$-x$", e.g. in a Theorem of Hironaka [50].

The topology of the set $T$ is certainly related to the set $S$ of Pisot numbers by the two Conjectures of Boyd:

$$ (1.3) \qquad \text{Conjecture : } S \cup T \text{ is closed} $$

and that the first derived set of $S \cup T$ satisfies

$$ (1.4) \qquad \text{Conjecture : } S = (S \cup T)^{(1)} $$

Only a part of them is proved. Indeed, Salem [93] proved that the set $S$ of Pisot numbers is closed, and $S \subset \bar{T}$. Its successive derived sets $S^{(i)}$, were extensively studied by Dufresnoy and Pisot [34], and their students (Amara [3], ...), by means of compact families of meromorphic functions, following ideas of Schur. This analytic approach is reported extensively in the book [16]. Conjecture (1.4), if true, would imply that all Salem numbers $< \Theta = 1.3247...$, would be isolated. The smallest Salem number known is Lehmer's number

$$ (1.5) \qquad \tau := 1.17628..., $$

discovered by Lehmer in 1933 [63], with minimal polynomial (named "Lehmer's polynomial"):

$$ (1.6) \qquad X^{10} + X^{9} - X^{7} - X^{6} - X^{5} - X^{4} - X^{3} + X + 1. $$

Lehmer's number was proved to be isolated in [99]. In [63] Lehmer discovered other small Salem numbers. There are now part of the list of Mossinghoff which contains the smallest Mahler measures of algebraic integers [74] known. For degrees up to 180, the list of Mossinghoff [75] (2001), with contributions of Boyd, Flammang, Grandcolas, Lisonek, Poulet, Rhin and Sac-Epée, Smyth, gives primitive, irreducible, noncyclotomic integer polynomials of degree at most 180 and of Mahler measure less than 1.3 [76]; this list is complete for
degrees less than 40, and, for Salem numbers, contains the list of the 47 known smallest Salem numbers, all of degree $\le 44$.

The present note is a contribution of the domain of numeration systems to these studies, with techniques of numeration in variable base. It is the proof that the Conjecture of Lehmer, restricted to Salem numbers, is true, i.e. it is a positive answer to the question Q. For every $n \ge 5$, let $\theta_n$ be the unique root of the trinomial $-1+x+x^n$ in $(0,1)$; its inverse $\theta_n^{-1} > 1$ is a Perron number. The sequence $(\theta_n^{-1})_{n \ge 5}$ is strictly decreasing, and satisfies: $\lim_{n \to \infty} \theta_n^{-1} = 1$. We report the reader to [110] for details on the family of trinomials $(-1+x+x^n)_{n \ge 5}$. In this note we prove the following result.

**Theorem 1.1** (ex-Lehmer conjecture for Salem numbers). *The set T is bounded from below:*

$$ \beta \in T \implies \beta > \theta_{31}^{-1} = 1.08544\dots $$

Lehmer's number 1.17628... belongs to the interval $(\theta_{12}^{-1}, \theta_{11}^{-1})$ (cf Table 1). This interval does not contain any other known Salem number. If there is another one, its degree should be greater than 180. After many attempts (by Denis Dutykh and the author), and a compilation of the literature [111] on small Salem numbers, to find Salem numbers smaller than Lehmer's number, we formulate:

**Conjecture 1.** *There is no Salem number in the interval*

$$ (\theta_{31}^{-1}, \theta_{12}^{-1}) = (1.08544\dots, 1.17295\dots). $$

The proof of Theorem 1.1 (in Section 4) uses the analytic function which is the dynamical zeta function, $\zeta_\beta(z)$, of the $\beta$-transformation, for $\beta > 1$ running over the set of reciprocal algebraic integers. In Section 2 the basic properties of the function $\zeta_\beta(z)$ and the Rényi-Parry numeration dynamical system are recalled. Indeed, if $1 < \beta < \theta_{31}^{-1}$, this function systematically admits a lenticular pole which is non-real, of modulus $< 1$, in an angular sector of the open unit disk containing 1; in Subsection 4.1 the existence of such a pole is proved. In Subsection 4.2 *rewriting trails*, in the numeration system in base $\beta$, are developed to allow to pass from $\beta$-representations of 1 from $\zeta_\beta(z)$ to $\beta$-representations of 1 deduced the minimal polynomial $P_\beta$ of $\beta$. Applying Kala-Vavra's Theorem in Subsection 4.3 implies the identification of this lenticular pole as a Galois conjugate of $1/\beta$, then of $\beta$. But the contradiction appears if we assume in particular that $\beta$ is a Salem number which is $< \theta_{31}^{-1}$, since a Salem number $\beta$ never admits a non-real conjugate in the open unit disk. The only conjugate of $\beta$ in the open unit disk is real and is $1/\beta$.

Section 3 gathers the Rényi $\beta$-expansions $d_\beta(1)$ of unity of the small Salem numbers known, those which can be found in Lehmer's paper [63] or those from the list of Mossinghoff [75]. In there it is shown how to compute readily $\zeta_\beta(z)$ from $d_\beta(1)$ for such Salem numbers $\beta$ in the interval $(\tau = 1.176280\dots, \Theta = \theta_5^{-1} = 1.32\dots)$. This provides examples of Salem numbers close to, and slightly greater than, Lehmer's number $\tau$ while the proof of Theorem 1.1 in Section 4 is concerned with hypothetical Salem numbers $< \tau$.
## 2. DYNAMICAL ZETA FUNCTION OF THE RÉNYI-PARRY NUMERATION DYNAMICAL SYSTEM

Let $\beta$ be a real number, $1 < \beta < 2$. Denote $\mathcal{A} := \{0, 1\}$. We refer the reader to Lothaire [66], Chap. 7 written by Christiane Frougny. Let us fix the notations. A representation in base $\beta$ (or $\beta$-representation) of a real number $x > 0$ is an infinite word $(x_i)_{i \ge 1}$ of $\mathcal{A}^\mathbb{N}$ such that

$$x = \sum_{i \ge 1} x_i \beta^{-i}.$$

A particular $\beta$-representation, called the $\beta$-expansion, or the greedy $\beta$-expansion, and denoted by $d_\beta(x)$, of $x$ can be computed either by the greedy algorithm, or equivalently by the $\beta$-transformation

$$T_{\beta} : x \mapsto \beta x \pmod{1} = \{\beta x\}.$$

The dynamical system $([0, 1], T_\beta)$ is called the Rényi-Parry numeration system in base $\beta$ [77] [82], the iterates of $T_\beta$ providing the successive digits $x_i$ of $d_\beta(x)$ [65]. Denoting $T_\beta^0 := \text{Id}, T_\beta^1 := T_\beta, T_\beta^i := T_\beta(T_\beta^{i-1})$ for all $i \ge 1$, we have:

$$d_{\beta}(x) = (x_i)_{i \ge 1} \quad \text{if and only if} \quad x_i = \lfloor \beta T_{\beta}^{i-1}(x) \rfloor$$

and we write the $\beta$-expansion of $x$ as

$$(2.1) \qquad x = \cdot x_1 x_2 x_3 \dots \quad \text{instead of} \quad x = \frac{x_1}{\beta} + \frac{x_2}{\beta^2} + \frac{x_3}{\beta^3} + \dots$$

The digits $x_i$ depend upon $\beta$. In particular, the $\beta$-expansion of 1 is by definition denoted by

$$(2.2) \qquad d_{\beta}(1) = 0.t_1 t_2 t_3 \dots \quad \text{and uniquely corresponds to} \quad 1 = \sum_{i=1}^{+\infty} t_i \beta^{-i},$$

where

$$(2.3) \qquad t_1 = \lfloor \beta \rfloor, t_2 = \lfloor \beta \{\beta\} \rfloor = \lfloor \beta T_{\beta}(1) \rfloor, t_3 = \lfloor \beta \{\beta \{\beta\} \} \rfloor = \lfloor \beta T_{\beta}^2(1) \rfloor, \dots$$

Denote by $\langle lex \rangle$ the lexicographical ordering relation on $\{0, 1\}^\mathbb{N}$. The $\beta$-expansion of 1 plays an important part in the following, by the fact that the map $\beta \to d_\beta(1)$ preserves the ordering as follows.

**Proposition 2.1.** Let $\alpha, \beta \in (1, 2)$ with $\alpha \neq \beta$. If the Rényi $\alpha$-expansion of 1 is

$$d_{\alpha}(1) = 0.t'_{1}t'_{2}t'_{3}\dots, \quad \text{i.e.} \quad 1 = \frac{t'_{1}}{\alpha} + \frac{t'_{2}}{\alpha^2} + \frac{t'_{3}}{\alpha^3} + \dots$$

and the Rényi $\beta$-expansion of 1 is

$$d_{\beta}(1) = 0.t_1 t_2 t_3 \dots, \quad \text{i.e.} \quad 1 = \frac{t_1}{\beta} + \frac{t_2}{\beta^2} + \frac{t_3}{\beta^3} + \dots,$$

then $\alpha < \beta$ if and only if $(t'_1, t'_2, t'_3, \dots) <_{lex} (t_1, t_2, t_3, \dots)$.

*Proof.* Lemma 3 in Parry [77]. $\square$

<figure><img src="image_1.png" /></figure>
Since the sequence $(\theta_n^{-1})_{n \ge 2}$ is (strictly) decreasing and tends to 1 if $n$ tends to infinity
[110], it induces the partitioning

$$
(2.4) \qquad \left(1, \frac{1+\sqrt{5}}{2}\right] = \bigcup_{n=2}^{\infty} \left[ \theta_{n+1}^{-1}, \theta_n^{-1} \right) \cup \{ \theta_2^{-1} \}.
$$

**Definition 2.2.** Let $\beta \in (1, \frac{1+\sqrt{5}}{2}]$ be a real number. The integer $n \ge 3$ such that $\theta_n^{-1} \le \beta < \theta_{n-1}^{-1}$ is called the dynamical degree of $\beta$, and is denoted by dyg($\beta$). By convention we put:

$\text{dyg}(\frac{1+\sqrt{5}}{2}) = 2$.

The function $\beta \to n = \text{dyg}(\beta)$ is locally constant on the interval $(1, \frac{1+\sqrt{5}}{2}]$, takes all values in $\mathbb{N} \setminus \{0, 1\}$, and satisfies: $\lim_{\beta>1, \beta \to 1} \text{dyg}(\beta) = +\infty$. We have:

$$
\beta \text{ tends to } 1 \quad \text{if and only if} \quad n = \text{dyg}(\beta) \text{ tends to } +\infty.
$$

**Definition 2.3.** Let $\beta \in (1, (1+\sqrt{5})/2]$ be a real number, and $d_{\beta}(1) = 0.t_1t_2t_3...$ its Rényi $\beta$-expansion of 1. The power series $f_{\beta}(z) := -1 + \sum_{i \ge 1} t_i z^i$ of the complex variable $z$ is called the *Parry Upper function* at $\beta$.

From $(t_i)_{i \ge 1} \in \mathcal{A}^\mathbb{N}$ is built $(c_i)_{i \ge 1} \in \mathcal{A}^\mathbb{N}$, defined by

$$
c_1 c_2 c_3 \dots := \begin{cases} t_1 t_2 t_3 \dots & \text{if } d_{\beta}(1) = 0.t_1 t_2 \dots \text{ is infinite,} \\ (t_1 t_2 \dots t_{q-1} (t_q - 1))^{\omega} & \text{if } d_{\beta}(1) \text{ is finite, } = 0.t_1 t_2 \dots t_q, \end{cases}
$$

where $()^\omega$ means that the word within $()$ is indefinitely repeated. The sequence $(c_i)_{i \ge 1}$ is
the unique element of $\mathcal{A}^\mathbb{N}$ which allows to obtain all the admissible $\beta$-expansions of all
the elements of $[0, 1)$.

