Topology of univoque sets in double-base expansions

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.