Dimension of the intersection of certain Cantor sets in the plane

Abstract

In this paper we consider a retained digits Cantor set $T$ based on digit expansions with Gaussian integer base. Let $F$ be the set all $x$ such that the intersection of $T$ with its translate by $x$ is non-empty and let $F_{\beta}$ be the subset of $F$ consisting of all $x$ such that the dimension of the intersection of $T$ with its translate by $x$ is $\beta$ times the dimension of $T$. We find conditions on the retained digits sets under which $F_{\beta}$ is dense in $F$ for all $0\leq\beta\leq 1$. The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.