On small bases which admit points with two expansions

Abstract 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 ∈ [ 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 = 2 m the smallest base is q 2 ( M ) = m + 1 + m 2 + 2 m + 5 2 , and for M = 2 m − 1 the smallest base q 2 ( M ) is the largest positive root of x 4 = ( m − 1 ) x 3 + 2 m x 2 + m x + 1 . Moreover, for M = 2 we show that q 2 ( 2 ) is also the smallest base of B k ( 2 ) for all k ≥ 3 .