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 $(\delta_i){i=1}^\infty\in{0,1}^{\mathbb{N}}$ satisfying $$ x=\sum{i=1}^\infty\frac{\delta_i}{q^i}.$$ 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 $q_1(\approx1.64541)$ 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 $q_3(\approx1.68042)$, 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 $q_2(\approx1.65462)$ is the positive root of $x^6-2x^4-x^3-1=0$.