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

Abstract

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.