# A non-trivial minoration for the set of Salem numbers

Jean-Louis Verger-Gaugry  Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, LAMA,
CNRS UMR 5127, F - 73000 Chambéry, France  verger-gaugry.jean-
<EMAIL_ADDRESS>
###### Abstract.

The set of Salem numbers is proved to be bounded from below by
$\theta_{31}^{-1}=1.08544\ldots$ where $\theta_{n}$, $n\geq 2$, is the unique
root in $(0,1)$ of the trinomial $-1+x+x^{n}$. Lehmer’s number
$1.176280\ldots$ belongs to the interval
$(\theta_{12}^{-1},\theta_{11}^{-1})$. We conjecture that there is no Salem
number in $(\theta_{31}^{-1},\theta_{12}^{-1})=(1.08544\ldots,1.17295\ldots)$.
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.

###### Key words and phrases:

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

###### 1991 Mathematics Subject Classification:

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

\- Dedicated to the 75th birthday of Christiane Frougny -

###### Contents

  1. 1 Introduction
  2. 2 Dynamical zeta function of the Rényi-Parry numeration dynamical system
  3. 3 Dynamics of the small Salem numbers known
  4. 4 Proof of the Main Theorem 1.1
    1. 4.1 A non-real lenticular pole of $\zeta_{\beta}(z)$ in a small angular sector
      1. 4.1.1 Roots of $G_{n}$ and the asymptotic expansion of the first root
      2. 4.1.2 Existence
    2. 4.2 Rewriting trails
    3. 4.3 Kala-Vavra Theorem and Galois identification

##  1\. Introduction

A Salem number is an algebraic integer $\beta>1$ such that its Galois
conjugates $\beta^{(i)}$ satisfy: $|\beta^{(i)}|\leq 1$ for all
$i=1,2,\ldots,m-1$, with $m={\rm deg}(\beta)\geq 4$, $\beta^{(0)}=\beta$ and
at least one conjugate $\beta^{(i)},i\neq 0$, 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 $\geq 4$. 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 ${\rm T}$ 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: $\beta\in{\rm T}\leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\ \beta\geq 1+c$? This question asks the
more general question of the topology of T and of its adherence
$\overline{{\rm T}}$: 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 $\theta^{(i)},i\neq 0$, 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 $i\neq 0$. The set of Pisot numbers admits the minorant
$\Theta=1.3247\ldots$, 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) $(X^{2}+1)P_{Salem}=XP_{Pisot}(X)+P^{*}_{Pisot}(X),$

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

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) $(z-1)P_{Salem}(z)=zP_{expansive}(z)-P^{*}_{expansive}(z),$

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) ${\rm Conjecture:}\qquad S\cup{\rm T}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{is closed}$

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

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

Only a part of them is proved. Indeed, Salem [93] proved that the set S of
Pisot numbers is closed, and ${\rm S}\subset\overline{{\rm T}}$. Its
successive derived sets ${\rm S}^{(i)}$, 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 $<\Theta=1.3247\ldots$, would be isolated.
The smallest Salem number known is Lehmer’s number

(1.5) $\tau:=1.17628\ldots,$

discovered by Lehmer in 1933 [63], with minimal polynomial (named “Lehmer’s
polynomial”):

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

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 $\leq 44$.

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 $n\geq 5$, 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 $(\theta_{n}^{-1})_{n\geq
5}$ 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 $(-1+x+x^{n})_{n\geq 5}$. In this note we
prove the following result.

######  Theorem 1.1 (ex-Lehmer conjecture for Salem numbers).

The set T is bounded from below:

$\beta\in{\rm
T}\qquad\Longrightarrow\qquad\beta>\theta_{31}^{-1}=1.08544\ldots$

Lehmer’s number $1.17628\ldots$ 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

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

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 developped 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 found in Lehmer’s paper [63] or
those from the list of Mossinghoff [75]. Inthere it is shown how to compute
readily $\zeta_{\beta}(z)$ from $d_{\beta}(1)$ for such Salem numbers $\beta$
in the interval $(\tau=1.176280\ldots,\Theta=\theta_{5}^{-1}=1.32\ldots)$.
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
$(x_{i})_{i\geq 1}$ of $\mathcal{A}^{\mathbb{N}}$ such that

$x=\sum_{i\geq 1}\,x_{i}\beta^{-i}\,.$

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

$T_{\beta}:\,\,x\,\mapsto\,\beta x\quad(\mod 1)=\\{\beta x\\}.$

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
$T_{\beta}^{0}:={\rm
Id},T_{\beta}^{1}:=T_{\beta},T_{\beta}^{i}:=T_{\beta}(T_{\beta}^{i-1})$ for
all $i\geq 1$, we have:

$d_{\beta}(x)=(x_{i})_{i\geq 1}\qquad\mbox{{\rm if and only if}}\qquad
x_{i}=\lfloor\beta T_{\beta}^{i-1}(x)\rfloor$

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

(2.1) $x\,=\,\cdot x_{1}x_{2}x_{3}\ldots\qquad\mbox{instead of}\qquad
x=\frac{x_{1}}{\beta}+\frac{x_{2}}{\beta^{2}}+\frac{x_{3}}{\beta^{3}}+\ldots.$

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

(2.2) $d_{\beta}(1)=0.t_{1}t_{2}t_{3}\ldots\qquad{\rm and\leavevmode\nobreak\
uniquely\leavevmode\nobreak\ corresponds\leavevmode\nobreak\ to}\qquad
1=\sum_{i=1}^{+\infty}t_{i}\beta^{-i}\,,$

where

(2.3)
$t_{1}=\lfloor\beta\rfloor,t_{2}=\lfloor\beta\\{\beta\\}\rfloor=\lfloor\beta
T_{\beta}(1)\rfloor,t_{3}=\lfloor\beta\\{\beta\\{\beta\\}\\}\rfloor=\lfloor\beta
T_{\beta}^{2}(1)\rfloor,\ldots$

Denote by $<_{lex}$ 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

$d_{\alpha}(1)=0.t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}\ldots,\qquad\leavevmode\nobreak\
i.e.\quad 1\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{t^{\prime}_{1}}{\alpha}+\frac{t^{\prime}_{2}}{\alpha^{2}}+\frac{t^{\prime}_{3}}{\alpha^{3}}+\ldots$

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

$d_{\beta}(1)=0.t_{1}t_{2}t_{3}\ldots,\qquad\leavevmode\nobreak\ i.e.\quad
1\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{t_{1}}{\beta}+\frac{t_{2}}{\beta^{2}}+\frac{t_{3}}{\beta^{3}}+\ldots,$

then $\alpha<\beta$ if and only if
$(t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3},\ldots)<_{lex}(t_{1},t_{2},t_{3},\ldots)$.

