Topology of univoque sets in double-base expansions
Vilmos Komornik, Yichang Li, Yuru Zou
TL;DR
This work extends the theory of univoque sets from equal-base representations to double-base expansions with $Q=(q_0,q_1)$ under $q_0+q_1\ge q_0q_1$. By deploying greedy, quasi-greedy, lazy, and quasi-lazy expansions and introducing the parameters $r_Q$ and $\ell_Q$ together with their $\alpha(Q)$ and $\mu(Q)$, the authors classify the topological and combinatorial structure of $\mathcal{U}_Q$ and $\mathcal{V}_Q$, including the partitions $A_Q$ and $B_Q$. They prove that for $Q$ in the doubly-infinite regime $\mathcal{V}$, the set $\mathcal{V}_Q\setminus\mathcal{U}_Q$ is countably infinite and dense in $\mathcal{V}_Q$, with detailed counts of expansions (2, 3, or $\aleph_0$) depending on the regime; they also analyze when $\mathcal{U}_Q$ is closed, when $\overline{\mathcal{U}}_Q=\mathcal{V}_Q$, and when $\mathcal{V}_Q$ fails to be a Cantor set. The paper culminates with a broad suite of examples that realize all twelve cases, revealing new phenomena unique to the double-base setting and laying groundwork for further generalizations to multi-base systems.
Abstract
Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*} x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{i_1}q_{i_2}\cdots q_{i_k}}. \end{equation*} We denote by $\mathcal{U}_{q_0,q_1}$ the set of numbers $x$ having a unique expansion. The topological properties of $\mathcal{U}_{q_0,q_1}$ have been investigated in the equal-base case $q_0=q_1$ for a long time. We extend this research to the case $q_0\neq q_1$. While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions.