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arXiv · 2105.11608

Metric results for numbers with multiple $q$-expansions

Simon Baker·Yuru Zou
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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.

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Simon Baker, Yuru Zou. 2021-05-25. Metric results for numbers with multiple $q$-expansions. https://arxiv.org/abs/2105.11608

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