Univoque bases and Hausdorff dimension

Given a positive integer M and a real number q> 1 , a q-expansion of a real number x is a sequence (c_i) = c₁c₂… with (c_i) ∈ { 0 , … , M} ^∞ such that (Formula presented.) It is well known that if q∈ (1 , M+ 1 ] , then each x∈ I_q: = [0 , M/ (q- 1) ] has a q-expansion. Let U= U(M) be the set of univoque basesq> 1 for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of U and to show that the Hausdorff dimension of the set of numbers x∈ I_q with a unique q-expansion changes the most if q “crosses” a univoque base. Denote by B₂= B₂(M) the set of q∈ (1 , M+ 1 ] such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741–754, 2009) and prove that (Formula presented.) where q^′= q^′(M) is the Komornik–Loreti constant.