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.
Categories: math.DS math.NT