Intersections of Cantor Sets Derived from Complex Radix Expansions

Abstract

Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $λ\in[0, 1]$, that the set of complex numbers $α$ for which $\dim(C\cap(C+α)) = λ\dim(C)$ is dense in the set of $α$ for which $C \cap (C + α) \neq \emptyset$ when $d \leq n^{2}/2$ for all $d\in D$ and $|δ- δ^{'}| > n$ for all $δ\neq δ^{'} \in D - D$. We show that this result still holds when we replace $|δ- δ^{'}| > n$ with $|δ- δ^{'}| > 1$. In fact, for sufficiently large $n$, the result even holds when we remove the assumption $d\leq n^{2}/2$ and replace $|δ- δ^{'}| > n$ by $|δ- δ^{'}| > 2$. Additionally, we make similar statements where $\dim$ denotes the Hausdorff dimension or packing dimension. Our insights also find application in classifying the self-similarity of $C\cap(C+α)$. Namely we connect the occurrence of self-similarity to the notion of strongly eventually periodic sequences seen for analogous objects on the real line. We also provide a new proof of a result of W. Gilbert that inspired this work.