**Definition 2.4 (Conditions of Parry).** A sequence $(y_i)_{i \ge 0}$ of elements of $\mathcal{A}$ (finite or not) is said *admissible* if

$$
(2.5) \quad \sigma^j (y_0, y_1, y_2, \ldots) := (y_j, y_{j+1}, y_{j+2}, \ldots) <_{\text{lex}} (c_1, c_2, c_3, \ldots) \quad \text{for all } j \ge 0.
$$

The operator $\sigma$ on $\mathcal{A}^\mathbb{N}$ is the one-sided shift.

**Definition 2.5.** A sequence $(a_i)_{i \ge 0} \in \mathcal{A}^\mathbb{N}$ satisfying (2.6) is said to be *Lyndon (or self-admissible)*:

$$
(2.6) \quad \sigma^n(a_0, a_1, a_2, \dots) = (a_n, a_{n+1}, a_{n+2}, \dots) <_{\text{lex}} (a_0, a_1, a_2, \dots) \quad \text{for all } n \ge 1.
$$

The terminology comes from the introduction of such words by Lyndon in [67], in hon-
our of his work.

**Theorem 2.6.** Let $n \ge 2$. A real number $\beta \in (1, \frac{1+\sqrt{5}}{2}]$ belongs to $[\theta_{n+1}^{-1}, \theta_n^{-1})$ if and only if the Rényi $\beta$-expansion of unity is of the form

$$
(2.7) \qquad d_{\beta}(1) = 0.10^{n-1}10^{n_1}10^{n_2}10^{n_3} \dots,
$$
with $n_k \ge n-1$ for all $k \ge 1$.

*Proof.* Since $d_{\theta_{n+1}^{-1}}(1) = 0.10^{n-1}1$ and $d_{\theta_n^{-1}}(1) = 0.10^{n-2}1$, Proposition 2.1 implies that the condition is sufficient. It is also necessary: $d_{\beta}(1)$ begins as $0.10^{n-1}1$ for all $\beta$ such that
$\theta_{n+1}^{-1} \le \beta < \theta_n^{-1}$. For such $\beta$s we write $d_\beta(1) = 0.10^{n-1}1u$ with digits in the alphabet $\mathcal{A}_\beta = \{0,1\}$ common to all $\beta$s, that is

$$u = 1^{h_0}0^{n_1}1^{h_1}0^{n_2}1^{h_2}\dots$$

and $h_0, n_1, h_1, n_2, h_2, \dots$ integers $\ge 0$. The self-admissibility lexicographic condition (2.6) applied to the sequence $(1, 0^{n-1}, 1^{1+h_0}, 0^{n_1}, 1^{h_1}, 0^{n_2}, 1^{h_3}, \dots)$, which characterizes uniquely the base of numeration $\beta$, readily implies $h_0 = 0$ and $h_k = 1$ and $n_k \ge n-1$ for all $k \ge 1$. $\square$

**Definition 2.7.** A power series $\sum_{j=0}^{+\infty} a_j z^j$, with $a_j \in \{0,1\}$ for all $j \ge 0$, $z$ the complex variable, is said to be *Lyndon (or self-admissible)* if its coefficient vector $(a_i)_{i \ge 0}$ is Lyndon.

In Fredholm theory, the Parry Upper function at $\beta$ is the generalized Fredholm determinant (up to the sign) of the transfer operator of the $\beta$-transformation (cf Ruelle [83] [84] [85] [86] [87] [88] [89] [90] [91] and more recently Baladi [6] [7] [8], Baladi and Keller [9], Hofbauer [51], Hofbauer and Keller [52], Milnor and Thurston [71], Parry and Pollicott [78], Pollicott [79] [80], Takahashi [103] [104]).

For $n \ge 2$, denote: $G_n(X) := -1 + X + X^n$.

**Proposition 2.8.** For $1 < \beta < (1 + \sqrt{5})/2$ any real number, with $d_\beta(1) = 0.t_1t_2t_3\dots$, the Parry Upper function $f_\beta(z)$ is such that $f_\beta(1/\beta) = 0$. It is such that $f_\beta(z) + 1$ has coefficients in the alphabet $\{0,1\}$ and its coefficient vector is Lyndon. It takes the form

$$(2.8) \qquad f_{\beta}(z) = G_{\text{dyg}(\beta)}(z) + z^{m_1} + z^{m_2} + \dots + z^{m_q} + z^{m_{q+1}} + \dots$$

with $m_1 - \text{dyg}(\beta) \ge \text{dyg}(\beta) - 1$, $m_{q+1} - m_q \ge \text{dyg}(\beta) - 1$ for $q \ge 1$. Conversely, given a power series

$$(2.9) \qquad -1 + z + z^n + z^{m_1} + z^{m_2} + \dots + z^{m_q} + z^{m_{q+1}} + \dots$$

with $n \ge 3$, $m_1 - n \ge n - 1$, $m_{q+1} - m_q \ge n - 1$ for $q \ge 1$, then there exists an unique $\beta \in (1, (1 + \sqrt{5})/2)$ for which $n = \text{dyg}(\beta)$ with $f_\beta(z)$ equal to (2.9).

Moreover, if $\beta$, $1 < \beta < (1 + \sqrt{5})/2$, is a reciprocal algebraic integer, in particular a Salem number, the power series (2.8) is never a polynomial.

*Proof.* The expression of $f_\beta(z)$ readily comes from Theorem 2.6. Let us prove the last claim. Assume that $\beta$ is a reciprocal algebraic integer and that $f_\beta(z)$ is a polynomial. The polynomial $f_\beta(z)$ would vanish at the two real zeroes $\beta$ and $1/\beta$. But the sequence $-1t_1t_2t_3\dots$ has only one sign change. By Descartes's rule we obtain a contradiction. $\square$

**Definition 2.9.** A real number $\beta > 1$ is said to be a simple Parry number if $d_\beta(1)$ is finite (i.e. if it ends in infinitely many zeroes). It is a Parry number if it is simple or if $d_\beta(1)$ is eventually periodic, with a preperiod of length $\ge 0$ and a nonzero period.

Parry [77] proved that the set of simple Parry numbers is dense in $(1, +\infty)$. Salem numbers are never simple Parry numbers.

In number theory, inequalities are often associated to collections of half-spaces in euclidean or adelic Geometry of Numbers (e.g. Minkowski's Theorem, etc). The conditions of Parry (at $\beta$) are of totally different nature since they refer to a reasonable control, order-preserving, of the gappiness (lacunarity) of the coefficient vectors of the power series which are the Parry Upper functions, when $\beta$ is close to 1.
**Theorem 2.10.** Let $\beta \in (1, \theta_2^{-1})$. Then, the Artin-Mazur dynamical zeta function $\zeta_\beta(z)$ defined by

$$ (2.10) \qquad \zeta_{\beta}(z) := \exp\left(\sum_{n=1}^{\infty} \frac{\#\{x \in [0, 1] \mid T_{\beta}^{n}(x) = x\}}{n} z^n\right), $$

counting the number of periodic points of period dividing $n$, has a sense, is nonzero and meromorphic in $\{z: |z| < 1\}$, and such that $1/\zeta_\beta(z)$ is holomorphic in $\{z: |z| < 1\}$,

*Proof.* Theorem 2 in [9], assuming that the set of intervals $([0, a_1], [a_1, 1])$ forming the partition of $[0, 1]$ is generating; In [87] [90] Ruelle shows that this assumption is not necessary, showing how to remove this obstruction. $\square$

Theorem 2.10 has been stated in Baladi and Keller [9] under more general assumptions. Theorem 2.10 had been previously conjectured by Hofbauer and Keller [52] for piecewise monotone maps, for the case where the function $g$ occuring in the transfer operators is piecewise constant. (cf also Mori [72] [73]). The case $g = 1$ in the transfer operators was studied by Milnor and Thurston [71], Hofbauer [51].

The relations between the poles of the dynamical zeta function $\zeta_\beta(z)$ and the zeroes of the Parry Upper function $f_\beta(z)$ come from Theorem 2.10 and from the following theorem.

**Theorem 2.11.** Let $\beta \in (1, \theta_2^{-1})$. Then the Parry Upper function $f_\beta(z)$ satisfies

$$ (2.11) \qquad (i) \quad f_{\beta}(z) = -\frac{1}{\zeta_{\beta}(z)} \quad \text{if } \beta \text{ is not a simple Parry number,} $$

and

$$ (2.12) \qquad (ii) \quad f_{\beta}(z) = -\frac{1-z^N}{\zeta_{\beta}(z)} \quad \text{if } \beta \text{ is a simple Parry number} $$

where $N$, which depends upon $\beta$, is the minimal positive integer such that $T_\beta^N(1) = 0$. It is holomorphic in the open unit disk $\{z: |z| < 1\}$. It has no zero in $\{z: |z| \le 1/\beta\}$ except $z = 1/\beta$ which is a simple zero.

*Proof.* Theorem 2.3 and Appendix A in Flatto, Lagarias and Poonen [41]; Theorem 1.2 in Flatto and Lagarias [40], I; Theorem 3.2 in Lagarias [55]. From Takahashi [102], Ito and Takahashi [53], these authors deduce

$$ (2.13) \qquad \zeta_{\beta}(z) = \frac{1-z^N}{(1-\beta z)(\sum_{n=0}^{\infty} T_{\beta}^{n}(1) z^n)} $$

where “$z^N$” has to be replaced by “0” if $\beta$ is not a simple Parry number. Since $\beta T_\beta^n(1) = \lceil \beta T_\beta^n(1) \rceil + \{\beta T_\beta^n(1)\} = t_{n+1} + T_\beta^{n+1}(1)$ by (2.3), for $n \ge 1$, expanding the power series of the denominator (2.13) readily gives:

$$ (2.14) \qquad -1 + t_1 z + t_2 z^2 + \dots = f_{\beta}(z) = -(1 - \beta z) \left( \sum_{n=0}^{\infty} T_{\beta}^{n}(1) z^n \right). $$

The zeroes of smallest modulus are characterized in Lemma 5.2, Lemma 5.3 and Lemma 5.4 in [41]. $\square$
Let $\beta \in (1, \theta_2^{-1})$. Since the power series $f_\beta(z)$ has coefficients in the finite set $\mathcal{A} \cup \{-1\}$, its radius of convergence is $\ge 1$ by Hadamard's formula, and it obeys the Carlson-Polya dichotomy (Bell and Chen [11], Bell, Miles and Ward [12], Carlson [26], Pólya [81], Szegő [101]). Applying the Carlson-Polya dichotomy gives the following equivalence.

**Theorem 2.12.** Let $\beta \in (1, \theta_2^{-1})$. The real number $\beta$ is a Parry number if and only if the Parry Upper function $f_\beta(z)$ is a rational function, equivalently if and only if $\zeta_\beta(z)$ is a rational function.

Moreover, the set of nonParry numbers in $(1, \infty)$ is not empty. If $\beta$ is not a Parry number, then the unit circle $\{z : |z| = 1\}$ is the natural boundary of $f_\beta(z)$, equivalently of $\zeta_\beta(z)$.

