On small bases which admit countably many expansions

Abstract

Let $q\in (1,2)$ and $x\in [0,\frac{1}{q-1}]$. We say that a sequence $(\hat{I}{\mu }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if x=\sum_{i=1}^{\infty}ε_iq^{-i}. Let ${\mathcal{B}}_{{\aleph }_0}$ denote the set of $q$ for which there exists $x$ with exactly ${\aleph }_0$ expansions in base $q$. In \cite{EHJ} it was shown that $\min {\mathcal{B}}_{{\aleph }_0}=\frac{1+\sqrt{5}}{2}.$ In this paper we show that the smallest element of ${\mathcal{B}}_{{\aleph }_0}$ strictly greater than $\frac{1+\sqrt{5}}{2}$ is $q_{{\aleph }_0}\approx 1.64541$, the appropriate root of $x^6=x^4+x^3+2x^2+x+1$. This leads to a full dichotomy for the number of possible $q$-expansions for $q\in (\frac{1+\sqrt{5}}{2},q_{{\aleph }_0})$. We also prove some general results regarding ${\mathcal{B}}_{{\aleph }_0}\cap [\frac{1+\sqrt{5}}{2},q_f],$ where $q_f\approx 1.75488$ is the appropriate root of $x^3=2x^2-x+1.$ Moreover, the techniques developed in this paper imply that if $x\in [0,\frac{1}{q-1}]$ has uncountably many $q$-expansions then the set of $q$-expansions for $x$ has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case.