Intersections of Cantor Sets Derived from Complex Radix Expansions

TL;DR

This paper investigates the intersection properties of Cantor sets derived from complex radix expansions, extending known results to broader conditions and linking self-similarity to periodic sequences.

Contribution

It generalizes previous intersection dimension results for complex Cantor sets by relaxing key assumptions and connects self-similarity to periodicity in complex radix expansions.

Findings

The intersection dimension results hold under weaker spacing conditions.

The results extend to Hausdorff and packing dimensions.

Self-similarity relates to strongly eventually periodic sequences.

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 $\lambda\in[0, 1]$, that the set of complex numbers $\alpha$ for which $\dim(C\cap(C+\alpha)) = \lambda\dim(C)$ is dense in the set of $\alpha$ for which $C \cap (C + \alpha) \neq \emptyset$ when $d \leq n^{2}/2$ for all $d\in D$ and $|\delta - \delta^{'}| > n$ for all $\delta \neq \delta^{'} \in D - D$. We show that this result still holds when we replace $|\delta - \delta^{'}| > n$ with $|\delta - \delta^{'}| > 1$. In fact, for sufficiently large $n$, the result even holds when we remove the assumption $d\leq n^{2}/2$ and replace $|\delta - \delta^{'}| > n$ by $|\delta - \delta^{'}| > 2$. 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 $C\cap(C+\alpha)$. 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.