On the Intersections of Homogeneous Self-similar Sets with their Translates in Rn and a Formulation of Multiplicative Invariance in Zn

Abstract

This thesis generalizes the study of C ∩(C+α) where C is the middle third Cantor set to self-affine sets in R n. We present sufficient and necessary conditions for when the translation α 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 α to the dimension of the intersection. This lends itself to a case study of the complex number system (−n + i, {0, 1, . . . , n2}), when n is an integer greater than or equal to 2. Lastly, we present a definition of multiplicative invariance for subsets of Z n and establish a connection, known in the one-dimensional case, between them and invariants sets of the n-dimensional torus.