###### Proof.

Lemma 3 in Parry [77]. ∎

Since the sequence $(\theta_{n}^{-1})_{n\geq 2}$ is (strictly) decreasing and
tends to $1$ if $n$ tends to infinity [110], it induces the partitioning

(2.4) $\bigl{(}1,\frac{1+\sqrt{5}}{2}\,\bigr{]}\leavevmode\nobreak\
=\leavevmode\nobreak\
\bigcup_{n=2}^{\infty}\left[\,\theta_{n+1}^{-1},\theta_{n}^{-1}\,\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \bigcup\leavevmode\nobreak\ \leavevmode\nobreak\
\left\\{\theta_{2}^{-1}\right\\}.$

######  Definition 2.2.

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

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

$\beta\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{tends to
1}\qquad\mbox{if and only if}\qquad\quad n={\rm
dyg}(\beta)\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{tends to}+\infty.$

######  Definition 2.3.

Let $\beta\in(1,(1+\sqrt{5})/2]$ be a real number, and
$d_{\beta}(1)=0.t_{1}t_{2}t_{3}\ldots$ its Rényi $\beta$-expansion of 1. The
power series $f_{\beta}(z):=-1+\sum_{i\geq 1}t_{i}z^{i}$ of the complex
variable $z$ is called the Parry Upper function at $\beta$.

From $(t_{i})_{i\geq 1}\in\mathcal{A}^{\mathbb{N}}$ is built $(c_{i})_{i\geq
1}\in\mathcal{A}^{\mathbb{N}}$, defined by

$c_{1}c_{2}c_{3}\ldots:=\left\\{\begin{array}[]{ll}t_{1}t_{2}t_{3}\ldots&\quad\mbox{if
\leavevmode\nobreak\ $d_{\beta}(1)=0.t_{1}t_{2}\ldots$\leavevmode\nobreak\ is
infinite,}\\\ (t_{1}t_{2}\ldots t_{q-1}(t_{q}-1))^{\omega}&\quad\mbox{if
\leavevmode\nobreak\ $d_{\beta}(1)$\leavevmode\nobreak\ is finite,
\leavevmode\nobreak\ $=0.t_{1}t_{2}\ldots t_{q}$,}\end{array}\right.$

where $(\,)^{\omega}$ means that the word within $(\,)$ is indefinitely
repeated. The sequence $(c_{i})_{i\geq 1}$ 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 $(y_{i})_{i\geq 0}$ of elements of $\mathcal{A}$ (finite or not) is
said admissible if

(2.5)
$\sigma^{j}(y_{0},y_{1},y_{2},\ldots):=(y_{j},y_{j+1},y_{j+2},\ldots)<_{lex}\leavevmode\nobreak\
(c_{1},\,c_{2},\,c_{3},\,\ldots)\quad\mbox{for all}\leavevmode\nobreak\ j\geq
0.$

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

######  Definition 2.5.

A sequence $(a_{i})_{i\geq 0}\in\mathcal{A}^{\mathbb{N}}$ satisfying (2.6) is
said to be Lyndon (or self-admissible):

(2.6)
$\sigma^{n}(a_{0},a_{1},a_{2},\ldots)=(a_{n},a_{n+1},a_{n+2},\ldots)<_{lex}(a_{0},a_{1},a_{2},\ldots)\qquad\mbox{for
all}\leavevmode\nobreak\ n\geq 1.$

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

######  Theorem 2.6.

Let $n\geq 2$. 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) $d_{\beta}(1)=0.10^{n-1}10^{n_{1}}10^{n_{2}}10^{n_{3}}\ldots,$

with $n_{k}\geq n-1$ for all $k\geq 1$.

###### 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
$\theta_{n+1}^{-1}\leq\beta<\theta_{n}^{-1}$. 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

$u=1^{h_{0}}0^{n_{1}}1^{h_{1}}0^{n_{2}}1^{h_{2}}\ldots$

and $h_{0},n_{1},h_{1},n_{2},h_{2},\ldots$ integers $\geq 0$. The self-
admissibility lexicographic condition (2.6) applied to the sequence
$(1,0^{n-1},1^{1+h_{0}},0^{n_{1}},1^{h_{1}},0^{n_{2}},1^{h_{3}},\ldots)$,
which characterizes uniquely the base of numeration $\beta$, readily implies
$h_{0}=0$ and $\leavevmode\nobreak\ h_{k}=1$ and $n_{k}\geq n-1$ for all
$k\geq 1$. ∎

######  Definition 2.7.

A power series $\sum_{j=0}^{+\infty}a_{j}z^{j}$, with $a_{j}\in\\{0,1\\}$ for
all $j\geq 0$, $z$ the complex variable, is said to be Lyndon (or self-
admissible) if its coefficient vector $(a_{i})_{i\geq 0}$ 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 $n\geq 2$, denote: $G_{n}(X):=-1+X+X^{n}$.

######  Proposition 2.8.

For $1<\beta<(1+\sqrt{5})/2$ any real number, with
$d_{\beta}(1)=0.t_{1}t_{2}t_{3}\ldots$, 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) $f_{\beta}(z)=G_{{\rm
dyg}(\beta)}(z)+z^{m_{1}}+z^{m_{2}}+\ldots+z^{m_{q}}+z^{m_{q+1}}+\ldots$

with $m_{1}-{\rm dyg}(\beta)\geq{\rm dyg}(\beta)-1$, $m_{q+1}-m_{q}\geq{\rm
dyg}(\beta)-1$ for $q\geq 1$. Conversely, given a power series

(2.9) $-1+z+z^{n}+z^{m_{1}}+z^{m_{2}}+\ldots+z^{m_{q}}+z^{m_{q+1}}+\ldots$

with $n\geq 3$, $m_{1}-n\geq n-1$, $m_{q+1}-m_{q}\geq n-1$ for $q\geq 1$, then
there exists an unique $\beta\in(1,(1+\sqrt{5})/2)$ for which $n={\rm
dyg}(\beta)$ 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
$-1\,t_{1}\,t_{2}\,t_{3}\ldots$ has only one sign change. By Descartes’s rule
we obtain a contradiction. ∎

######  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 $\geq 0$ 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)
$\zeta_{\beta}(z):=\exp\Bigl{(}\sum_{n=1}^{\infty}\,\frac{\\#\\{x\in[0,1]\mid
T_{\beta}^{n}(x)=x\\}}{n}\,z^{n}\Bigr{)},$

