Bases which admit exactly two expansions

Abstract: For a positive integer $m$ let $\Omega _m={0,1, \cdots , m}$ and \begin{align*} \mathcal B_2(m)=&\left {q\in(1,m+1]: \text{$\exists\; x\in [0, m/(q-1)]$ has exactly }\right. \ &\left. \text{two different $q$-expansions w.r.t. $\Omega _m$}\right }. \end{align*} Sidorov \cite{S} firstly studied the set $\mathcal B_2(1)$ and raised some questions. Komornik and Kong \cite{KK} further studied the set $\mathcal B_2(1)$ and answered partial 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.