Topological and dimensional properties of univoque bases in double-base expansions

TL;DR

This paper investigates the topological and dimensional characteristics of univoque bases in double-base expansions, revealing that such sets are meagre yet possess full Hausdorff dimension, indicating complex structure.

Contribution

It introduces new methods to analyze the topological and dimensional properties of univoque bases in double-base expansions, extending previous single-base studies.

Findings

The set of univoque bases is meagre.

The set of univoque bases has full Hausdorff dimension.

New research strategies are developed for this complex problem.

Abstract

Given two real numbers $q_0,q_1$ with $q_0, q_1 > 1$ satisfying $q_0+q_1 \ge q_0q_1$, we call a sequence $(d_i)$ with $d_i\in \{0,1\}$ a $(q_0,q_1)$-expansion or a double-base expansion of a real number $x$ if \[ x=\mathop{\sum}\limits_{i=1}^{\infty} \frac{d_{i}}{q_{d_1}q_{d_2}\cdots q_{d_i}}. \] When $q_0=q_1=q$, the set of univoque bases is given by the set of $q$'s such that $x = 1$ has exactly one $(q, q)$-expansion. The topological, dimensional and symbolic properties of such sets and their corresponding sequences have been intensively investigated. In our research, we study the topological and dimensional properties of the set of univoque bases for double-base expansions. This problem is more complicated, requiring new research strategies. Several new properties are uncovered. In particular, we show that the set of univoque bases in the double base setting is a meagre set with full Hausdorff dimension.