Metric results for numbers with multiple q-expansions

Let M be a positive integer and q ∈ (1,M +1]. A q-expansion of a real number x is a sequence (c_i) = c₁c₂ · · · with c_i ∈ {0, 1, . . . ,M} such that x = [equation here]. In this paper we study the set U^j_q consisting of those real numbers having exactly j q-expansions. Our main result is that for Lebesgue almost every q ∈ (_qKL,M + 1), we have 

dim_H U^j_q ≤ max{0, 2 dim_H U_q − 1} for all j ∈ {2, 3, . . .}.

Here _qKL is the Komornik-Loreti constant. As a corollary of this result, we show that for any j ∈ {2, 3, . . .}, the function mapping q to dim_H U^j_q is not continuous.