# Arithmetic on self-similar sets

## TL;DR

This paper establishes conditions under which the continuous image of the product of two self-similar sets contains interior points, extending previous results by removing the need for a thickness product condition, with applications to q-expansions.

## Contribution

It provides a new sufficient condition for the image of two self-similar sets under a continuous function to have interior, generalizing prior results by relaxing the thickness requirement.

## Key findings

- The image contains interior points under the new condition.
- The result does not require the product of thicknesses to exceed one.
- Application to univoque sets in q-expansions.

## 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.