Bases in which some numbers have exactly two expansions

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.