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. ∎

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

and

(2.12) $(ii)\quad f_{\beta}(z)=-\frac{1-z^{N}}{\zeta_{\beta}(z)}\qquad\mbox{if
$\beta$ 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 $\\{z:|z|\leq 1/\beta\\}$ 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) $\zeta_{\beta}(z)=\frac{1-z^{N}}{(1-\beta
z)\Bigl{(}\sum_{n=0}^{\infty}T_{\beta}^{n}(1)\,z^{n}\Bigr{)}}$

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

(2.14) $-1+t_{1}z+t_{2}z^{2}+\ldots=f_{\beta}(z)=-(1-\beta
z)\Bigl{(}\sum_{n=0}^{\infty}T_{\beta}^{n}(1)\,z^{n}\Bigr{)}.$

The zeroes of smallest modulus are characterized in Lemma 5.2, Lemma 5.3 and
Lemma 5.4 in [41]. ∎

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 $\geq 1$ 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 non empty, in $(1,\infty)$ has a sense. ∎

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 $\geq 6$,
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 $\leq 8$ are
all greater than $1.280638\ldots$ 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 $\leq 44$, within the intervals of
extremities the Perron numbers $\theta_{n}^{-1},\,n=5,6,\ldots,12$. In each
interval $(\theta_{n}^{-1},\theta_{n-1}^{-1})$ the Salem numbers $\beta$ have
the same dynamical degree ${\rm dyg}(\beta)$, 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 $(\theta_{n}^{-1},\theta_{n-1}^{-1}),\,n\geq 6$, if any, should
be of degrees $>180$.

${\rm dyg}$ | $\deg$ | $\beta$ | $P_{\beta,P}$ | $d_{\beta}(1)$
---|---|---|---|---
5 | 3 | $\theta_{5}^{-1}=1.324717$ | 5 |  $0.10^{3}1$ smallest Pisot number - Siegel
6 | 18 | $1.29567$ | 22 | $0.1(0^{4}10^{9}10^{6})^{\omega}$
6 | 10 | $1.293485$ | 12 | $0.1(0^{4}10^{6})^{\omega}$
6 | 24 | $1.291741$ | 24 irr. | $0.1(0^{4}10^{11}10^{6})^{\omega}$
6 | 26 | $1.286730$ | 30 | $0.1(0^{4}10^{17}10^{6})^{\omega}$
6 | 34 | $1.285409$ | 38 | $0.1(0^{4}10^{25}10^{6})^{\omega}$
6 | 30 | $1.285235$ | 45 | $0.1(0^{4}10^{32}10^{6})^{\omega}$
6 | 44 | $1.285199$ | 66 | $0.1(0^{4}10^{54}10^{6})^{\omega}$
6 | 6 | $\theta_{6}^{-1}=1.285199$ | 6 irr. |  $0.10^{4}1$ Perron number
7 | 26 | $1.285196$ | 44 | $0.1(0^{5}10^{5}10^{5}10^{5}10^{5}10^{5}10^{7})^{\omega}$
7 | 26 | $1.281691$ | .. | $0.1(0^{5}10^{5}10^{9}10^{5}10^{17}10^{7}10^{6}10^{6}10^{7}10^{12})^{\omega}$
7 | 8 | $1.280638$ | 20 | $0.1(0^{5}10^{5}10^{7})^{\omega}$
7 | 10 | $1.261230$ | 14 | $0.1(0^{5}10^{7})^{\omega}$
7 | 24 | $1.260103$ | 28 | $0.1(0^{5}10^{13}10^{7})^{\omega}$
7 | 18 | $1.256221$ | 36 | $0.1(0^{5}10^{21}10^{7})^{\omega}$
7 | 7 | $\theta_{7}^{-1}=1.255422$ | 7 irr. |  $0.10^{5}1$ Perron number
8 | 18 | $1.252775$ | 120 | $0.1(0^{6}10^{6}10^{10}10^{16}10^{12}10^{7}10^{12}0^{16}10^{10}10^{6}10^{8})^{\omega}$
8 | 12 | $1.240726$ | 48 | $0.1(0^{6}10^{11}10^{7}10^{11}10^{8})^{\omega}$
8 | 20 | $1.232613$ | 41 | $0.1(0^{6}10^{24}10^{8})^{\omega}$
8 | 8 | $\theta_{8}^{-1}=1.232054$ | 8 irr. |  $0.10^{6}1$ Perron number
9 | 10 | $1.216391$ | 18 | $0.1(0^{7}10^{9})^{\omega}$
9 | 9 | $\theta_{9}^{-1}=1.213149$ | 9 irr. |  $0.10^{7}1$ Perron number
10 | 14 | $1.200026$ | 20 | $0.1(0^{8}10^{10})^{\omega}$
10 | 10 | $\theta_{10}^{-1}=1.197491$ | 10 irr. |  $0.10^{8}1$ Perron number
11 | 9 | $\theta_{11}^{-1}=1.184276$ | 11 |  $0.10^{9}1$ Perron number
12 | 10 | $1.176280$ | 75 |  Lehmer’s number $:0.1(0^{10}10^{18}10^{12}10^{18}10^{12})^{\omega}$
12 | 12 | $\theta_{12}^{-1}=1.172950$ | 12 irr. |  $0.10^{10}1$ Perron number

Table 1. Smallest Salem numbers $\beta<1.3$ of degree $\leq 44$, which are
Parry numbers, computed from the “list of Mossinghoff” [75] of irreducible
monic integer polynomials of degree $\leq 180$. 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.

The Salem numbers $\beta$, $\theta_{12}^{-1}<\beta<\theta_{5}^{-1}$, in Table
1, of degree $\leq 44$, are Parry numbers. The Parry polynomial
$P_{\beta,P}(X)$ of a Salem number $\beta$ is determined by the equation

$f_{\beta}(z)=\frac{-P^{*}_{\beta,P}(z)}{1-z^{p+1}}\qquad\mbox{if}\leavevmode\nobreak\
d_{\beta}(1)=0.t_{1}t_{2}\ldots t_{m}(t_{m+1}t_{m+2}\ldots
t_{m+p+1})^{\omega}\leavevmode\nobreak\ \mbox{has period
length}\leavevmode\nobreak\ p+1\geq 1.$

$\deg(\beta)$ | $\beta$ | minimal polynomial of $\beta$ | $d_{\beta}(1)$
---|---|---|---
4 | $1.722\ldots$ | $X^{4}-X^{3}-X^{2}-X+1$ | $0.1(100)^{\omega}$
6 | $1.401\ldots$ | $X^{6}-X^{4}-X^{3}-X^{2}+1$ | $0.1(0^{2}10^{4})^{\omega}$
8 | $1.2806\ldots$ | $X^{8}-X^{5}-X^{4}-X^{3}+1$ | $0.1(0^{5}10^{5}10^{7})^{\omega}$
10 | $1.17628\ldots$ | $X^{10}+X^{9}-X^{7}-X^{6}-X^{5}$ | $0.1(0^{10}10^{18}10^{12}10^{18}10^{12})^{\omega}$
|  | $-X^{4}-X^{3}+X+1$ |

Table 2. The small Salem numbers, with their minimal polynomial, discovered by
Lehmer in [63].

