  In this paper we answer several questions raised by Sidorov on the set
$\mathcal B_2$ of bases in which there exist numbers with exactly two
expansions. In particular, we prove that the set $\mathcal B_2$ is closed, and
it contains both infinitely many isolated and accumulation points in $(1,
q_{KL})$, where $q_{KL}\approx 1.78723$ is the Komornik-Loreti constant.
Consequently we show that the second smallest element of $\mathcal B_2$ is the
smallest accumulation point of $\mathcal B_2$. We also investigate the higher
order derived sets of $\mathcal B_2$. Finally, we prove that there exists a
$\delta>0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL},
q_{KL}+\delta))<1, \end{equation*} where $\dim_H$ denotes the Hausdorff
dimension.
