Bases which admit exactly two expansions

Given a positive integer $m$, let $\Omega_m=\{0,1,\dots,m\}$, and let ${\mathcal B}_2(m)$ denote the set of bases $q\in(1,m+1]$ in which there exist numbers having precisely two $q$-expansions over the alphabet $\Omega _m$. Sidorov [23] firstly studied the set ${\mathcal B}_2(1)$ and raised some questions. Komornik and Kong [15] further investigated the set ${\mathcal B}_2(1)$ and partially answered Sidorov's questions. In the present paper, we consider the set ${\mathcal B}_2(m)$ for general positive integer $m$, and generalise the results obtained by Komornik and Kong.