*Proof.* [110]. The set of nonParry numbers $\beta$ in $(1, \infty)$ is not empty as a consequence of Fekete-Szegő's Theorem [39] since the radius of convergence of $f_\beta(z)$ is equal to 1 in any case, and that its domain of definition always contains the open unit disk which has a transfinite diameter equal to 1. Since the set of Parry numbers $\beta$ in $(1, \infty)$ is nonempty, the dichotomy between the set of Parry numbers and the set of nonParry numbers, both nonempty, in $(1, \infty)$ has a sense. $\square$

Recall that the set of simple Parry numbers is dense [77] in $(1, +\infty)$ and that the set of Parry numbers which are not simple is infinite since it contains intinitely many Pisot numbers by Schmidt [95], Bertrand-Mathis [17].

### 3. DYNAMICS OF THE SMALL SALEM NUMBERS KNOWN

The set $\mathbb{P}$ of Perron numbers is dense in $[1, +\infty)$. It contains the subset $\mathbb{P}_P$ of the Parry numbers by a result of Lind [64] (Blanchard [18], Boyle [25], Denker, Grillenberger and Sigmund [32], Frougny in [66] chap.7). The set $\mathbb{P} \setminus \mathbb{P}_P$ is not empty (Akiyama [2], and as a consequence of Fekete-Szegő's Theorem [39]); it would contain all Salem numbers of large degrees, by Thurston [107] p. 11. Parry ([77], Theorem 5) proved that the subcollection of simple Parry numbers is dense in $[1, \infty)$.

In the opposite direction a Conjecture of K. Schmidt [95] asserts that Salem numbers are all Parry numbers. For Salem numbers $\beta$ of degree $\ge 6$, Boyd [23] established a simple probabilistic model, based on the frequencies of digits occurring in the Rényi $\beta$-expansions of unity, to conjecture that, more realistically, Salem numbers are dispatched into the two sets of Parry numbers and nonParry numbers, each of them with densities $> 0$. This model, coherent with Thurston's one ([107], p. 11), is in contradiction with the conjecture of K. Schmidt.

This dichotomy of Salem numbers was verified by Hichri [47] [48] [49] for Salem numbers of degree 8. Hichri further developped the heuristic approach of Boyd for Salem numbers of degree 8. The Salem numbers of degree $\le 8$ are all greater than $1.280638...$ from [75]. Salem numbers of degree 4 are Parry numbers [21] [22]. Boyd's model covers the set of Salem numbers smaller than Lehmer's number, if any.

The small Salem numbers found by Lehmer in [63], reported in Table 2, either given by their minimal polynomial or equivalently by their $\beta$-expansion, are Parry numbers whose Rényi $\beta$-expansion of unity has a preperiod length equal to 1. Table 1 gives the subcollection of those Salem numbers $\beta$ which are Parry numbers, of degree $\le 44$, within the intervals of extremities the Perron numbers $\theta_n^{-1}$, $n = 5, 6, ..., 12$. In each interval $(\theta_n^{-1}, \theta_{n-1}^{-1})$
the Salem numbers $\beta$ have the same dynamical degree $dyg(\beta)$, while a certain disparity of the degrees $deg(\beta)$ occurs. The remaining Salem numbers in [75] are very probably nonParry numbers though proofs are not available yet; they are not included in Table 1. Apart from them, the other Salem numbers which exist in the intervals $(\theta_n^{-1}, \theta_{n-1}^{-1})$, $n \ge 6$, if any, should be of degrees > 180.

<table><caption>Table 1. Smallest Salem numbers $\beta < 1.3$ of degree $\le 44$, which are Parry numbers, computed from the “list of Mossinghoff” [75] of irreducible monic integer polynomials of degree $\le 180$. In Column 1 is reported the dynamical degree of $\beta$. Column 4 gives the degree $d_P$ of the Parry polynomial $P_{\beta,P}$ of $\beta$; $P_{\beta,P}$ is reducible except if “irr.” is mentioned.</caption><thead><tr><th>dyg</th><th>deg</th><th>β</th><th>P<sub>β,P</sub></th><th colspan="2">d<sub>β</sub>(1)</th></tr></thead><tbody><tr><td>5</td><td>3</td><td>θ<sub>5</sub><sup>-1</sup> = 1.324717</td><td>5</td><td>0.10<sup>3</sup>1</td><td>smallest Pisot number - Siegel</td></tr><tr><td>6</td><td>18</td><td>1.29567</td><td>22</td><td colspan="2">0.1(0<sup>4</sup>10<sup>9</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>10</td><td>1.293485</td><td>12</td><td colspan="2">0.1(0<sup>4</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>24</td><td>1.291741</td><td>24 irr.</td><td colspan="2">0.1(0<sup>4</sup>10<sup>11</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>26</td><td>1.286730</td><td>30</td><td colspan="2">0.1(0<sup>4</sup>10<sup>17</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>34</td><td>1.285409</td><td>38</td><td colspan="2">0.1(0<sup>4</sup>10<sup>25</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>30</td><td>1.285235</td><td>45</td><td colspan="2">0.1(0<sup>4</sup>10<sup>32</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>44</td><td>1.285199</td><td>66</td><td colspan="2">0.1(0<sup>4</sup>10<sup>54</sup>10<sup>6</sup>)<sup>ω</sup></td></tr><tr><td>6</td><td>6</td><td>θ<sub>6</sub><sup>-1</sup> = 1.285199</td><td>6 irr.</td><td>0.10<sup>4</sup>1</td><td>Perron number</td></tr><tr><td>7</td><td>26</td><td>1.285196</td><td>44</td><td colspan="2">0.1(0<sup>5</sup>10<sup>5</sup>10<sup>5</sup>10<sup>5</sup>10<sup>5</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>26</td><td>1.281691</td><td>..</td><td colspan="2">0.1(0<sup>5</sup>10<sup>5</sup>10<sup>9</sup>10<sup>5</sup>10<sup>17</sup>10<sup>7</sup>10<sup>6</sup>10<sup>6</sup>10<sup>7</sup>10<sup>12</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>8</td><td>1.280638</td><td>20</td><td colspan="2">0.1(0<sup>5</sup>10<sup>5</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>10</td><td>1.261230</td><td>14</td><td colspan="2">0.1(0<sup>5</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>24</td><td>1.260103</td><td>28</td><td colspan="2">0.1(0<sup>5</sup>10<sup>13</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>18</td><td>1.256221</td><td>36</td><td colspan="2">0.1(0<sup>5</sup>10<sup>21</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>7</td><td>7</td><td>θ<sub>7</sub><sup>-1</sup> = 1.255422</td><td>7 irr.</td><td>0.10<sup>5</sup>1</td><td>Perron number</td></tr><tr><td>8</td><td>18</td><td>1.252775</td><td>120</td><td colspan="2">0.1(0<sup>6</sup>10<sup>6</sup>10<sup>10</sup>10<sup>16</sup>10<sup>12</sup>10<sup>7</sup>10<sup>12</sup>0<sup>16</sup>10<sup>10</sup>10<sup>6</sup>10<sup>8</sup>)<sup>ω</sup></td></tr><tr><td>8</td><td>12</td><td>1.240726</td><td>48</td><td colspan="2">0.1(0<sup>6</sup>10<sup>11</sup>10<sup>7</sup>10<sup>11</sup>10<sup>8</sup>)<sup>ω</sup></td></tr><tr><td>8</td><td>20</td><td>1.232613</td><td>41</td><td colspan="2">0.1(0<sup>6</sup>10<sup>24</sup>10<sup>8</sup>)<sup>ω</sup></td></tr><tr><td>8</td><td>8</td><td>θ<sub>8</sub><sup>-1</sup> = 1.232054</td><td>8 irr.</td><td>0.10<sup>6</sup>1</td><td>Perron number</td></tr><tr><td>9</td><td>10</td><td>1.216391</td><td>18</td><td colspan="2">0.1(0<sup>7</sup>10<sup>9</sup>)<sup>ω</sup></td></tr><tr><td>9</td><td>9</td><td>θ<sub>9</sub><sup>-1</sup> = 1.213149</td><td>9 irr.</td><td>0.10<sup>7</sup>1</td><td>Perron number</td></tr><tr><td>10</td><td>14</td><td>1.200026</td><td>20</td><td colspan="2">0.1(0<sup>8</sup>10<sup>10</sup>)<sup>ω</sup></td></tr><tr><td>10</td><td>10</td><td>θ<sub>10</sub><sup>-1</sup> = 1.197491</td><td>10 irr.</td><td>0.10<sup>8</sup>1</td><td>Perron number</td></tr><tr><td>11</td><td>9</td><td>θ<sub>11</sub><sup>-1</sup> = 1.184276</td><td>11</td><td colspan="2">0.10<sup>9</sup>1</td></tr><tr><td>12</td><td>10</td><td>1.176280</td><td>75</td><td colspan="2">Lehmer's number : 0.1(0<sup>10</sup>10<sup>18</sup>10<sup>12</sup>10<sup>18</sup>10<sup>12</sup>)<sup>ω</sup></td></tr><tr><td>12</td><td>12</td><td>θ<sub>12</sub><sup>-1</sup> = 1.172950</td><td>12 irr.</td><td>0.10<sup>10</sup>1</td><td>Perron number</td></tr></tbody></table>
The Salem numbers $\beta$, $\theta_{12}^{-1} < \beta < \theta_5^{-1}$, in Table 1, of degree $\le 44$, are Parry numbers. The Parry polynomial $P_{\beta,P}(X)$ of a Salem number $\beta$ is determined by the equation

$$f_{\beta}(z) = \frac{-P_{\beta,P}^{*}(z)}{1-z^{p+1}} \quad \text{if } d_{\beta}(1) = 0.t_1t_2\dots t_m(t_{m+1}t_{m+2}\dots t_{m+p+1})^{\omega} \text{ has period length } p+1 \ge 1.$$

<table><caption>Table 2. The small Salem numbers, with their minimal polynomial, discovered by Lehmer in [63].</caption><thead><tr><th>deg(β)</th><th>β</th><th>minimal polynomial of β</th><th>d<sub>β</sub>(1)</th></tr></thead><tbody><tr><td>4</td><td>1.722...</td><td>X<sup>4</sup> - X<sup>3</sup> - X<sup>2</sup> - X + 1</td><td>0.1(100)<sup>ω</sup></td></tr><tr><td>6</td><td>1.401...</td><td>X<sup>6</sup> - X<sup>4</sup> - X<sup>3</sup> - X<sup>2</sup> + 1</td><td>0.1(0<sup>2</sup>10<sup>4</sup>)<sup>ω</sup></td></tr><tr><td>8</td><td>1.2806...</td><td>X<sup>8</sup> - X<sup>5</sup> - X<sup>4</sup> - X<sup>3</sup> + 1</td><td>0.1(0<sup>5</sup>10<sup>5</sup>10<sup>7</sup>)<sup>ω</sup></td></tr><tr><td>10</td><td>1.17628...</td><td>X<sup>10</sup> + X<sup>9</sup> - X<sup>7</sup> - X<sup>6</sup> - X<sup>5</sup> - X<sup>4</sup> - X<sup>3</sup> + X + 1</td><td>0.1(0<sup>10</sup>10<sup>18</sup>10<sup>12</sup>10<sup>18</sup>10<sup>12</sup>)<sup>ω</sup></td></tr></tbody></table>

Using the “Construction of Salem”, Hare and Tweedle [46] obtain convergent families of Salem numbers, all Parry numbers, having as limit points the limit points of the set S of Pisot numbers in the interval (1, 2) (characterized by Amara [3]). These families of Salem numbers which are Parry numbers do not contain Salem numbers smaller than Lehmer’s number.

