Let M be a positive integer and q ∈ (1,M +1]. A q-expansion of a real number x is a sequence (ci) = c1c2 · · · with ci ∈ {0, 1, . . . ,M} such that x = [equation here]. In this paper we study the set Ujq 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
dimH Ujq ≤ max{0, 2 dimH Uq − 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 dimH Ujq is not continuous.
Funding
Fractal Properties of Some Sets in Real Non-Integer Base Representation and Related Problems