On small bases which admit points with two expansions

Given two positive integers M and k, let B-k(M) be the set of bases q > 1 such that there exists a real number x epsilon [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 q(2)(M) of B-2(M). We prove that for M=2m the smallest base is q(2)(M) = m+1+root m(2)+2m+5/2, and for M = 2m - 1 the smallest base q(2)(M) is the largest positive root of x(4)=(m - 1)x(3) + 2mx(2) + mx+1. Moreover, for M = 2 we show that q(2)(2) is also the smallest base of B-k(2) for all k >= 3. (C) 2016 Elsevier Inc. All rights reserved.