The relations between the digits ($t_i$) in the Rényi $\beta$-expansion of unity $d_{\beta}(1)$ of a Salem number $\beta$ and the coefficient vector of its minimal polynomial are obscure in general. However we have the following general result on the gappiness, which is never “extreme”.

**Theorem 3.1.** Let $\beta > 1$ be a Salem number. We assume that $d_{\beta}(1)$ is gappy in the sense that there exist two infinite sequences $\{m_n\}_{n \ge 1}$ and $\{s_n\}_{n \ge 0}$ such that

$$1 = s_0 \le m_1 < s_1 \le m_2 < s_2 \le \dots \le m_n < s_n \le m_{n+1} < s_{n+1} \le \dots$$

with $(s_n - m_n) \ge 2, t_{m_n} \ne 0, t_{s_n} \ne 0 \text{ and } t_i = 0 \text{ if } m_n < i < s_n \text{ for all } n \ge 1$. Then

$$(3.1) \qquad \limsup_{n \to +\infty} \frac{s_n}{m_n} = 1$$

*Proof.* There are two cases: (i) $\beta$ is a Parry number: we readily obtain the result. (ii) $\beta$ is not a Parry number: then $M(\beta) = \beta$, where M denotes the Mahler measure of an algebraic number. Then we apply Theorem 1.1 in [108], and deduce the result. Another way is to obtain (ii) as a consequence of Theorem 2 in [1]. □

### 4. PROOF OF THE MAIN THEOREM 1.1

For proving Theorem 1.1, we assume the existence of a Salem number $\beta$ in an interval $(\theta_n^{-1}, \theta_{n-1}^{-1})$, $n \ge 32$, and arrive at a contradiction.

**4.1. A non-real lenticular pole of $\zeta_{\beta}(z)$ in a small angular sector.** This Salem number $\beta \in (\theta_n^{-1}, \theta_{n-1}^{-1})$, $n \ge 32$, would admit a Rényi $\beta$-expansion of unity $d_{\beta}(1)$ which is infinite, by Proposition 2.8:

$$d_{\beta}(1) = 0.10^{n-2}10^{n_1}10^{n_2}10^{n_3}\dots,$$
with $n_k \ge n-2$ for all $k \ge 1$. The gappiness is controlled by the integer $n-2$, in the sense that any string of zeroes has length $\ge n-2$, and is never asymptotically important by Theorem 3.1. Then the Parry Upper function $f_\beta(z)$ at $\beta$ takes the following form and is characterized by the sequence of exponents $(m_q)_{q \ge 0}$:

$$ (4.1) \qquad f_{\beta}(z) = -1 + z + z^n + z^{m_1} + z^{m_2} + z^{m_3} + \dots = G_n(z) + \sum_{q \ge 1} z^{m_q}, $$

where $m_0 := n$, with the fundamental minimal gappiness condition:

$$ (4.2) \qquad m_{q+1} - m_q \ge n - 1 \quad \text{for all } q \ge 0. $$

Inside the open unit disk, the zeroes of $f_\beta(z)$ are exactly the poles of $\zeta_\beta(z)$, with the same multiplicities, by Theorem 2.11. For convenience, we will deal with the zeroes of $f_\beta(z)$ instead of the poles of $\zeta_\beta(z)$, in the sequel. The terminology “lenticular poles”, in the title of this section, comes from the fact that the first zero (cf the definition below) naturally belongs to a lenticulus of zeroes, as shown in Figure 2, and the existence of lenticuli is the case in general [112].

The method which will be used to detect a non-real zero of $f_\beta(z)$ in the open unit disk is the method of Rouché applied to (4.1) and to the first zero of $G_n(z)$, in which $\sum_{q \ge 1} z^{m_q}$ is a perturbation of $G_n(z) = -1+z+z^n$. Let us first recall the notations of the zeroes of the trinomials $G_n$ and their geometry, from the factorization of $G_n(X) := -1+X+X^n$ by Selmer [96] (cf [110] Section 2).

### 4.1.1. Roots of $G_n$ and the asymptotic expansion of the first root.
Let $n \ge 2$. Summing in pairs over complex conjugated imaginary roots, the indexation of the roots and the factorization of $G_n(X)$ are taken as follows:

$$ (4.3) \qquad G_n(X) = (X - \theta_n) \left( \prod_{j=1}^{\lfloor n/6 \rfloor} (X - z_{j,n}) (X - \overline{z_{j,n}}) \right) \times q_n(X), $$

where $\theta_n$ is the only (real) root of $G_n(X)$ in the interval $(0, 1)$, where

$$ q_n(X) = \begin{cases} \left( \prod_{j=1+\lfloor n/6 \rfloor}^{\frac{n-2}{2}} (X-z_{j,n})(X-\overline{z_{j,n}}) \right) \times (X-z_{\frac{n}{2},n}) & \text{if } n \text{ is even, with} \\ z_{\frac{n}{2},n} & \text{real} < -1, \\ \prod_{j=1+\lfloor n/6 \rfloor}^{\frac{n-1}{2}} (X-z_{j,n})(X-\overline{z_{j,n}}) & \text{if } n \text{ is odd,} \end{cases} $$

where the index $j = 1, 2, \dots$ is such that $z_{j,n}$ is a (nonreal) complex zero of $G_n(X)$, except if $n$ is even and $j = n/2$, such that the argument $\arg(z_{j,n})$ of $z_{j,n}$ is $\approx 2\pi j/n$ and that the family of arguments $(\arg(z_{j,n}))_{1 \le j < \lfloor n/2 \rfloor}$ forms a strictly increasing sequence with $j$:

$$ 0 < \arg(z_{1,n}) < \arg(z_{2,n}) < \dots < \arg(z_{\lfloor n/2 \rfloor,n}) \le \pi. $$

For $n \ge 2$ all the roots of $G_n(X)$ are simple, and the roots of $G_n^*(X) = 1 + X^{n-1} - X^n$, as inverses of the roots of $G_n(X)$, are classified in the reversed order.
**Proposition 4.1.** Let $n \ge 2$. If $n \ne 5$ (mod 6), then $G_n(X)$ is irreducible over $\mathbb{Q}$. If $n \equiv 5 \pmod{6}$, then the polynomial $G_n(X)$ admits $X^2 - X + 1$ as irreducible factor in its factorization and $G_n(X)/(X^2 - X + 1)$ is irreducible.

*Proof.* Selmer [96]. □

**Proposition 4.2.** For all $n \ge 2$, all zeros $z_{j,n}$ and $\theta_n$ of the polynomials $G_n(X)$ have a modulus in the interval

$$ (4.4) \qquad \left[ 1 - \frac{2\mathrm{Log}n}{n}, 1 + \frac{2\mathrm{Log}2}{n} \right], $$

(ii) the trinomial $G_n(X)$ admits a unique real root $\theta_n$ in the interval $(0,1)$. The sequence $(\theta_n)_{n \ge 2}$ is strictly increasing, $\lim_{n \to +\infty} \theta_n = 1$, with $\theta_2 = \frac{2}{1+\sqrt{5}} = 0.618\dots$,

(iii) the root $\theta_n$ is the unique root of smallest modulus among all the roots of $G_n(X)$; if $n \ge 6$, the roots of modulus $< 1$ of $G_n(z)$ in the closed upper half-plane have the following properties:

(iii-1) $\theta_n < |z_{1,n}|$,

(iii-2) for any pair of successive indices $j, j+1$ in $\{1, 2, \dots, \lfloor n/6 \rfloor\}$,

$$ |z_{j,n}| < |z_{j+1,n}|. $$

*Proof.* (i)(ii) Selmer [96], pp 291–292; (iii-1) Flatto, Lagarias and Poonen [41], (iii-2) Verger-Gaugry [110]. □

**Proposition 4.3.** Let $n \ge 2$. Then (i) the number $p_n$ of roots of $G_n(X)$ which lie inside the open sector $\mathcal{S} = \{z \mid |\arg(z)| < \pi/3\}$ is equal to

$$ (4.5) \qquad 1 + 2\lfloor \frac{n}{6} \rfloor, $$

(ii) the correlation between the geometry of the roots of $G_n(X)$ which lie inside the unit disk and the upper half-plane and their indexation is given by:

$$ (4.6) \qquad j \in \{1, 2, \dots, \lfloor \frac{n}{6} \rfloor\} \iff \Re(z_{j,n}) > \frac{1}{2} \iff |z_{j,n}| < 1. $$

*Proof.* Proposition 3.7 in [110]. □

The roots $z_{j,n}$ of $G_n$ are given in [110] by their (Poincaré) asymptotic expansions [109], as a function of $n$ and $j$. They are generically written: $\mathrm{Re}(z_{j,n}) = \mathrm{D}(\mathrm{Re}(z_{j,n})) + \mathrm{tl}(\mathrm{Re}(z_{j,n}))$, $\mathrm{Im}(z_{j,n}) = \mathrm{D}(\mathrm{Im}(z_{j,n})) + \mathrm{tl}(\mathrm{Im}(z_{j,n}))$, $\theta_n = \mathrm{D}(\theta_n) + \mathrm{tl}(\theta_n)$, where "D" and "tl" stands for "development" (or "limited expansion", or "lowest order terms") and "tl" for "tail" (or "remainder", or "terminant").

They are given at a sufficiently high order which allows to make sufficiently precise their positioning inside the unit disk and to deduce the asymptotic expansions of the Mahler measures $M(G_n)$. The terminology *order* comes from the general theory (Borel [19], Copson [30], Dingle [33], Erdélyi [38]). Here we do not need to compute the Mahler measures.

We are only concerned with the first root $z_{1,n}$ of $G_n$ around which a small Rouché circle (centered at $z_{1,n}$) will be set up to deduce the existence of a zero of $f_\beta(z)$ inside it, as soon as $n \ge 32$.
**Proposition 4.4.** Let $n \ge 2$. The root $\theta_n$ can be expressed as: $\theta_n = D(\theta_n) + \text{tl}(\theta_n)$ with $D(\theta_n) = 1$

$$ (4.7) \quad \frac{\text{Log} n}{n} \left( 1 - \left( \frac{n - \text{Log} n}{n \text{Log} n + n - \text{Log} n} \right) \left( \text{Log} \text{Log} n - n \text{Log} \left( 1 - \frac{\text{Log} n}{n} \right) - \text{Log} n \right) \right) $$

and

$$ (4.8) \quad \text{tl}(\theta_n) = \frac{1}{n} O \left( \left( \frac{\text{Log} \text{Log} n}{\text{Log} n} \right)^2 \right), $$

with the constant $1/2$ involved in $O()$.

*Proof.* [110] Proposition 3.1.

<figure><img src="image_2.png" /></figure>

**Lemma 4.5.** Given the limited expansion $D(\theta_n)$ of $\theta_n$ as in (4.7), denote

$$ \lambda_n := 1 - (1 - D(\theta_n)) \frac{n}{\text{Log} n}. $$

Then $\lambda_n = D(\lambda_n) + \text{tl}(\lambda_n)$, with

$$ (4.9) \quad D(\lambda_n) = \frac{\text{Log} \text{Log} n}{\text{Log} n} \left( \frac{1}{1 + \frac{1}{\text{Log} n}} \right), \quad \text{tl}(\lambda_n) = O \left( \frac{\text{Log} \text{Log} n}{n} \right) $$

with the constant 1 in the Big O.

