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

TL;DR

This work extends the theory of univoque bases from single-base to double-base expansions with bases $(q_0,q_1)$ under the constraint $q_0+q_1\,\ge\,q_0q_1$. By developing a robust symbolic-analytic framework, the authors establish a bijection between the base-pair domain and symbolic pairs, and dissect the topology and geometry of the associated univoque sets. They prove that the set of univoque bases in the double-base setting is meagre yet possesses full Hausdorff dimension, with the main sets $_2$, $_2$-bar, and $_2$ forming an unbounded, nowhere dense hierarchy: $_2\subsetneq\overline{_2}\subsetneq\mathcal{V}_2$, where $_2$ is a countable union of Cantor sets and $\,_2$-bar has dimension $2$, while $_2$-bar$\,\backslash\u0002_2$ has dimension at least $1$. These results illuminate the intricate structure of exceptional bases in multi-base systems and pave the way for further dimension and measure investigations in generalized expansion settings.

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.

Figures (2)

• Figure 1: The maps $T_{q_0,0}$ and $T_{q_1,1}$ generate double-base expansions for $x \in [0,1/(q_1-1)]$ when $q_0 = 2$ and $q_1 = 3/2$.
• Figure 2: The red region represents $\mathcal{B}$, and the black curve represents $\mathcal{C}$.

Theorems & Definitions (80)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1
• Remark 1.7
• Theorem 2
• Theorem 3