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.

中文翻译：

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自相似集上的算术

令$K_1$和$K_2$为两个一维齐次自相似集。令 $f$ 是定义在开集 $U\subset \mathbb{R}^{2}$ 上的连续函数。用$$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in (K_1\times K_2)\cap U\}表示$f$的连续图像。$$ 在本文中，我们给出了一个充分条件，保证 $f_{U}(K_1,K_2)$ 包含一些内部。我们的结果与 Simon 和 Taylor 的 \cite[Proposition 2.9]{ST} 不同，因为我们不需要 $K_1$ 和 $K_2$ 的厚度乘积严格大于 $1$ 的条件。因此，我们在 $q$-expansions 的设置中对 univoque 集进行了应用。