Periodic representations in Salem bases

Abstract

We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb{Q}(\beta )$ with a finite alphabet of digits $\mathcal{A}$. Moreover, the classification of bases allowing that those representations have the so-called weak greedy property is given. The decision problem whether a given pair $(\beta ,\mathcal{A})$ allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.