Periodic representations in algebraic bases

Abstract

We study periodic representations in number systems with an algebraic base $\beta$ (not a rational integer). We show that if $\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal{A}$ such that every element of $\mathbb{Q}(\beta )$ admits an eventually periodic representation with base $\beta$ and digits in $\mathcal{A}$.