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.

Figures (5)

• Figure 1: The functions $\mathcal{G}(q_0)$ (blue), $\mathcal{K}(q_0)$ (red), and the curves $(q_0{-}1)(q_1{-}1) = \frac{1}{q_0+1}, \frac{1}{q_1+1}, \frac{1}{2}, \frac{q_1}{q_1+1}, \frac{q_0}{q_0+1}, 1$.
• Figure 2: The maps $(q_0{-}1)(\mathcal{G}(q_0){-}1)$ (blue), $(q_0{-}1)(\mathcal{K}(q_0){-}1)$ (red).
• Figure 3: The cardinality of $\Omega_{\mathbf{a},\mathbf{b}}$ according to Theorem \ref{['t:lex']}, for $\sigma = \tilde{\sigma} M$ with $\tilde{\sigma} \in \{L,R\}^*$ (left) and $\tilde{\sigma} \in \{L,M,R\}^*$ (right). In the regions with question marks, we have to consider substitutions starting with $\tilde{\sigma} L$ (in the lower left corners), with $\tilde{\sigma} R$ (in the upper right corners) and with $\tilde{\sigma} M$ (in the middle of the right picture).
• Figure 4: The maps $T_{0,q_0}$ and $T_{1,q_1}$; here, $q_0=2$ and $q_1=3/2$.
• Figure 5: Some values of $s$ and some intervals $[\sigma(\overline{0}), \sigma(0\overline{1})]$, $[\sigma(1\overline{0}), \sigma(\overline{1})]$, $\sigma \in \{L,R\}^*M$.

Theorems & Definitions (42)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Example 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3