Intersections of Cantor Sets Derived from Complex Radix Expansions

TL;DR

This work investigates intersections of Cantor-like sets arising from complex radix expansions with base $b=-n+i$ and digits $D\subset\{0,1,...,n^{2}\}$. By refining neighbor separation bounds and leveraging Moran/SEP structures, the authors prove that the level sets of the dimension function $\Phi_{n,D}(\alpha)=\dim(C_{n,D}\cap(C_{n,D}+\alpha))$ are dense in the translation set under weaker conditions than previously known, and extend results to box-counting, Hausdorff, and packing dimensions. A key contribution is connecting self-similarity of intersections to SEP sequences, providing new proofs of Gilbert’s results, and giving explicit dimension computations (including Moran-based equalities) for many intersections. The results advance understanding of how complex radix fractals intersect under translations and offer practical tools for classifying self-similarity via combinatorial/digit-structure properties. Overall, the paper broadens the scope of fractal intersection theory in the complex plane and links radix-expansion dynamics to dimension-theoretic properties.

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.

Theorems & Definitions (76)

• Theorem 1.1: I. Katai, J. Szabo, KS74, theorem 1
• Definition 1.2
• Theorem 1.3: I. Katai, J. Szabo, KS74, theorem 2
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8: S. Pedersen, V. Shaw, PS21, corollary 7.5
• Theorem 1.9
• Definition 1.10