Metric results for numbers with multiple $q$-expansions

Abstract: Let $M$ be a positive integer and $q\in (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $c_i\in {0,1,\ldots, M}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. In this paper we study the set $\mathcal{U}q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $q\in (q{KL}, M+1), $ we have $$\dim_{H}\mathcal{U}{q}^{j}\leq \max{0, 2\dim_H\mathcal{U}_q-1}\text{ for all } j\in{2,3,\ldots}.$$ Here $q{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $j\in{2,3,\ldots},$ the function mapping $q$ to $\dim_{H}\mathcal{U}_{q}^{j}$ is not continuous.