On universal and periodic $\beta$-expansions, and the Hausdorff dimension of the set of all expansions

TL;DR

This paper investigates the topological and fractal properties of sets of $eta$-expansions, especially for Pisot bases, establishing conditions for finiteness and linking growth rates to Hausdorff dimension.

Contribution

It provides new necessary and sufficient conditions for the finiteness of certain $eta$-expansion sets and generalizes a theorem relating growth rate and Hausdorff dimension.

Findings

Characterization of when the set of $eta$-expansions is finite for Pisot bases

Conditions under which growth rate equals Hausdorff dimension

Explicit calculation methods for Hausdorff dimension of $eta$-expansion sets

Abstract

In this paper we study the topology of a set naturally arising from the study of $\beta$-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of $\beta$-expansions are equal and explicitly calculable.