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 $(c_i)=c_1{c}_2\cdots$ with $c_i\in \{0,1,\dots ,M\}$ such that $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$. 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 $${\dim }_H{\mathcal{U}}_q^j\le \max \{0,2{\dim }_H{\mathcal{U}}_q-1\}\text{ for all }j\in \{2,3,\dots \}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $j\in \{2,3,\dots \},$ the function mapping $q$ to ${\dim }_H{\mathcal{U}}_q^j$ is not continuous.

20pages