Topology of univoque sets in double-base expansions

Vilmos Komornik; Yichang Li; Yuru Zou

paper

Topology of univoque sets in double-base expansions

arXiv:2504.21374

Abstract

Given two real numbers $q_{0}, q_{1} > 1$ satisfying $q_{0} + q_{1} \geq q_{0} q_{1}$ and two real numbers $d_{0} \neq = d_{1}$ , 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 $U_{q_{0}, q_{1}}$ the set of numbers $x$ having a unique expansion. The topological properties of $U_{q_{0}, q_{1}}$ 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.