*Proof.* [110] Lemma 3.2.

<figure><img src="image_3.png" /></figure>

The asymptotic expansions of those roots $z_{j,n}$ of $G_n(z)$ which lie close to the real axis and $\{z=1\}$ are (divergent) sums of functions of a couple of two variables which is: $(n, j/\text{Log} n)$ in the angular sector $2\pi \text{Log} n/n > \arg z > 0$. It is the case for $j=1$.

**Proposition 4.6.** Let $n \ge 18$. The first root $z_{1,n}$ of $G_n$ is given by:

$$ D(\text{Re}(z_{1,n})) = \theta_n + \frac{2\pi^2}{n} \left( \frac{1}{\text{Log} n} \right)^2 (1 + 2\lambda_n), $$

$$ D(\text{Im}(z_{1,n})) = \frac{2\pi \text{Log} n}{n} \left( \frac{1}{\text{Log} n} \right) \left[ 1 - \frac{1}{\text{Log} n} (1 + \lambda_n) \right], $$

with

$$ \text{tl}(\text{Re}(z_{1,n})) = \frac{1}{n \text{Log} n} \left( \frac{1}{\text{Log} n} \right)^2 O \left( \left( \frac{\text{Log} \text{Log} n}{\text{Log} n} \right)^2 \right), $$

$$ \text{tl}(\text{Im}(z_{1,n})) = \frac{1}{n \text{Log} n} \left( \frac{1}{\text{Log} n} \right)^2 O \left( \left( \frac{\text{Log} \text{Log} n}{\text{Log} n} \right)^2 \right), $$

*Proof.* [110].

<figure><img src="image_4.png" /></figure>
4.1.2. Existence. The Salem number $\beta \in (\theta_n^{-1}, \theta_{n-1}^{-1})$, $n \ge 32$, is uniquely characterized in (4.1) by the sequence of exponents $(m_q)_{q \ge 0}$ in

$$ (4.10) \qquad f_{\beta}(z) = -1 + z + z^n + z^{m_1} + z^{m_2} + z^{m_3} + \dots = G_n(z) + \sum_{q \ge 1} z^{m_q}, $$

with $m_0 := n$ and the distanciation condition: $m_{q+1} - m_q \ge n - 1$ for all $q \ge 0$. Since we do not know exactly the sequence $(m_q)$, the problem is to make a direct test of the Rouché condition using the inequality

$$ (4.11) \qquad |f_{\beta}(z) - G_n(z)| = \left| \sum_{q \ge 1} z^{m_q} \right| < |G_n(z)| \quad \text{for } z \in C_{1,n}, $$

for some circle $C_{j,n} := \{z : |z - z_{j,n}| = \frac{t_{1,n}}{n}\}$ of center $z_{1,n}$ and radius $t_{1,n}/n > 0$ small enough. To overcome this difficulty, we proceed by considering the general inequality:

$$ |f_{\beta}(z) - G_n(z)| = \left| \sum_{q \ge 1} z^{m_q} \right| \le \sum_{q \ge 1} |z^{m_q}| \le \frac{|z|^{2(n-1)+1}}{1 - |z|^{n-1}}, \quad |z| < 1. $$

Then we test the Rouché condition by the following inequality:

$$ (4.12) \qquad \frac{|z|^{2(n-1)+1}}{1 - |z|^{n-1}} = \frac{|z|^{2n-1}}{1 - |z|^{n-1}} < |G_n(z)| \quad \text{for } z \in C_{1,n}, $$

instead of (4.11), too complicated to handle.

The problem of the choice of the radius $t_{1,n}/n$ is a true problem. The circle $C_{1,n}$ should not intersect the unit circle $z = 1$, nor the real axis, and should not contain the second root $z_{2,n}$ of $G_n$. On one hand, a too small radius would lead to make impossible the application of the Rouché condition. On the other hand, taking larger values of $t_{1,n}/n$ would readily lead to a bad or impossible localization of the zero of $f_{\beta}(z)$. Indeed, we do not know a priori whether the unit circle is a natural boundary or not for $f_{\beta}(z)$; locating zeroes close to a natural boundary is a difficult problem in general.

The radius of $C_{1,n}$ is chosen to be

$$ \frac{t_{1,n}}{n} = \frac{\pi |z_{1,n}|}{n a_{\max}} $$

where $a_{\max}$ is determined by the following easy lemma.

**Lemma 4.7.** *The function*

$$ a \to h(1, a) := \frac{|1 - \exp(\frac{\pi}{a})| \exp(-\frac{\pi}{a})}{\exp(\frac{\pi}{a}) + |1 - \exp(\frac{\pi}{a})|} $$

defined on $[1, +\infty)$ reaches its maximum $h(1, a_{\max}) := 0.171784 \dots$ at $a_{\max} = 5.8743 \dots$ (Figure 1).
<figure><img src="image_5.png" /><figcaption>FIGURE 1. Curve of the Rouché condition $a \to h(1, a)$.</figcaption></figure>

**Theorem 4.8.** Let $n \ge 32$. Denote by $C_{1,n} := \{z \mid |z - z_{1,n}| = \frac{\pi|z_{1,n}|}{na_{\max}}\}$ the circle centered at the first root $z_{1,n}$ of $G_n(z) = -1 + z + z^n$. Then the condition of Rouché

$$ (4.13) \qquad \frac{|z|^{2n-1}}{1-|z|^{n-1}} < |-1+z+z^n|, \quad \text{for all } z \in C_{1,n}, $$

holds true.

*Proof.* Let $a \ge 1$ and $n \ge 18$. Denote by $\varphi := \arg(z_{1,n})$ the argument of the first root $z_{1,n}$ (in Im($z$) > 0). Since $-1 + z_{1,n} + z_{1,n}^n = 0$, we have $|z_{1,n}|^n = |-1 + z_{1,n}|$. Let us write $z = z_{1,n} + \frac{\pi|z_{1,n}|}{na}e^{i\psi} = z_{1,n}(1 + \frac{\pi}{an}e^{i(\psi-\varphi)})$ the generic element belonging to $C_{1,n}$, with $\psi \in [0, 2\pi]$. Let $X := \cos(\psi - \varphi)$. Let us show that if the inequality (4.13) of Rouché holds true for $X = +1$, then it holds true for all $X \in [-1, +1]$, that is for every argument $\psi \in [0, 2\pi]$, i.e. for every $z \in C_{1,n}$. We have

$$ \left|1 + \frac{\pi}{an} e^{i(\psi - \varphi)}\right|^n = \exp\left(\frac{\pi X}{a}\right) \times \left(1 - \frac{\pi^2}{2a^2n}(2X^2 - 1) + O\left(\frac{1}{n^2}\right)\right) $$

and

$$ \arg\left(\left(1 + \frac{\pi}{an} e^{i(\psi - \varphi)}\right)^n\right) = \operatorname{sgn}\left(\sin(\psi - \varphi)\right) \times \left(\frac{\pi\sqrt{1-X^2}}{a}\left[1 - \frac{\pi X}{an}\right] + O\left(\frac{1}{n^2}\right)\right). $$

Moreover,

$$ \left|1 + \frac{\pi}{an} e^{i(\psi - \varphi)}\right| = \left|1 + \frac{\pi}{an}(X \pm i\sqrt{1-X^2})\right| = 1 + \frac{\pi X}{an} + O\left(\frac{1}{n^2}\right). $$

with

$$ \arg\left(1 + \frac{\pi}{an} e^{i(\psi - \varphi)}\right) = \operatorname{sgn}\left(\sin(\psi - \varphi)\right) \times \frac{\pi\sqrt{1-X^2}}{an} + O\left(\frac{1}{n^2}\right). $$

For all $n \ge 18$, from Proposition 4.6, we have

$$ (4.14) \qquad |z_{1,n}| = 1 - \frac{\operatorname{Log}n - \operatorname{Log}\operatorname{Log}n}{n} + \frac{1}{n}O\left(\frac{\operatorname{Log}\operatorname{Log}n}{\operatorname{Log}n}\right). $$

from which we deduce the following equality, up to $O(\frac{1}{n})$ - terms,

$$ |z_{1,n}| \left|1 + \frac{\pi}{an} e^{i(\psi - \varphi)}\right| = |z_{1,n}|. $$
Then the left-hand side term of (4.13) is

$$ \frac{|z|^{2n-1}}{1-|z|^{n-1}} = \frac{|-1+z_{1,n}|^2 |1+\frac{\pi}{an}e^{i(\psi-\varphi)}|^{2n}}{|z_{1,n}| |1+\frac{\pi}{an}e^{i(\psi-\varphi)}| - |-1+z_{1,n}| |1+\frac{\pi}{an}e^{i(\psi-\varphi)}|^n} $$
$$ (4.15) \qquad = \frac{|-1+z_{1,n}|^2 \left(1-\frac{\pi^2}{an}(2X^2-1)\right) \exp\left(\frac{2\pi X}{a}\right)}{|z_{1,n}| |1+\frac{\pi}{an}e^{i(\psi-\varphi)}| - |-1+z_{1,n}| \left(1-\frac{\pi^2}{2an}(2X^2-1)\right) \exp\left(\frac{\pi X}{a}\right)} $$

up to $\frac{1}{n}O(\frac{\log \log n}{\log n})$-terms (in the terminant). The right-hand side term of (4.13) is

$$ \begin{aligned} |-1+z+z^n| &= |-1+z_{1,n}\left(1+\frac{\pi}{na}e^{i(\psi-\varphi)}\right) + z_{1,n}^n\left(1+\frac{\pi}{na}e^{i(\psi-\varphi)}\right)^n| \\ &= |-1+z_{1,n}\left(1\pm i\frac{\pi\sqrt{1-X^2}}{an}\right)\left(1+\frac{\pi X}{an}\right) + (1-z_{1,n})\left(1-\frac{\pi^2}{2a^2n}(2X^2-1)\right)| \\ &\quad \times \exp\left(\frac{\pi X}{a}\right) \exp\left(\pm i\left(\frac{\pi\sqrt{1-X^2}}{a}[1-\frac{\pi X}{an}]\right)\right) + O\left(\frac{1}{n^2}\right) \end{aligned} $$

Let us consider (4.15) and (4.16) at the first order for the asymptotic expansions, i.e. up to $O(1/n)$-terms instead of up to $O(\frac{1}{n}(\log \log n/\log n))$-terms or $O(1/n^2)$-terms. (4.15) becomes:

$$ \frac{|-1+z_{1,n}|^2 \exp\left(\frac{2\pi X}{a}\right)}{|z_{1,n}| |-1+z_{1,n}| \exp\left(\frac{\pi X}{a}\right)} $$

and (4.16) is equal to:

$$ |-1+z_{1,n}| \left| 1 - \exp\left(\frac{\pi X}{a}\right) \exp\left(\pm i \frac{\pi \sqrt{1-X^2}}{a}\right) \right| $$

and is independent of the sign of $\sin(\psi - \varphi)$. Then the inequality (4.13) is equivalent to

$$ (4.17) \qquad \frac{|-1+z_{1,n}|^2 \exp\left(\frac{2\pi X}{a}\right)}{|z_{1,n}| |-1+z_{1,n}| \exp\left(\frac{\pi X}{a}\right)} < |-1+z_{1,n}| \left| 1 - \exp\left(\frac{\pi X}{a}\right) \exp\left(\pm i \frac{\pi \sqrt{1-X^2}}{a}\right) \right|, $$

and (4.17) to

$$ (4.18) \qquad \frac{|-1+z_{1,n}|}{|z_{1,n}|} < \frac{\left| 1 - \exp\left(\frac{\pi X}{a}\right) \exp\left(i \frac{\pi \sqrt{1-X^2}}{a}\right) \right| \exp\left(-\frac{\pi X}{a}\right)}{\exp\left(\frac{\pi X}{a}\right) + \left| 1 - \exp\left(\frac{\pi X}{a}\right) \exp\left(i \frac{\pi \sqrt{1-X^2}}{a}\right) \right|} = \kappa(X, a). $$