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 $\\{m_{n}\\}_{n\geq 1}$ and
$\\{s_{n}\\}_{n\geq 0}$ such that

$1=s_{0}\leq m_{1}<s_{1}\leq m_{2}<s_{2}\leq\ldots\leq m_{n}<s_{n}\leq
m_{n+1}<s_{n+1}\leq\ldots$

with $(s_{n}-m_{n})\geq 2$, $t_{m_{n}}\neq 0$, $t_{s_{n}}\neq 0$ and $t_{i}=0$
if $m_{n}<i<s_{n}$ for all $n\geq 1$. Then

(3.1) $\limsup_{n\to+\infty}\frac{s_{n}}{m_{n}}=1$

###### Proof.

There are two cases: (i) $\beta$ is a Parry number: we readily obtain the
result. (ii) $\beta$ is not a Parry number: then ${\rm M}(\beta)=\beta$, 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 $(\theta_{n}^{-1},\theta_{n-1}^{-1})$, $n\geq 32$, and arrive at a
contradiction.

###  4.1. A non-real lenticular pole of $\zeta_{\beta}(z)$ in a small angular
sector

This Salem number $\beta\in(\theta_{n}^{-1},\theta_{n-1}^{-1})$, $n\geq 32$,
would admit a Rényi $\beta$-expansion of unity $d_{\beta}(1)$ which is
infinite, by Proposition 2.8:

$d_{\beta}(1)=0.10^{n-2}10^{n_{1}}10^{n_{2}}10^{n_{3}}\ldots,$

with $n_{k}\geq n-2$ for all $k\geq 1$. The gappiness is controlled by the
integer $n-2$, in the sense that any string of zeroes has length $\geq n-2$,
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 $(m_{q})_{q\geq 0}$:

(4.1)
$f_{\beta}(z)=-1+z+z^{n}+z^{m_{1}}+z^{m_{2}}+z^{m_{3}}+\ldots=G_{n}(z)+\sum_{q\geq
1}z^{m_{q}},$

where $m_{0}:=n$, with the fundamental minimal gappiness condition:

(4.2) $m_{q+1}-m_{q}\geq n-1\qquad\mbox{for all}\leavevmode\nobreak\ q\geq 0.$

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 $\sum_{q\geq 1}z^{m_{q}}$ 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 $n\geq 2$. 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)
$G_{n}(X)=(X-\theta_{n})\,\left(\prod_{j=1}^{\lfloor\frac{n}{6}\rfloor}(X-z_{j,n})(X-\overline{z_{j,n}})\right)\times
q_{n}(X),$

where $\theta_{n}$ is the only (real) root of $G_{n}(X)$ in the interval
$(0,1)$, where

$q_{n}(X)\leavevmode\nobreak\ =\leavevmode\nobreak\
\left\\{\begin{array}[]{ll}\displaystyle\left(\prod_{j=1+\lfloor\frac{n}{6}\rfloor}^{\frac{n-2}{2}}(X-z_{j,n})(X-\overline{z_{j,n}})\right)\times(X-z_{\frac{n}{2},n})&\mbox{if
$n$ is even, with}\\\ &\mbox{$z_{\frac{n}{2},n}$ real $<-1$},\\\
\displaystyle\prod_{j=1+\lfloor\frac{n}{6}\rfloor}^{\frac{n-1}{2}}(X-z_{j,n})(X-\overline{z_{j,n}})&\mbox{if
$n$ is odd,}\end{array}\right.$

where the index $j=1,2,\ldots$ 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 $(\arg(z_{j,n}))_{1\leq j<\lfloor n/2\rfloor}$ forms a strictly
increasing sequence with $j$:

$0<\arg(z_{1,n})<\arg(z_{2,n})<\ldots<\arg(z_{\lfloor\frac{n}{2}\rfloor,n})\leq\pi.$

For $n\geq 2$ 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 $n\geq 2$. If $n\not\equiv 5\leavevmode\nobreak\ ({\rm
mod}\leavevmode\nobreak\ 6)$, then $G_{n}(X)$ is irreducible over
$\mathbb{Q}$. If $n\equiv 5\leavevmode\nobreak\ ({\rm mod}\leavevmode\nobreak\
6)$, 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 $n\geq 2$, all zeros $z_{j,n}$ and $\theta_{n}$ of the polynomials
$G_{n}(X)$ have a modulus in the interval

(4.4) $\Bigl{[}\leavevmode\nobreak\ 1-\frac{2\,{\rm
Log\,}n}{n},\leavevmode\nobreak\ 1+\frac{2\,{\rm
Log\,}2}{n}\leavevmode\nobreak\ \Bigr{]},$

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

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

(iii-1) $\theta_{n}\leavevmode\nobreak\ <\leavevmode\nobreak\ |z_{1,n}|$,

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

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

###### Proof.

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

######  Proposition 4.3.

Let $n\geq 2$. 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) $1+2\lfloor\frac{n}{6}\rfloor,$

(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) $j\in\\{1,2,\ldots,\lfloor\frac{n}{6}\rfloor\\}\leavevmode\nobreak\
\Longleftrightarrow\leavevmode\nobreak\
\Re(z_{j,n})>\frac{1}{2}\leavevmode\nobreak\
\Longleftrightarrow\leavevmode\nobreak\ |z_{j,n}|<1.$

###### 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: ${\rm Re}(z_{j,n})={\rm D}({\rm Re}(z_{j,n}))+{\rm
tl}({\rm Re}(z_{j,n}))$, ${\rm Im}(z_{j,n})={\rm D}({\rm Im}(z_{j,n}))+{\rm
tl}({\rm Im}(z_{j,n}))$, $\theta_{n}={\rm D}(\theta_{n})+{\rm
tl}(\theta_{n})$, 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 ${\rm M}(G_{n})$. 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 $n\geq 32$.

