Intersections of Cantor Sets Derived from Complex Radix Expansions

Abstract

Let $C$ be the attractor of the IFS $\{f_d(z)=(-n+i{)}^{-1}(z+d):d\in D\}$, $D\subset \{0,1,\dots ,n^2\}$ 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 $C\cap (C+\alpha )\ne \varnothing$ when $d\le n^2/2$ for all $d\in D$ and $|\delta -{\delta }^{{}^{\prime }}|>n$ for all $\delta \ne {\delta }^{{}^{\prime }}\in D-D$. We show that this result still holds when we replace $|\delta -{\delta }^{{}^{\prime }}|>n$ with $|\delta -{\delta }^{{}^{\prime }}|>1$. In fact, for sufficiently large $n$, the result even holds when we remove the assumption $d\le n^2/2$ and replace $|\delta -{\delta }^{{}^{\prime }}|>n$ by $|\delta -{\delta }^{{}^{\prime }}|>2$. 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.