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arXiv · 1908.00224

Arithmetic on self-similar sets

Bing Zhao·Xiaomin Ren·Jiali Zhu·Kan Jiang
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Abstract

Let $K_1$ and $K_2$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by $$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in (K_1\times K_2)\cap U\}. $$ In this paper we give an sufficient condition which guarantees that $f_{U}(K_1,K_2)$ contains some interiors. Our result is different from Simon and Taylor's \cite[Proposition 2.9]{ST} as we do not need the condition that the multiplication of the thickness of $K_1$ and $K_2$ is strictly greater than $1$. As a consequence, we give an application to the univoque sets in the setting of $q$-expansions.

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Bing Zhao, Xiaomin Ren, Jiali Zhu, Kan Jiang. 2019-08-01. Arithmetic on self-similar sets. https://arxiv.org/abs/1908.00224

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