Soit $\beta$ un nombre de Pisot ; nous montrons que pour tout entier $n$ assez grand il existe une matrice carrée à coefficients positifs ou nuls dont l’ordre est égal au degré de $\beta$ et dont ${\beta }^n$ est valeur propre.Soit $\beta =a_1/\beta +a_2/{\beta }^2+\cdots +a_n/{\beta }^n+\cdots$ le $\beta$-développement de $\beta$ ; si $\beta$ est un nombre de Pisot, alors la suite ${\left(a_n\right)}_{n\ge 1}$ est périodique après un certain rang $n_0$ (pour $n\ge n_0$, $a_{n+k}=a_n$) et le polynôme$$X^{n_0+k}-(a_1{X}^{n_0+k-1}+\cdots +a_{n_0+k})-(X^{n_0}-(a_1{X}^{n_0}+\cdots +a_{n_0}))$$est appelé polynôme de Parry. Nous montrons qu’il existe un ensemble relativement dense d’entiers $n$ tels que le polynôme minimal de ${\beta }^n$ est égal à son polynôme...

We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1+x^{r₁}+\cdots +x^{r₅}$, where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2r_j<r_{j+1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct value of...

The main result of this paper implies that for every positive integer $d⩾2$ there are at least ${(d-3)}^2/2$ nonconjugate algebraic numbers which have their Mahler measures lying in the interval $(1,2)$. These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.

Let $q$ be a complex number, $m$ be a positive rational integer and $l_m\left(q\right)=inf\{\left|P\left(q\right)\right|,P\in {\mathbb{Z}}_m\left[X\right],P\left(q\right)\ne 0\}$, where ${\mathbb{Z}}_m\left[X\right]$ denotes the set of polynomials with rational integer coefficients of absolute value $\le m$. We determine in this note the maximum of the quantities $l_m\left(q\right)$ when $q$ runs through the interval $]m,m+1[$. We also show that if $q$ is a non-real number of modulus $\&gt;1$, then $q$ is a complex Pisot number if and only if $l_m\left(q\right)\&gt;0$ for all $m$.

Let $\beta \&gt;1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $\&gt;1$ with classes called Q...

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers $\beta$ and $\gamma$ respectively, such that $\beta$ and $\gamma$ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.