arXiv subjects

Neil MacVicar

Publications and source records attributed to Neil MacVicar.

3 recordsLinked to original sources

arXiv2026

On the intersections of homogeneous self-similar sets with their translates in $\mathbb{R}^{n}$ and a formulation of multiplicative invariance in $\mathbb{Z}^{n}$

This thesis generalizes the study of $C\cap(C + \alpha)$ where $C$ is the middle third Cantor set to self-affine sets in $\mathbb{R}^{n}$. We present sufficient and necessary conditions for when the translation $\alpha$ produces a self-affine intersection for a particular class of self-affine sets. In the case where the attractor is self-similar, we improve results concerning the function from $\alpha$ to the fractal dimension of the intersection. This lends itself to a case study of the complex number system $(-n + i, \{0, 1, . . . , n^{2}\})$, when $n$ is an integer greater than or equal to $2$. Lastly, we present a definition of multiplicative invariance for subsets of $\mathbb{Z}^{n}$ and establish a connection, known in the one-dimensional case, between them and invariant sets of the $n$-dimensional torus.

math.DS↗

arXiv2024

Intersections of Cantor Sets Derived from Complex Radix Expansions

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.

math.DS↗

arXiv2023

Multiplicative Invariance for a Class of Subsets of the Complex Plane

Multiplicative invariance is a well-studied property of subsets of the unit interval. The theory in the complex plane is less developed. This paper introduces an analogous definition for multiplicative invariance in the complex plane coinciding with a more general definition concerning subsets of attractors of iterated function systems satisfying the strong separation condition. We establish similar results to those of Furstenberg's in the unit interval. Namely, that the Hausdorff and box-counting dimensions of a multiplicatively invariant set are equal and, furthermore, are equal to the normalized topological entropy of an underlying subshift. We also extend results concerning the box-counting dimension of intersections of base-$b$ restricted digit sets with their translates where $b$ is a suitably chosen Gaussian integer.

math.DS↗