Arithmetic on Moran sets

Abstract

Let $(\mathcal{M},c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in (\mathcal{M},c_k,n_k)$ is $[0,1]$. Let $A,B$ be two non-empty sets in $\mathbb{R}$. Suppose that $f$ is a continuous function defined on an open set $U\subset {\mathbb{R}}^2$. Denote the continuous image of $f$ by \begin{equation*} f_{U}(A,B)=\{f(x,y):(x,y)\in (A\times B)\cap U\}. \end{equation*} In this paper, we prove the following result. Let $E_1,E_2\in (\mathcal{M},c_k,n_k)$. If there exists some $(x_0,y_0)\in (E_1\times E_2)\cap U$ such that $$\underset{k\ge 1}{\sup }\left\{1-c_k{n}_k\right\}<\left|\frac{{\partial }_{y}f{|}_{(x_0,y_0)}}{{\partial }_{x}f{|}_{(x_0,y_0)}}\right|<\underset{k\ge 1}{\inf }\left\{\frac{c_k}{1-n_k{c}_k}\right\},$$ then $f_U(E_1,E_2)$ contains an interior.

8 pages