Purely periodic beta-expansions in the Pisot non-unit case

TL;DR

This paper extends the characterization of purely periodic expansions from decimal to Pisot non-unit bases, using a generalized Rauzy fractal in Euclidean and p-adic spaces.

Contribution

It introduces a new characterization of purely periodic beta-expansions for Pisot non-unit bases via an explicit generalized Rauzy fractal.

Findings

The generalized Rauzy fractal is a graph-directed self-affine compact set.

The set has positive measure in the combined Euclidean and p-adic spaces.

The characterization generalizes known results for Pisot units.

Abstract

It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and p-adic spaces.