Unique double base expansions

Abstract

For two real bases $q_0, q_1 > 1$, we consider expansions of real numbers of the form $\sum_{k=1}^{\infty} i_k/(q_{i_1}q_{i_2}\cdots q_{i_k})$ with $i_k \in \{0,1\}$, which we call $(q_0,q_1)$-expansions. A sequence $(i_k)$ is called a unique $(q_0,q_1)$-expansion if all other sequences have different values as $(q_0,q_1)$-expansions, and the set of unique $(q_0,q_1)$-expansions is denoted by $U_{q_0,q_1}$. In the special case $q_0 = q_1 = q$, the set $U_{q,q}$ is trivial if $q$ is below the golden ratio and uncountable if $q$ is above the Komornik--Loreti constant. The curve separating pairs of bases $(q_0, q_1)$ with trivial $U_{q_0,q_1}$ from those with non-trivial $U_{q_0,q_1}$ is the graph of a function $\mathcal{G}(q_0)$ that we call generalized golden ratio. Similarly, the curve separating pairs $(q_0, q_1)$ with countable $U_{q_0,q_1}$ from those with uncountable $U_{q_0,q_1}$ is the graph of a function $\mathcal{K}(q_0)$ that we call generalized Komornik--Loreti constant. We show that the two curves are symmetric in $q_0$ and $q_1$, that $\mathcal{G}$ and $\mathcal{K}$ are continuous, strictly decreasing, hence almost everywhere differentiable on $(1,\infty)$, and that the Hausdorff dimension of the set of $q_0$ satisfying $\mathcal{G}(q_0)=\mathcal{K}(q_0)$ is zero. We give formulas for $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ for all $q_0 > 1$, using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of $S$-adic sequences including Sturmian and the Thue--Morse sequences are simpler than those of Labarca and Moreira (2006) and Glendinning and Sidorov (2015), and are relevant also for other open dynamical systems.