On small bases for which $1$ has countably many expansions

Abstract

Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(\hat{I}{´}_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ satisfying $$x=\sum _{i=1}^{\infty }\frac{\hat{I}{´}_i}{q^i}.$$ Let ${\mathcal{B}}_{{\aleph }_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let ${\mathcal{B}}_{1,{\aleph }_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \cite{Sidorov6} it was shown that $\min {\mathcal{B}}_{{\aleph }_0}=\min {\mathcal{B}}_{1,{\aleph }_0}=\frac{1+\sqrt{5}}{2},$ and in \cite{Baker} it was shown that ${\mathcal{B}}_{{\aleph }_0}\cap (\frac{1+\sqrt{5}}{2},q_1]=\{q_1\}$, where $q_1(\approx 1.64541)$ is the positive root of $x^6-x^4-x^3-2x^2-x-1=0$. In this paper we show that the second smallest point of ${\mathcal{B}}_{1,{\aleph }_0}$ is $q_3(\approx 1.68042)$, the positive root of $x^5-x^4-x^3-x+1=0$. Enroute to proving this result we show that ${\mathcal{B}}_{{\aleph }_0}\cap (q_1,q_3]=\{q_2,q_3\}$, where $q_2(\approx 1.65462)$ is the positive root of $x^6-2x^4-x^3-1=0$.

14 pages, 2 figures