Une preuve de la conjecture d’Erdős, Joò et Komornik dans le cas des séries formelles

Abstract

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdös, Joò and Komornik in (Bull Soc Math France 118:377–390, 1990), is the study of the set $\Lambda _{m}(\beta )$ the spectrum of $\beta$ and the determination of $l^{m}(\beta )$ for Pisot number $\beta$, where $\Lambda _{m}(\beta )$ denotes the set of numbers having at least one representation of the form $\omega =\varepsilon _{n} \beta ^{n}+\varepsilon _{n-1}\beta ^{n-1}+\cdots +\varepsilon _{1}\beta +\varepsilon _{0},$ such that the $\varepsilon _{i}\in \{-m,\ldots ,0,\ldots ,m\}$, for all $0\le i\le n$, and $l^{m}(\beta )=\inf \{|\omega |:\omega \in \Lambda _{m},\omega \ne 0\}.$ In this paper, we consider $\Lambda _{m}(\beta )$, where $\beta$ is a formal power series over a finite field and the $\varepsilon _{i}$ are polynomials of degree at most $m$ for all $0\le i\le n$. Our main result is to give a full answer in the Laurent series case, to an old question of Erdős and Komornik (Acta Math Hungar 79:57–83, 1998), as to whether $l^{1}(\beta )=0$ for all non-Pisot numbers. More generally, we characterize the inequalities $l^{m}(\beta )>0$.