Topology of univoque sets in real base expansions

TL;DR

This paper studies the topological structure of numbers with unique base-$q$ expansions and their digit sequences, generalizing previous results and providing simplified proofs for known theorems.

Contribution

It extends the understanding of univoque sets in real base expansions for arbitrary integer alphabets and offers shorter proofs of existing main results.

Findings

Characterization of the topological properties of $U_q$

Analysis of the combinatorial structure of $U_q'$

Simplified proofs of Baker's main results

Abstract

Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting the method given in [EJK] for the case $M=1$.