The right-hand side function $\kappa(X, a)$ is a function of $(X, a)$, on $[-1, +1] \times [1, +\infty)$. which is strictly decreasing for any fixed $a$, and reaches its minimum at $X = 1$; this minimum is always strictly positive. Consequently the inequality of Rouché (4.13) will be satisfied on $C_{1,n}$ once it is satisfied at $X = 1$, as claimed.
Hence, up to $O(1/n)$-terms, the Rouché condition (4.18), for any fixed $a$, will be satisfied (i.e. for any $X \in [-1, +1]$) by the set of integers $n = n(a)$ for which $z_{1,n}$ satisfies:

$$ (4.19) \qquad \frac{|-1+z_{1,n}|}{|z_{1,n}|} < \kappa(1,a) = \frac{|1-\exp(\frac{\pi}{a})|\exp(-\frac{\pi}{a})}{\exp(\frac{\pi}{a}) + |1-\exp(\frac{\pi}{a})|}, $$

equivalently, from Proposition 4.6,

$$ (4.20) \qquad \frac{\mathrm{Log}n - \mathrm{Log}\mathrm{Log}n}{n} < \frac{\kappa(1, a)}{1 + \kappa(1, a)}. $$

In order to obtain the largest possible range of values of $n$, the value of $a \ge 1$ has to be chosen such that $a \to \kappa(1, a)$ is maximal in (4.20). From Lemma 4.7 we take $a = a_{\max}$.

The slow decrease of the functions of the variable $n$ involved in the terminants when $n$ tends to infinity, as a factor of uncertainty on (4.20), has to be taken into account in (4.20). It amounts to check numerically whether (4.13) is satisfied for the small values $18 \le n \le 100$ for $a = a_{\max}$, or not. Indeed, for the large enough values of $n$, the inequality (4.20) is satisfied since $\lim_{n \to \infty} \frac{\mathrm{Log}n - \mathrm{Log}\mathrm{Log}n}{n} = 0$. On the computer, the critical threshold of $n = 32$ is easily calculated, with $(\mathrm{Log}32 - \mathrm{Log}\mathrm{Log}32)/32 = 0.0694628...$. Then

$$ \frac{\mathrm{Log}n - \mathrm{Log}\mathrm{Log}n}{n} < \frac{\kappa(1, a_{\max})}{1 + \kappa(1, a_{\max})} = 0.146447... \quad \text{for all } n \ge 32. $$

Let us note that the last inequality also holds for some values of $n$ less than 32. $\square$

**Corollary 4.9.** Let $\beta > 1$ be a Salem number such that $\mathrm{dyg}(\beta) \ge 32$ (if any). Then the Parry Upper function $f_{\beta}(z)$ admits a simple nonreal zero of modulus $< 1$ in the open disk $\{z : |z - z_{1,n}| < \frac{\pi|z_{1,n}|}{na_{\max}}\}$.

*Proof.* The polynomial $G_n(z)$ has simple roots. Since (4.13) is satisfied, the Theorem of Rouché states that $f_{\beta}(z)$ and $G_n(z) = -1 + z + z^n$ have the same number of roots, counted with multiplicities, in the open disk $\{z : |z - z_{1,n}| < \frac{\pi|z_{1,n}|}{na_{\max}}\}$. We deduce the existence of an unique simple zero of $f_{\beta}(z)$ inside this disk. $\square$

Let us denote by $\omega_{1,n}$ this zero of $f_{\beta}(z)$. To summarize, for $n \ge 32$, $\theta_n^{-1} < \beta < \theta_{n-1}^{-1}$, taking the complex-conjugates, we have the 2 tri-uplets (represented schematically in Fig. 2):

$$ \{\overline{z_{1,n}}, \theta_n, z_{1,n}\}, \quad \{\overline{\omega_{1,n}}, \beta^{-1}, \omega_{1,n}\} $$

which are very close in a narrow symmetrical angular sector containing $\{z = 1\}$ and satisfy:

$$ f_{\beta}(1/\beta) = f_{\beta}(\omega_{1,n}) = f_{\beta}(\overline{\omega_{1,n}}) = 0 $$

and

$$ f_{\theta_n^{-1}}(\theta_n) = f_{\theta_n^{-1}}(z_{1,n}) = f_{\theta_n^{-1}}(\overline{z_{1,n}}) = 0. $$

In the sequel we prove that $\omega_{1,n}$ is a conjugate of $\beta^{-1}$, then of $\beta$.
<figure><img src="image_6.png" /><figcaption>FIGURE 2. In a) and b) the zeroes of $G_n$ and $f_β$, resp., are represented by black bullets, with lenticularity appearing (symmetrically with respect to $\mathbb{R}$) in the angular sector $|\arg z| < \pi/18$, about the unit circle in $\mathbb{C}$. In a) the first zero $z_{1,n}$ of $G_n$ (here $n = 37$) is the first one in this sector having positive imaginary part. In b) the zero $\omega_{1,n}$ is obtained by a very slight deformation of $z_{1,n}$ according to Theorem 4.8. The other roots of $f_β$ can be found in a narrow annular neighbourhood of $|z| = 1$.</figcaption></figure>

**4.2. Rewriting trails.** Let us assume that $f_β(\omega_{1,n}) = 0$ with the property that the minimal polynomial $P_β$ of $β$ satisfies $P_β(\omega_{1,n}) \neq 0$, and show the contradiction. Denote $v := |P_β(\omega_{1,n})| > 0$. Let us consider the $s$-th polynomial section $S_s(z) = -1 + \sum_{j=1}^s t_j z^j$ of $f_β(z)$, where the integer $s$, taken large enough, will be fixed below. The vector coefficient $(t_j)$ is as in (4.10).

The $s$-th polynomial section $S_s(z)$ admits a unique real zero in $(0, 1)$. Indeed $S_s(0) = -1$, $S_s(1) > 1$, and the derivative of the restriction of $S_s(z)$ to $[0, 1]$ is positive on $[0, 1]$. The polynomial $S_s^*(z)$, reciprocal polynomial of $S_s(z)$, admits a unique real zero, say $\gamma_s > 1$. We have: $\deg(S_s) \leq s$ and $\lim_{s \to \infty} \gamma_s^{-1} = \beta^{-1}$. The real number $\gamma_s$ is a nonreciprocal algebraic integer which is such that $1 < \gamma_s < \beta$: indeed, $y \to S_s(y)$ is strictly increasing on $(0, 1)$ and $S_s(\beta^{-1}) = -1 + \sum_{j=1}^s t_j \beta^{-j} = f_β(\beta^{-1}) - \sum_{j=s+1}^\infty t_j \beta^{-j} = -\sum_{j=s+1}^\infty t_j \beta^{-j} < 0$, so that $\beta^{-1} < \gamma_s^{-1}$. There exists an integer, say $W_v$, such that: $s \geq W_v \implies |P_β(\gamma_s^{-1})| < \min\{1, v/2\}$. In the following we take $s \geq W_v$.

Before going further, let us recall the general factorization of a Parry Upper function $f_α(z)$ for $α$ any simple Parry number.

**Theorem 4.10.** Let $α$ be a simple Parry number, with dyg($α$) $\geq 3$. Let $f_α$ denote its Parry Upper function with factorization in $\mathbb{Z}[x]$:

$$f_{\alpha}(x) = A(x)B(x)C(x) = -1 + x + x^n + x^{m_1} + x^{m_2} + \dots + x^{m_s},$$

where $s \geq 1$, $m_1 - n \geq n - 1$, $m_{j+1} - m_j \geq n - 1$ for $1 \leq j < s$, where

* A is the cyclotomic component,

* B the reciprocal noncyclotomic component,

* C the nonreciprocal part.

Then C is irreducible. Moreover C has no root of modulus 1.

*Proof.* [35] [37]. □

Then, inside the circle $C_{1,n}$ the polynomial section $S_s(z)$ has a unique zero (same proof as for Theorem 4.8); let us denote it by $r_s$. We have: $\lim_{s \to \infty} r_s = \omega_{1,n}$ and $r_s$ is equal to the conjugate $\sigma_s(\gamma_s^{-1})$ of $\gamma_s^{-1}$ for some $\sigma_s$ which is the conjugation relative to the irreducible nonreciprocal (never trivial) part of $S_s$ by Proposition 4.10. We have: $C(\gamma_s^{-1}) = C(r_s) =$
0. Denote by $s_c = \deg(C)$ the degree of the component $C$. The irreducible polynomials $P_\beta(X)$ and $C(X)$ are coprime: indeed the first one is reciprocal while the second one is nonreciprocal. The integer $s_c$ is a function of $s$. Denote

$$d := \deg(P_{\beta}) \quad \text{and} \quad H := \max_{j=1,\dots,d-1} \{|a_j|\} \ge 1$$

the (naïve) height of $P_\beta(X) = 1 + \sum_{j=1}^{d-1} a_j X^j + X^d$.

Let us now define the notion of *rewriting trail* and justify its use. On one side, the Salem number $\beta$ is entirely determined by its minimal polynomial $P_\beta(X)$, from Commutative Algebra. On the other side, $\beta$ is completely determined by the Parry Upper function $f_\beta(z)$ coming from the dynamical system. A priori, the two analytic functions $z \to P_\beta(z)$ and $z \to f_\beta(z)$ have very different coefficient vectors, the second one never being a polynomial, by Proposition 2.8. But the “object” $\beta$ is the same in both cases.

These two functions give rise to two distinct $\beta$-representations, from the equations:

$$P_{\beta}(1/\beta) = P_{\beta}(\beta) = 0 \quad \text{by the minimal polynomial}$$

and

$$f_{\beta}(1/\beta) = 0 \quad \text{by the numeration dynamical system and } \zeta_{\beta}(z).$$

To prove that $\omega_{1,n}$ is a conjugate of $\beta^{-1}$, to pass from the first $\beta$-representation to the second $\beta$-representation is needed. We cannot proceed directly, and a sequence of intermediate $\beta$-representations is required.

In this subsection, below, we substitute $\beta$ by $\gamma_s$ and $f_\beta(z)$ by $S_s(z)$, and construct step by step the intermediate $\gamma_s$-representations. These intermediate $\gamma_s$-representations constitute a rewriting trail. There is no unicity. Then, in the next subsection, the limit when $s$ tends to infinity is taken to conclude.

