Metric results for numbers with multiple $q$-expansions

TL;DR

This paper investigates the Hausdorff dimension of real numbers with exactly j q-expansions, showing that for almost all q in a certain interval, the dimension is bounded and that the dimension function is discontinuous.

Contribution

It establishes bounds on the Hausdorff dimension of sets of numbers with fixed numbers of q-expansions and proves the discontinuity of the dimension function with respect to q.

Findings

For almost every q in (q_{KL}, M+1), the Hausdorff dimension of al{U}_q^j is bounded by , 2im_Hal{U}_q - 1.

The function q im_Hal{U}_q^j is discontinuous for all j ,3,...

The study extends understanding of the structure of numbers with multiple q-expansions and their dimensional properties.

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.