######  Proposition 4.4.

Let $n\geq 2$. The root $\theta_{n}$ can be expressed as: $\theta_{n}={\rm
D}(\theta_{n})+{\rm tl}(\theta_{n})$ with ${\rm D}(\theta_{n})=1-$

(4.7) $\frac{{\rm Log\,}n}{n}\left(1-\bigl{(}\frac{n-{\rm Log\,}n}{n\,{\rm
Log\,}n+n-{\rm Log\,}n}\bigr{)}\Bigl{(}{\rm Log\,}{\rm Log\,}n-n{\rm
Log\,}\Bigl{(}1-\frac{{\rm Log\,}n}{n}\Bigr{)}-{\rm Log}n\Bigr{)}\right)$

and

(4.8) ${\rm tl}(\theta_{n})\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{n}\,O\left(\left(\frac{{\rm Log\,}{\rm Log\,}n}{{\rm
Log\,}n}\right)^{2}\right),$

with the constant $1/2$ involved in $O\left(\leavevmode\nobreak\ \right)$.

###### Proof.

[110] Proposition 3.1. ∎

######  Lemma 4.5.

Given the limited expansion D$(\theta_{n})$ of $\theta_{n}$ as in (4.7),
denote

$\lambda_{n}:=1-(1-{\rm D}(\theta_{n}))\frac{n}{{\rm Log\,}n}.$

Then $\lambda_{n}={\rm D}(\lambda_{n})+{\rm tl}(\lambda_{n})$, with

(4.9) ${\rm D}(\lambda_{n})=\frac{{\rm Log\,}{\rm Log\,}n}{{\rm
Log\,}n}\left(\frac{1}{1+\frac{1}{{\rm Log\,}n}}\right),\qquad{\rm
tl}(\lambda_{n})=O\left(\frac{{\rm Log\,}{\rm Log\,}n}{n}\right)$

with the constant 1 in the Big O.

###### Proof.

[110] Lemma 3.2. ∎

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: $(n,j/{\rm Log\,}n)$ in the angular sector
$2\pi\,{\rm Log\,}n/n>\arg z>0$. It is the case for $j=1$.

######  Proposition 4.6.

Let $n\geq 18$. The first root $z_{1,n}$ of $G_{n}$ is given by:

${\rm D}({\rm Re}(z_{1,n}))=\theta_{n}+\frac{2\pi^{2}}{n}\left(\frac{1}{{\rm
Log\,}n}\right)^{2}\bigl{(}1+2\lambda_{n}\bigr{)},$ ${\rm D}({\rm
Im}(z_{1,n}))=\frac{2\pi{\rm Log\,}n}{n}\left(\frac{1}{{\rm
Log\,}n}\right)\left[1-\frac{1}{{\rm Log\,}n}(1+\lambda_{n})\right],$

with

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

###### Proof.

[110]. ∎

####  4.1.2. Existence

The Salem number $\beta\in(\theta_{n}^{-1},\theta_{n-1}^{-1})$, $n\geq 32$, is
uniquely characterized in (4.1) by the sequence of exponents $(m_{q})_{q\geq
0}$ in

(4.10)
$f_{\beta}(z)=-1+z+z^{n}+z^{m_{1}}+z^{m_{2}}+z^{m_{3}}+\ldots=G_{n}(z)+\sum_{q\geq
1}z^{m_{q}},$

with $m_{0}:=n$ and the distanciation condition: $m_{q+1}-m_{q}\geq
n-1\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{for
all}\leavevmode\nobreak\ q\geq 0.$ 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) $\left|f_{\beta}(z)-G_{n}(z)\right|=\left|\sum_{q\geq
1}z^{m_{q}}\right|<\left|G_{n}(z)\right|\qquad\mbox{for}\leavevmode\nobreak\
z\in C_{1,n},$

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:

$\left|f_{\beta}(z)-G_{n}(z)\right|=\left|\sum_{q\geq
1}z^{m_{q}}\right|\leq\sum_{q\geq
1}|z^{m_{q}}|\leq\frac{|z|^{2(n-1)+1}}{1-|z|^{n-1}},\qquad\quad|z|<1.$

Then we test the Rouché condition by the following inequality:

(4.12)
$\frac{|z|^{2(n-1)+1}}{1-|z|^{n-1}}=\frac{|z|^{2n-1}}{1-|z|^{n-1}}\leavevmode\nobreak\
<\leavevmode\nobreak\ \left|G_{n}(z)\right|\quad\mbox{for}\leavevmode\nobreak\
z\in C_{1,n},$

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

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

defined on $[1,+\infty)$ reaches its maximum $h(1,a_{\max}):=0.171784\ldots$
at $a_{\max}=5.8743\ldots$ (Figure 1).

Figure 1.  Curve of the Rouché condition $a\to h(1,a)$.

######  Theorem 4.8.

Let $n\geq 32$. Denote by
$C_{1,n}:=\\{z\mid|z-z_{1,n}|=\frac{\pi|z_{1,n}|}{n\,a_{\max}}\\}$ 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) $\frac{\left|z\right|^{2n-1}}{1-|z|^{n-1}}\leavevmode\nobreak\
<\leavevmode\nobreak\ \left|-1+z+z^{n}\right|,\qquad\mbox{for
all}\leavevmode\nobreak\ z\in C_{1,n},$

holds true.

###### Proof.

Let $a\geq 1$ and $n\geq 18$. Denote by $\varphi:=\arg(z_{1,n})$ the argument
of the first root $z_{1,n}$ (in ${\rm Im}(z)>0$). Since
$-1+z_{1,n}+z_{1,n}^{n}=0$, we have $|z_{1,n}|^{n}=|-1+z_{1,n}|$. Let us write
$z=z_{1,n}+\frac{\pi|z_{1,n}|}{n\,a}e^{i\psi}=z_{1,n}(1+\frac{\pi}{a\,n}e^{i(\psi-\varphi)})$
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

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

and

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

Moreover,

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

with

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

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

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

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

$|z_{1,n}|\,\left|1+\frac{\pi}{a\,n}e^{i(\psi-\varphi)}\right|=|z_{1,n}|.$

Then the left-hand side term of (4.13) is

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

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

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

$\left|-1+z+z^{n}\right|=\left|-1+z_{1,n}\Bigl{(}1+\frac{\pi}{n\,a}e^{i(\psi-\varphi)}\Bigr{)}+z_{1,n}^{n}\Bigl{(}1+\frac{\pi}{n\,a}e^{i(\psi-\varphi)}\Bigr{)}^{n}\right|$
$=\left|-1+z_{1,n}(1\pm
i\frac{\pi\sqrt{1-X^{2}}}{a\,n})(1+\frac{\pi\,X}{a\,n})+(1-z_{1,n})\left(1-\frac{\pi^{2}}{2a^{2}\,n}(2X^{2}-1)\right)\right.\hskip
85.35826pt\mbox{}$ (4.16)
$\left.\times\exp\bigl{(}\frac{\pi\,X}{a}\bigr{)}\,\exp\Bigl{(}\pm\,i\,\Bigl{(}\leavevmode\nobreak\
\frac{\pi\,\sqrt{1-X^{2}}}{a}[1-\frac{\pi\,X}{a\,n}]\Bigr{)}\Bigr{)}+O(\frac{1}{n^{2}})\right|$