Let us construct the rewriting trail from “$S_s$” to “$P_\beta$”, at $\gamma_s^{-1}$ [36]. The starting point is the identity $1=1$, to which we add $0 = S_\gamma(\gamma_s^{-1})$ in the (rhs) right hand side. Then we define the rewriting trail from the Rényi $\gamma_s^{-1}$-expansion of 1

$$(4.21) \qquad 1 = 1 + S_{\gamma_s}(\gamma_s^{-1}) = t_1 \gamma_s^{-1} + t_2 \gamma_s^{-2} + \dots + t_{s-1} \gamma_s^{-(s-1)} + t_s \gamma_s^{-s}$$

(with $t_1 = 1, t_2 = t_3 = \dots = t_{n-1} = 0, t_n = 1$, etc) to

$$(4.22) \qquad -a_1 \gamma_s^{-1} - a_2 \gamma_s^{-2} + \dots - a_{d-1} \gamma_s^{-(d-1)} - \gamma_s^{-d} = 1 - P_{\beta}(\gamma_s^{-1}),$$

by “restoring” the digits of $1-P_\beta(X)$ one after the other, from the left.

We obtain a sequence $(A'_q(X))_{q \ge 1}$ of rewriting polynomials involved in this rewriting trail; for $q \ge 1$, $A'_q \in \mathbb{Z}[X]$, $\deg(A'_q) \le q$ and $A'_q(0) = 1$.

At the first step we add $0 = -(-a_1 - t_1)\gamma_s^{-1}S_{\gamma_s}^*(\gamma_s^{-1})$; and we obtain

$$1 = -a_1 \gamma_s^{-1} + (-(-a_1 - t_1)t_1 + t_2)\gamma_s^{-2} + (-(-a_1 - t_1)t_2 + t_3)\gamma_s^{-3} + \dots$$

so that the height of the polynomial

$$(-(-a_1 - t_1)t_1 + t_2)X^2 + (-(-a_1 - t_1)t_2 + t_3)X^3 + \dots$$
is $\le H+2$. At the second step we add $0 = -(-a_2 - (-(-a_1-t_1)t_1+t_2))\gamma_s^{-2}S_{\gamma_s^*}(\gamma_s^{-1})$. Then we obtain

$$1 = -a_1\gamma_s^{-1} - a_2\gamma_s^{-2} - [(-a_2 - (-(-a_1-t_1)t_1+t_2))t_1 + (-(-a_1-t_1)t_2+t_3)]\gamma_s^{-3} + \dots$$

where the height of the polynomial

$$-[( -a_2 - ( -(-a_1 - t_1)t_1 + t_2) ) t_1 + ( -(-a_1 - t_1)t_2 + t_3) ] X^3 + \dots$$

is $\le H + (H+2) + (H+2) = 3H+4$. Iterating this process $d$ times we obtain

$$1 = -a_1\gamma_s^{-1} - a_2\gamma_s^{-2} - \dots - a_d\gamma_s^{-d} + \text{polynomial remainder in } \gamma_s^{-1}.$$

Denote by $V(\gamma_s^{-1})$ this polynomial remainder in $\gamma_s^{-1}$, for some $V(X) \in \mathbb{Z}[X]$, and $X$ specializing in $\gamma_s^{-1}$. If we denote the upper bound of the height of the polynomial remainder $V(X)$, at step $q$, by $\lambda_q H + v_q$, we readily deduce: $v_q = 2^q$, and $\lambda_{q+1} = 2\lambda_q + 1$, $q \ge 1$, with $\lambda_1 = 1$; then $\lambda_q = 2^q - 1$.

To summarize, the first rewriting polynomials of the sequence $(A'_q(X))_{q \ge 1}$ involved in this rewriting trail are

$$A'_1(X) = -1 - (-a_1 - t_1)X, \\ A'_2(X) = -1 - (-a_1 - t_1)X - (-a_2 - (-(-a_1 - t_1)t_1 + t_2))X^2, \text{ etc.}$$

For $q \ge \text{deg}(P_\beta)$, all the coefficients of $P_\beta$ are “restored”; denote by $(h_{q,j})_{j=0,1,\dots,s-1}$ the $s$-tuple of integers produced by this rewriting trail, at step $q$. It is such that

$$(4.23) \qquad A'_q(\gamma_s^{-1}) S_{\gamma_s^*}(\gamma_s^{-1}) = -P(\gamma_s^{-1}) + \gamma_s^{-q-1} \left( \sum_{j=0}^{s-1} h_{q,j} \gamma_s^{-j} \right).$$

Then take $q=d$. The (lhs) left-and side of (4.23) is equal to 0. Thus

$$P(\gamma_s^{-1}) = \gamma_s^{-d-1} \left( \sum_{j=0}^{s-1} h_{d,j} \gamma_s^{-j} \right) \implies P(\gamma_s) = \sum_{j=0}^{s-1} h_{d,j} \gamma_s^{-j-1}.$$

The height of the polynomial

$$(4.24) \qquad W(X) := \sum_{j=0}^{s-1} h_{d,j} X^{j+1} \quad \text{is} \quad \le (2^d - 1)H + 2^d,$$

and is independent of $s \ge W_v$.

For any $s \ge W_v$, let us observe that $-P_\beta(\gamma_s^{-1}) > 0$, and that the sequence $(\gamma_s^{-1})_s$ is decreasing. Indeed, the polynomial function $x \to P_\beta(x)$ is positive on $(0, \beta^{-1})$, vanishes at $\beta^{-1}$, and changes its sign for $x > \beta^{-1}$, so that $P_\beta(\gamma_s^{-1}) < 0$. We have: $\lim_{s \to \infty} P_\beta(\gamma_s^{-1}) = P_\beta(\beta^{-1}) = 0$.

Let us use the $\gamma_s$-transformation and the greedy (Rényi) $\gamma_s$-expansion of $-P_\beta(\gamma_s^{-1})$: there exists a unique sequence of integers $(\hat{t}_i)_{i \ge 1} \ne (0)$ in the alphabet $\{0,1\}$ such that

$$(4.25) \qquad -P_{\beta}(\gamma_{s}^{-1}) = \frac{\hat{t}_{1}}{\gamma_{s}} + \frac{\hat{t}_{2}}{\gamma_{s}^{2}} + \frac{\hat{t}_{3}}{\gamma_{s}^{3}} + \dots$$
The integers $\hat{t}_i$ are given by the $\gamma_s$-transformation $T_{\gamma_s} : [0, 1] \to [0, 1], x \to \{\gamma_s x\}$. Explicitly, the digits are

$$ \hat{t}_1 = [\gamma_s(-P_\beta(\gamma_s^{-1}))], $$

$$ \hat{t}_2 = [\gamma_s\{\gamma_s(-P_\beta(\gamma_s^{-1}))\}], $$

$$ \hat{t}_3 = [\gamma_s\{\gamma_s\{\gamma_s(-P_\beta(\gamma_s^{-1}))\}\}], \dots, \text{ and depend upon } \gamma_s. $$

Since $\lim_{s \to \infty} P_\beta(\gamma_s^{-1}) = P_\beta(\beta^{-1}) = 0$ there exists an increasing sequence $(u_s)_{s \ge W_v}$ of positive integers, satisfying $\hat{t}_1 = \hat{t}_2 = \dots = \hat{t}_{u_s-1} = 0$, $\hat{t}_{u_s} = 1$, such that the identity between $-P_\beta(\gamma_s^{-1})$ and its greedy expansion holds, as:

$$ (4.26) \qquad -P_{\beta}(\gamma_s^{-1}) = \frac{\hat{t}_{u_s}}{\gamma_s^{\mu_s}} + \frac{\hat{t}_{u_s+1}}{\gamma_s^{\mu_s+1}} + \frac{\hat{t}_{u_s+2}}{\gamma_s^{\mu_s+2}} + \dots $$

The sequence $(u_s)$ is defined by the bounds

$$ (4.27) \qquad |\beta^{u_s}(P_\beta(\gamma_s^{-1}))| \ge |\gamma_s^{\mu_s}(P_\beta(\gamma_s^{-1}))| \ge 1 $$

and

$$ |\gamma_s^{\mu_s}(P_\beta(\gamma_s^{-1}))| \le \frac{1}{1 - \gamma_s^{-1}}. $$

Therefore, in (4.26),

$$ (4.28) \qquad \hat{t}_i \in \{0, 1\}, \quad i \ge 1, \quad \text{and} \quad \lim_{s \to +\infty} u_s = +\infty. $$

Now the rhs of (4.26) belongs to $\mathbb{Q}(\gamma_s)$. For conjugating (4.26) by $\sigma_s$, if the image by $\sigma_s$ of the rhs of (4.26) belongs to $\mathbb{Q}(r_s)$, there are three cases for the conjugation of the rhs of (4.26):

(i) the rhs of (4.26) is finite (ends in infinitely many zeroes),

(ii-1) the rhs of (4.26) is eventually periodic (infinite and ultimately periodic),

(ii-2) the rhs of (4.26) is infinite and not eventually periodic.

**Case (i):** say that $\frac{\hat{t}_{u_s}}{\gamma_s^{\mu_s}} + \frac{\hat{t}_{u_s+1}}{\gamma_s^{\mu_s+1}} + \dots + \frac{\hat{t}_{u_s+N}}{\gamma_s^{\mu_s+N}}$ is the rhs of (4.26). Then its image by $\sigma_s$ is $\hat{t}_{u_s} r_s^{u_s} + \hat{t}_{u_s+1} r_s^{u_s+1} + \dots + \hat{t}_{u_s+N} r_s^{u_s+N}$ and we have the equality

$$ P_{\beta}(r_s) = \sigma_s \left( - \sum_{j=0}^{s_c-1} h_j'' \gamma_s^{-j-d-2} \right) = - \sum_{j=0}^{s_c-1} h_j'' r_s^{j+d+2} $$

$$ = \sigma_s \left( \frac{\hat{t}_{u_s}}{\gamma_s^{\mu_s}} + \frac{\hat{t}_{u_s+1}}{\gamma_s^{\mu_s+1}} + \dots + \frac{\hat{t}_{u_s+N}}{\gamma_s^{\mu_s+N}} \right) = \hat{t}_{u_s} r_s^{u_s} + \hat{t}_{u_s+1} r_s^{u_s+1} + \dots + \hat{t}_{u_s+N} r_s^{u_s+N}. $$

Conjugation by $\sigma_s$ is done term by term.

