On a problem of countable expansions

For a real number $q\in (1,2)$ and $x\in [0,1\mathrm{/}(q-1)]$, the infinite sequence $(d_i)$ is called a \emph{$q$-expansion} of $x$ if $$x=\sum _{i=1}^{\mathrm{\infty }}\frac{d_i}{q^i},d_i\in \{0,1\}\text{for all }i\ge 1.$$ For $m=1,2,\cdots$ or ${\mathrm{\aleph }}_0$ we denote by ${\mathcal{B}}_m$ the set of $q\in (1,2)$ such that there exists $x\in [0,1\mathrm{/}(q-1)]$ having exactly $m$ different $q$-expansions. It was shown by Sidorov (2009) that $q_2:=\min $\mathcal{B}$2\approx1.71064$,andlateraskedbyBaker(2015)whether$q_2\in$\mathcal{B}${\aleph_0}$? In this paper we provide a negative answer to this question and conclude that ${\mathcal{B}}_{{\mathrm{\aleph }}_0}$ is not a closed set. In particular, we give a complete description of $x\in [0,1\mathrm{/}(q_2-1)]$ having exactly two different $q_2$-expansions.