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 $O(\frac{1}{n}({\rm
Log\,}{\rm Log\,}n/{\rm Log\,}n))$ \- terms or $O(1/n^{2})$ \- terms. (4.15)
becomes:

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

and (4.16) is equal to:

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

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

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

and (4.17) to

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

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)
$\frac{|-1+z_{1,n}|}{|z_{1,n}|}<\kappa(1,a)=\frac{\left|1-\exp\bigl{(}\frac{\pi}{a}\bigr{)}\right|\exp\bigl{(}\frac{-\pi}{a}\bigr{)}}{\exp\bigl{(}\frac{\pi}{a}\bigr{)}+\left|1-\exp\bigl{(}\frac{\pi}{a}\bigr{)}\right|},$

equivalently, from Proposition 4.6,

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

In order to obtain the largest possible range of values of $n$, the value of
$a\geq 1$ 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 $18\leq n\leq 100$ for
$a=a_{\max}$, or not. Indeed, for the large enough values of $n$, the
inequality (4.20) is satisfied since $\lim_{n\to\infty}\frac{{\rm Log\,}n-{\rm
Log\,}{\rm Log\,}n}{n}=0$. On the computer, the critical threshold of $n=32$
is easily calculated, with $({\rm Log\,}32-{\rm Log\,}{\rm
Log\,}32)/32=0.0694628\ldots$. Then

$\frac{{\rm Log\,}n-{\rm Log\,}{\rm
Log\,}n}{n}<\frac{\kappa(1,a_{\max})}{1+\kappa(1,a_{\max})}=0.146447\ldots\qquad\mbox{for
all $n\geq 32$}.$

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

######  Corollary 4.9.

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

###### 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
$\\{z:|z-z_{1,n}|<\frac{\pi|z_{1,n}|}{n\,a_{\max}}\\}$. We deduce the
existence of an unique simple zero of $f_{\beta}(z)$ inside this disk. ∎

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

$\\{\overline{z_{1,n}},\theta_{n},z_{1,n}\\},\leavevmode\nobreak\
\leavevmode\nobreak\ \\{\overline{\omega_{1,n}},\beta^{-1},\omega_{1,n}\\}$

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 $\beta$.

a)  b)

Figure 2. In a) and b) the zeroes of $G_{n}$ and $f_{\beta}$, 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 $f_{\beta}$ can be found in a
narrow annular neighbourhood of $|z|=1$.

###  4.2. Rewriting trails

Let us assume that $f_{\beta}(\omega_{1,n})=0$ with the property that the
minimal polynomial $P_{\beta}$ of $\beta$ satisfies
$P_{\beta}(\omega_{1,n})\neq 0$, and show the contradiction. Denote
$\nu:=|P_{\beta}(\omega_{1,n})|>0$. Let us consider the $s$-th polynomial
section $S_{s}(z)=-1+\sum_{j=1}^{s}t_{j}z^{j}$ of $f_{\beta}(z)$, 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
$S_{s}(\beta^{-1})=-1+\sum_{j=1}^{s}t_{j}\beta^{-j}=f_{\beta}(\beta^{-1})-\sum_{j=s+1}^{\infty}t_{j}\beta^{-j}=-\sum_{j=s+1}^{\infty}t_{j}\beta^{-j}<0$,
so that $\beta^{-1}<\gamma_{s}^{-1}$. There exists an integer, say $W_{\nu}$,
such that: $s\geq
W_{\nu}\Longrightarrow|P_{\beta}(\gamma_{s}^{-1})|<\min\\{1,\nu/2\\}.$ In the
following we take $s\geq W_{\nu}$.

Before going further, let us recall the general factorization of a Parry Upper
function $f_{\alpha}(z)$ for $\alpha$ any simple Parry number.

######  Theorem 4.10.

Let $\alpha$ be a simple Parry number, with ${\rm dyg}(\alpha)\geq 3$. Let
$f_{\alpha}$ denote its Parry Upper function with factorization in
$\mathbb{Z}[x]$:

$f_{\alpha}(x)=A(x)B(x)C(x)=-1+x+x^{n}+x^{m_{1}}+x^{m_{2}}+\ldots+x^{m_{s}},$

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 $C$ 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

$d:=\deg(P_{\beta})\qquad{\rm and}\qquad
H:=\max_{j=1,\ldots,d-1}\\{|a_{j}|\\}\geq 1$

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:

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

and

$f_{\beta}(1/\beta)=0\qquad\mbox{by the numeration dynamical system
and}\leavevmode\nobreak\ \zeta_{\beta}(z).$

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 $0=S_{\gamma_{s}}(\gamma_{s}^{-1})$ in the (rhs) right hand side. Then we
define the rewriting trail from the Rényi $\gamma_{s}^{-1}$-expansion of 1

(4.21)
$1=1+S_{\gamma_{s}}(\gamma_{s}^{-1})=t_{1}\gamma_{s}^{-1}+t_{2}\gamma_{s}^{-2}+\ldots+t_{s-1}\gamma_{s}^{-(s-1)}+t_{s}\gamma_{s}^{-s}$

(with $t_{1}=1,t_{2}=t_{3}=\ldots=t_{n-1}=0,t_{n}=1,$ etc) to

(4.22) $-a_{1}\gamma_{s}^{-1}-a_{2}\gamma_{s}^{-2}+\ldots-
a_{d-1}\gamma_{s}^{-(d-1)}-\gamma_{s}^{-d}=1-P_{\beta}(\gamma_{s}^{-1}),$

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

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

At the first step we add
$0=-(-a_{1}-t_{1})\gamma_{s}^{-1}S_{\gamma_{s}}^{*}(\gamma_{s}^{-1})$; and we
obtain

$1=-a_{1}\gamma_{s}^{-1}$
$+(-(-a_{1}-t_{1})t_{1}+t_{2})\gamma_{s}^{-2}+(-(-a_{1}-t_{1})t_{2}+t_{3})\gamma_{s}^{-3}+\ldots$

so that the height of the polynomial

$(-(-a_{1}-t_{1})t_{1}+t_{2})X^{2}+(-(-a_{1}-t_{1})t_{2}+t_{3})X^{3}+\ldots$

is $\leq H+2$. 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

$1=-a_{1}\gamma_{s}^{-1}-a_{2}\gamma_{s}^{-2}$
$-[(-a_{2}-(-(-a_{1}-t_{1})t_{1}+t_{2}))t_{1}+(-(-a_{1}-t_{1})t_{2}+t_{3})]\gamma_{s}^{-3}+\ldots$

where the height of the polynomial

$-[(-a_{2}-(-(-a_{1}-t_{1})t_{1}+t_{2}))t_{1}+(-(-a_{1}-t_{1})t_{2}+t_{3})]X^{3}+\ldots$

