Univoque graphs and multiple expansions

TL;DR

This paper characterizes the sequences generated by univoque graphs for non-integer bases and studies the properties of numbers with multiple expansions, revealing that certain sets are empty or have dimension independent of the number of expansions.

Contribution

It provides a detailed description of sequences from univoque graphs and analyzes the structure and Hausdorff dimension of sets of numbers with multiple expansions.

Findings

Sets of numbers with exactly j expansions are empty or have specific properties.

Hausdorff dimension of sets with j expansions is independent of j for many bases.

Characterization of sequences generated by univoque graphs.

Abstract

Unique expansions in non-integer bases $q$ have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description of the set of sequences generated by these graphs. Using the description of univoque graphs, the second part of the paper is devoted to the study of multiple expansions. Contrary to the unique expansions, we prove for each $j\ge 2$ that the set $U_q^j$ of numbers having exactly $j$ expansions is closed only if it is empty. Furthermore, generalizing an important example of Sidorov, we prove for a large class of bases that the Hausdorff dimension of $U_q^j$ is independent of $j$. In the last two sections our results are illustrated by many examples.