**Case (ii-1):** the rhs of (4.26) is eventually periodic. Let us write it

$$ = \frac{\hat{t}_{u_s}}{\gamma_s^{\mu_s}} + \frac{\hat{t}_{u_s+1}}{\gamma_s^{\mu_s+1}} + \dots + \frac{\hat{t}_{u_s+N}}{\gamma_s^{\mu_s+N}} + \sum_{i=0}^{\infty} \left( \frac{\hat{t}_{u_s+N+1}}{\gamma_s^{\mu_s+N+iq+1}} + \frac{\hat{t}_{u_s+N+2}}{\gamma_s^{\mu_s+N+iq+2}} + \dots + \frac{\hat{t}_{u_s+N+q}}{\gamma_s^{\mu_s+N+iq+q}} \right). $$
The period is not equal to zero. The period length is $q$. We have: $|r_s| = |\sigma_s(\gamma_s^{-1})| < 1$. Then it is equal to

$$ 
\begin{aligned}
&= \frac{\hat{t}_{u_s}}{\gamma_s^{u_s}} + \dots + \frac{\widehat{t_{u_s+N}}}{\gamma_s^{u_s+N}} + \sum_{i=0}^{\infty} \gamma_s^{-iq} \left( \frac{\widehat{t_{u_s+N+1}}}{\gamma_s^{u_s+N+1}} + \dots + \frac{\widehat{t_{u_s+N+q}}}{\gamma_s^{u_s+N+q}} \right) \\
&= \frac{\hat{t}_{u_s}}{\gamma_s^{u_s}} + \dots + \frac{\widehat{t_{u_s+N}}}{\gamma_s^{u_s+N}} + \frac{1}{1-\gamma_s^{-q}} \left( \frac{\widehat{t_{u_s+N+1}}}{\gamma_s^{u_s+N+1}} + \dots + \frac{\widehat{t_{u_s+N+q}}}{\gamma_s^{u_s+N+q}} \right)
\end{aligned}
$$

and its image by $\sigma_s$ is

$$ 
\begin{aligned}
&= \hat{t}_{u_s} r_s^{u_s} + \dots + \widehat{t_{u_s+N}} r_s^{u_s+N} + \frac{1}{1-r_s^q} \left( \widehat{t_{u_s+N+1}} r_s^{u_s+N+1} + \dots + \widehat{t_{u_s+N+q}} r_s^{u_s+N+q} \right) \\
&= \hat{t}_{u_s} r_s^{u_s} + \dots + \widehat{t_{u_s+N}} r_s^{u_s+N} + \sum_{i=0}^{\infty} \left( \widehat{t_{u_s+N+1}} r_s^{u_s+N+iq+1} + \dots + \widehat{t_{u_s+N+q}} r_s^{u_s+N+iq+q} \right).
\end{aligned}
$$

The series can be conjugated term by term by $\sigma_s$; in this case we have the identity

$$ P_{\beta}(r_s) = \sigma_s \left( \frac{\widehat{t_{u_s}}}{\gamma_s^{u_s}} + \frac{\widehat{t_{u_s+1}}}{\gamma_s^{u_s+1}} + \dots + \widehat{t_{u_s+q}} \right) = \widehat{t_{u_s}} r_s^{u_s} + \widehat{t_{u_s+1}} r_s^{u_s+1} + \dots $$

**Case (ii-2):** if the rhs of (4.26) is not eventually periodic its conjugation by $\sigma_s$ cannot be done term by term. This difficulty is overcome by enlarging the alphabet $\mathcal{A} = \{0, 1\}$ to a bigger alphabet $\mathcal{B}$ and by replacing the Rényi expansion by a $(\gamma_s, \mathcal{B})$-eventually periodic representation of $-P_{\beta}(\gamma_s^{-1})$. This method of enlargement of the alphabet [36] is made possible as a consequence of the Theorem of Kala-Vavra recalled in the next subsection. It has to be noted that Theorem 4.10 implies that we are in the domain of applicability of Kala-Vavra's Theorem, because $\gamma_s$ has no conjugates on the unit circle. It is the key point.

**4.3. Kala-Vavra Theorem and Galois identification.** Let us recall the definitions. The $(\delta, \mathcal{B})$-representations for a given $\delta \in \mathbb{C}$, $|\delta| > 1$ and a given alphabet $\mathcal{B} \subset \mathbb{C}$ finite, are expressions of the form $\sum_{k \ge -L} a_k \delta^{-k}$, $a_k \in \mathcal{B}$, for some integer $L$. We denote

$$ \mathrm{Per}_{\mathcal{B}}(\delta) := \{x \in \mathbb{C} : x \text{ has an eventually periodic } (\delta, \mathcal{B})\text{-representation}\}. $$

**Theorem 4.11 (Kala - Vavra [54], [5]).** Let $\delta \in \mathbb{C}$ be an algebraic number of degree $d$, $|\delta| > 1$, and $a_d x^d - a_{d-1} x^{d-1} - \dots - a_1 x - a_0 \in \mathbb{Z}[x]$, $a_0 a_d \ne 0$, be its minimal polynomial. Suppose that $|\delta'| \ne 1$ for any conjugate $\delta'$ of $\delta$, Then there exists a finite alphabet $\mathcal{B} \subset \mathbb{Z}$ such that

$$ \mathbb{Q}(\delta) = \mathrm{Per}_{\mathcal{B}}(\delta). $$

Let us apply Theorem 4.11 to $\delta = \gamma_s$. By Proposition 5 in [35] $\gamma_s$ has no conjugate of modulus 1. Therefore there exists a finite alphabet $\mathcal{B} \subset \mathbb{Z}$ such that the rhs of (4.26) be identified with a $(\gamma_s, \mathcal{B})$- representation which is eventually periodic, for some integer $u_s \in \mathbb{Z}$:

$$ (4.29) \qquad -P_{\beta}(\gamma_s^{-1}) = \frac{\widehat{t_{u_s}}}{\gamma_s^{u_s}} + \frac{\widehat{t_{u_s+1}}}{\gamma_s^{u_s+1}} + \frac{\widehat{t_{u_s+2}}}{\gamma_s^{u_s+2}} + \dots $$

Being eventually periodic, the representation (4.29) can now be conjugated term by term by $\sigma_s$, since $|\sigma_s(\gamma_s^{-1})| < 1$, as in case (ii-1). In (4.29) the digits $\hat{t}_i$ belong to a symmetrical
alphabet $\mathcal{B} = \{-m, \dots, 0, \dots, m\}$; the integer $m$ is provided by the rewriting trail, given by the rewriting trail and (4.24): we have $m = \lceil 2((2^d - 1)H + 2^d)/3 \rceil$.

Indeed, by Theorem 4.11 there exist a preperiod $R(X) \in \mathcal{B}[X]$, a period $T(X) \in \mathcal{B}[X]$ such that

$$ \hat{W}(\gamma_s^{-1}) := -P_{\beta}(\gamma_s) = R(\gamma_s^{-1}) + \gamma_s^{-\deg R-1} \sum_{j=0}^{\infty} \frac{1}{\gamma_s^{j(\deg T+1)}} T(\gamma_s^{-1}), $$

both polynomials $R$ and $T$ depending upon $s$. Since the relation

$$ S_{\gamma_s}(\gamma_s^{-1}) = -1 + t_1\gamma_s^{-1} + t_2\gamma_s^{-2} + \dots + t_{s-1}\gamma_s^{-s+1} + t_s\gamma_s^{-s} = 0 $$

holds, we may assume $\deg R \le s - 1$, $\deg T \le s - 1$. Then, for $X$ specialized at $\gamma_s^{-1}$, we have the identity

$$ (4.30) \qquad \hat{W}(X) = R(X) + X^L \frac{T(X)}{1-X^r} $$

for some positive integers $L, r$. The height of $(1-X^r)\hat{W}(X)$ is $\le 2((2^d - 1)H + 2^d)$ and, with $\mathcal{B}$ assumed $= \{-m, \dots, 0, \dots, +m\}$, the height of $(1-X^r)R(X) + X^L T(X)$ is less than $3m$. Therefore $m$ is $\le 2((2^d - 1)H + 2^d)/3$. We can take

$$ m = \lceil 2((2^d - 1)H + 2^d)/3 \rceil. $$

The alphabet $\mathcal{B} = \{-m, \dots, m\}$ only depends upon the degree $d$ and the height $H$ of the polynomial $P_{\beta}$, and does not depend upon $s$.

We now assume $0 \ne |P_{\beta}(\gamma_s)| \ll 1$. The $(\gamma_s, \mathcal{B})$-eventually periodic representation of $-P_{\beta}(\gamma_s)$ starts as

$$ -P_{\beta}(\gamma_s) = \hat{W}(\gamma_s^{-1}) = \frac{\hat{t}_{u_s}}{\gamma_s^{\mu_s}} + \frac{\hat{t}_{u_s+1}}{\gamma_s^{\mu_s+1}} + \frac{\hat{t}_{u_s+2}}{\gamma_s^{\mu_s+2}} + \dots, \quad \text{with } |\hat{t}_j| \le m, j = u_s, u_s+1, \dots $$

with $\hat{t}_{u_s} \ne 0$. The exponent $u_s$ appearing in the first term is defined in [42]; by Theorem 4, Remarks 5 to 7, in [42], there exists a positive real number $\kappa_{\gamma_s, \mathcal{B}} > 0$ such that $u_s$ is the minimal integer such that

$$ \gamma_s^{\mu_s-1} \ge \frac{\kappa_{\gamma_s, \mathcal{B}}}{|P(\gamma_s)|}. $$

Since $\lim_{s \to \infty} \gamma_s = \beta > 1$ and that the alphabet $\mathcal{B}$ does not depend upon $s$, from Theorem 4, Remarks 5 to 7, in [42], we can replace $\kappa_{\gamma_s, \mathcal{B}}$ by a constant $\kappa > 0$, independent of $s$ (cf also [36]). Thus

$$ \lim_{s \to \infty} u_s = +\infty. $$

The sequence $(u_s)$ is here defined by the bounds

$$ (4.31) \qquad |\beta^{u_s-1}(P_{\beta}(\gamma_s^{-1}))| \ge |\gamma_s^{\mu_s-1}(P_{\beta}(\gamma_s^{-1}))| \ge \kappa $$

and

$$ |\gamma_s^{\mu_s}(P_{\beta}(\gamma_s^{-1}))| \le \frac{m}{1-\gamma_s^{-1}}. $$

To the collection $(\gamma_s)_{s \ge W_v}$ is associated the collection $(\sigma_s : \gamma_s \to r_s)_{s \ge W_v}$ of $\mathbb{Q}$-automorphisms of $\mathbb{C}$. Now, for any $s \ge W_v$, let us conjugate the eventually periodic representation of
$-P_{\beta}(\gamma_s^{-1})$ by $\sigma_s$, term by term. For the cases (i) and (ii-1) we consider (4.26), and in the case (ii-2) we consider (4.29).

**Lemma 4.12.** Denote $c_{lent} = \frac{\pi|z_{1,n}|}{a_{max}}$. Then

$$s \geq W_v \quad \implies \quad |r_s| < 1 - \frac{c_{lent}}{n}.$$

*Proof.* The inequality readily comes from the proof of Theorem 4.8 since the condition of Rouché holds true. □

Using Lemma 4.12 we deduce:

**Case (i) and (ii-1):** with the minimal alphabet $\{-1, 0, +1\}$,

$$(4.32) \qquad |P_{\beta}(r_s)| \le |r_s|^{u_s} \frac{1}{1-|r_s|} \le \frac{n}{c_{lent}} (1-\frac{c_{lent}}{n})^{u_s},$$

**Case (ii-2):** with the alphabet $\mathcal{B} = \{-m, \dots, +m\}$,

$$(4.33) \qquad |P_{\beta}(r_s)| \le |r_s|^{u_s} \frac{m}{1-|r_s|} \le \frac{nm}{c_{lent}} (1-\frac{c_{lent}}{n})^{u_s}.$$

In both cases, $\lim_{s \to \infty} u_s = +\infty$. The rhs of (4.32), resp. of (4.33), tends to 0 if $s$ tends to infinity. We have: $\lim_{s \to \infty} P_{\beta}(r_s) = 0$. But $\omega_{1,n} = \lim_{s \to \infty} r_s$ and $z \to P_{\beta}(z)$ is continuous. Therefore there exists $s_0 \ge W_v$ such that $s \ge s_0 \implies |P_{\beta}(r_s)| < v/2$. Contradiction.

The only limit possibility is $P_{\beta}(\omega_{1,n}) = 0$. To summarize,

$$f_{\beta}(\omega_{1,n}) = 0 \quad \implies \quad P_{\beta}(\omega_{1,n}) = 0.$$

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UNIV. GRENOBLE ALPES, UNIV. SAVOIE MONT BLANC, LAMA, CNRS UMR 5127, F-73000 CHAMBÉRY, FRANCE

E-mail address: verger-gaugry.jean-louis@unsechat-math.fr