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

$1=-a_{1}\gamma_{s}^{-1}-a_{2}\gamma_{s}^{-2}-\ldots-a_{d}\gamma_{s}^{-d}$
$+\leavevmode\nobreak\ \leavevmode\nobreak\ polynomial\leavevmode\nobreak\
\leavevmode\nobreak\ remainder\leavevmode\nobreak\ \leavevmode\nobreak\
in\leavevmode\nobreak\ \leavevmode\nobreak\ \gamma_{s}^{-1}.$

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
$\lambda_{q+1}=2\lambda_{q}+1$, $q\geq 1$, with $\lambda_{1}=1$; then
$\lambda_{q}=2^{q}-1$.

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

$A^{\prime}_{1}(X)=-1-(-a_{1}-t_{1})X,$
$A^{\prime}_{2}(X)=-1-(-a_{1}-t_{1})X-(-a_{2}-(-(-a_{1}-t_{1})t_{1}+t_{2}))X^{2},\quad{\rm
etc}.$

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

(4.23)
$A^{\prime}_{q}(\gamma_{s}^{-1})S_{\gamma_{s}}^{*}(\gamma_{s}^{-1})=-P(\gamma_{s}^{-1})+\gamma_{s}^{-q-1}\Bigl{(}\sum_{j=0}^{s-1}h_{q,j}\gamma_{s}^{-j}\Bigr{)}.$

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

$P(\gamma_{s}^{-1})=\gamma_{s}^{-d-1}\Bigl{(}\sum_{j=0}^{s-1}h_{d,j}\gamma_{s}^{-j}\Bigr{)}\qquad\Longrightarrow\qquad
P(\gamma_{s})=\sum_{j=0}^{s-1}h_{d,j}\gamma_{s}^{-j-1}.$

The height of the polynomial

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

and is independent of $s\geq W_{v}$.

For any $s\geq W_{\nu},$ let us observe that $-P_{\beta}(\gamma_{s}^{-1})$ is
$>0$, 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 an
unique sequence of integers $(\widehat{t_{i}})_{i\geq 1}\not\equiv(0)$ in the
alphabet $\\{0,1\\}$ such that

(4.25)
$-P_{\beta}(\gamma_{s}^{-1})=\frac{\widehat{t_{1}}}{\gamma_{s}}+\frac{\widehat{t_{2}}}{\gamma_{s}^{2}}+\frac{\widehat{t_{3}}}{\gamma_{s}^{3}}+\ldots.$

The integers $\widehat{t_{i}}$ are given by the $\gamma_{s}$-transformation
$T_{\gamma_{s}}:[0,1]\to[0,1],x\to\\{\gamma_{s}x\\}$. Explicitely, the digits
are

$\widehat{t_{1}}=\lfloor\gamma_{s}(-P_{\beta}(\gamma_{s}^{-1}))\rfloor$,

$\widehat{t_{2}}=\lfloor\gamma_{s}\\{\gamma_{s}(-P_{\beta}(\gamma_{s}^{-1}))\\}\rfloor$,

$\widehat{t_{3}}=\lfloor\gamma_{s}\\{\gamma_{s}\\{\gamma_{s}(-P_{\beta}(\gamma_{s}^{-1}))\\}\\}\rfloor,\,\ldots$
, 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 $(u_{s})_{s\geq W_{\nu}}$ of positive
integers, satisfying
$\widehat{t_{1}}=\widehat{t_{2}}=\ldots=\widehat{t_{u_{s}-1}}=0$,
$\widehat{t_{u_{s}}}=1$, such that the identity between
$-P_{\beta}(\gamma_{s}^{-1})$ and its greedy expansion holds, as:

(4.26)
$-P_{\beta}(\gamma_{s}^{-1})=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\frac{\widehat{t_{u_{s}+2}}}{\gamma_{s}^{u_{s}+2}}+\ldots.$

The sequence $(u_{s})$ is defined by the bounds

(4.27)
$\bigl{|}\beta^{u_{s}}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\geq\bigl{|}\gamma_{s}^{u_{s}}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\geq
1$

and

$\bigl{|}\gamma_{s}^{u_{s}}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\leq\frac{1}{1-\gamma_{s}^{-1}}.$

Therefore, in (4.26),

(4.28) $\widehat{t_{i}}\in\\{0,1\\},\quad i\geq 1,\qquad{\rm
and}\qquad\lim_{s\to+\infty}u_{s}=+\infty.$

Now the lhs of (4.26) belongs to $\mathbb{Q}(\gamma_{s})$. For conjugating
(4.26) by $\sigma_{s}$, if the image by $\sigma_{s}$ of the lhs of (4.26)
belongs to $\mathbb{Q}(r_{s})$, there are three cases for the conjugation of
the rhs of (4.26):

  1. (i)

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

  2. (ii-1)

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

  3. (ii-2)

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

Case (i): say that
$\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\ldots+\frac{\widehat{t_{u_{s}+N}}}{\gamma_{s}^{u_{s}+N}}$
is the rhs of (4.26). Then its image by $\sigma_{s}$ is
$\widehat{t_{u_{s}}}\,r_{s}^{u_{s}}+\widehat{t_{u_{s}+1}}\,r_{s}^{u_{s}+1}+\ldots+\widehat{t_{u_{s}+N}}\,r_{s}^{u_{s}+N}$
and we have the equality

$P_{\beta}(r_{s})=\sigma_{s}\left(-\sum_{j=0}^{s_{c}-1}h^{\prime\prime}_{j}\,\gamma_{s}^{-j-d-2}\right)=-\sum_{j=0}^{s_{c}-1}h^{\prime\prime}_{j}\,r_{s}^{j+d+2}$
$=\sigma_{s}\left(\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\ldots+\frac{\widehat{t_{u_{s}+N}}}{\gamma_{s}^{u_{s}+N}}\right)=\widehat{t_{u_{s}}}r_{s}^{u_{s}}+\widehat{t_{u_{s}+1}}r_{s}^{u_{s}+1}+\ldots+\widehat{t_{u_{s}+N}}r_{s}^{u_{s}+N}.$

Conjugation by $\sigma_{s}$ is done term by term.

