Arithmetic on self-similar sets

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.