On small bases which admit points with two expansions

Given two positive integers M and k, let Bk(M) be the set of bases q>1 such that there exists a real number x∈[0,M/(q−1)] having precisely k different q-expansions over the alphabet {0,1,…,M}. In this paper we consider k=2 and investigate the smallest base q2(M) of B2(M). We prove that for M=2m the smallest base isq2(M)=m+1+m2+2m+52, and for M=2m−1 the smallest base q2(M) is the largest positive root ofx4=(m−1)x3+2mx2+mx+1. Moreover, for M=2 we show that q2(2) is also the smallest base of Bk(2) for all k≥3.