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

$=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\ldots+\frac{\widehat{t_{u_{s}+N}}}{\gamma_{s}^{u_{s}+N}}+\sum_{i=0}^{\infty}\left(\frac{\widehat{t_{u_{s}+N+1}}}{\gamma_{s}^{u_{s}+N+iq+1}}+\frac{\widehat{t_{u_{s}+N+2}}}{\gamma_{s}^{u_{s}+N+iq+2}}+\ldots+\frac{\widehat{t_{u_{s}+N+q}}}{\gamma_{s}^{u_{s}+N+iq+q}}\right).$

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

$=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\ldots+\frac{\widehat{t_{u_{s}+N}}}{\gamma_{s}^{u_{s}+N}}+\sum_{i=0}^{\infty}\gamma_{s}^{-iq}\left(\frac{\widehat{t_{u_{s}+N+1}}}{\gamma_{s}^{u_{s}+N+1}}+\ldots+\frac{\widehat{t_{u_{s}+N+q}}}{\gamma_{s}^{u_{s}+N+q}}\right)$
$=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\ldots+\frac{\widehat{t_{u_{s}+N}}}{\gamma_{s}^{u_{s}+N}}+\frac{1}{1-\gamma_{s}^{-q}}\left(\frac{\widehat{t_{u_{s}+N+1}}}{\gamma_{s}^{u_{s}+N+1}}+\ldots+\frac{\widehat{t_{u_{s}+N+q}}}{\gamma_{s}^{u_{s}+N+q}}\right)$

and its image by $\sigma_{s}$ is

$=\widehat{t_{u_{s}}}r_{s}^{u_{s}}+\ldots+\widehat{t_{u_{s}+N}}r_{s}^{u_{s}+N}+\frac{1}{1-r_{s}^{q}}\left(\widehat{t_{u_{s}+N+1}}r_{s}^{u_{s}+N+1}+\ldots+\widehat{t_{u_{s}+N+q}}r_{s}^{u_{s}+N+q}\right)$
$=\widehat{t_{u_{s}}}r_{s}^{u_{s}}+\ldots+\widehat{t_{u_{s}+N}}r_{s}^{u_{s}+N}+\sum_{i=0}^{\infty}\left(\widehat{t_{u_{s}+N+1}}r_{s}^{u_{s}+N+iq+1}+\ldots+\widehat{t_{u_{s}+N+q}}r_{s}^{u_{s}+N+iq+q}\right).$

The series can be conjugated term by term by $\sigma_{s}$; in this case we
have the identity

$P_{\beta}(r_{s})=\sigma_{s}\left(\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\ldots+\right)=\widehat{t_{u_{s}}}r_{s}^{u_{s}}+\widehat{t_{u_{s}+1}}r_{s}^{u_{s}+1}+\ldots.$

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
$\sum_{k\geq-L}a_{k}\delta^{-k}$, $a_{k}\in\mathcal{B}$, for some integer $L$.
We denote

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

######  Theorem 4.11 (Kala - Vavra [54], [5]).

Let $\delta\in\mathbb{C}$ be an algebraic number of degree $d$, $|\delta|>1$,
and $a_{d}x^{d}-a_{d-1}x^{d-1}-\ldots-a_{1}x-a_{0}\in\mathbb{Z}[x]$,
$a_{0}a_{d}\neq 0$, be its minimal polynomial. Suppose that
$|\delta^{\prime}|\neq 1$ for any conjugate $\delta^{\prime}$ of $\delta$,
Then there exists a finite alphabet $\mathcal{B}\subset\mathbb{Z}$ such that

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

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 lhs 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)
$-P_{\beta}(\gamma_{s}^{-1})=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\frac{\widehat{t_{u_{s}+2}}}{\gamma_{s}^{u_{s}+2}}+\ldots\leavevmode\nobreak\
\leavevmode\nobreak\ .$

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 $\widehat{t_{i}}$ belong to a symmetrical
alphabet $\mathcal{B}=\\{-m,\ldots,0,\ldots,m\\}$; 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

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

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

$S_{\gamma_{s}}(\gamma_{s}^{-1})=-1+t_{1}\gamma_{s}^{-1}+t_{2}\gamma_{s}^{-2}+\ldots+t_{s-1}\gamma_{s}^{-s+1}+t_{s}\gamma^{-s}=0$

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

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

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

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

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

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

$-P_{\beta}(\gamma_{s})=\widehat{W}(\gamma_{s}^{-1})=\frac{\widehat{t_{u_{s}}}}{\gamma_{s}^{u_{s}}}+\frac{\widehat{t_{u_{s}+1}}}{\gamma_{s}^{u_{s}+1}}+\frac{\widehat{t_{u_{s}+2}}}{\gamma_{s}^{u_{s}+2}}+\ldots,\qquad{\rm
with}\leavevmode\nobreak\ |\widehat{t_{j}}|\leq m,j=u_{s},u_{s}+1,\ldots$

with $\,\widehat{t_{u_{s}}}\neq 0$. 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

$\gamma_{s}^{u_{s}-1}\geq\frac{\kappa_{\gamma,_{s}\mathcal{B}}}{|P(\gamma_{s})|}.$

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)
$\bigl{|}\beta^{u_{s}-1}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\geq\bigl{|}\gamma_{s}^{u_{s}-1}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\geq\kappa$

and

$\bigl{|}\gamma_{s}^{u_{s}}(P_{\beta}(\gamma_{s}^{-1}))\bigr{|}\leq\frac{m}{1-\gamma_{s}^{-1}}.$

To the collection $(\gamma_{s})_{s\geq W_{\nu}}$ is associated the collection
$(\sigma_{s}:\gamma_{s}\to r_{s})_{s\geq W_{\nu}}$ of
$\mathbb{Q}$-automorphisms of $\mathbb{C}$. Now, for any $s\geq W_{\nu}$, 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

$s\geq W_{\nu}\qquad\Longrightarrow\qquad|r_{s}|<1-\frac{c_{lent}}{n}.$

###### 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)
$\bigl{|}P_{\beta}(r_{s})\bigr{|}\leq|r_{s}|^{u_{s}}\frac{1}{1-|r_{s}|}\leq\frac{n}{c_{lent}}(1-\frac{c_{lent}}{n})^{u_{s}},$

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

(4.33)
$\bigl{|}P_{\beta}(r_{s})\bigr{|}\leq|r_{s}|^{u_{s}}\frac{m}{1-|r_{s}|}\leq\frac{n\,m}{c_{lent}}(1-\frac{c_{lent}}{n})^{u_{s}}.$

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 $s_{0}\geq W_{\nu}$ such that $s\geq
s_{0}\Longrightarrow\bigl{|}P_{\beta}(r_{s})\bigr{|}<\nu/2$. Contradiction.

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

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

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