UC Riverside

While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of these spaces. In this dissertation we will focus on the geometry of Ahlfors regular spaces, Metric Measure spaces with an additional regularity condition. Historically, fractals have been studied using different ideas of dimension which have all proven to be unsatisfactory to some degree. The theory of complex dimensions, developed by M.L. Lapidus and a number of collaborators, was developed in part to better understand fractality in the Euclidean case and seeks to overcome these problems. Of particular interest is the recent theory of complex dimensions in higher-dimensional Euclidean spaces, as studied by M.L.Lapidus, G. Radunovic, and D. Zubrinic, who introduced and studied the properties of the distance zeta function $\ze_A$. We will show that this theory of complex dimensions naturally generalizes to the case of Ahlfors regular spaces, as the distance zeta function can be modified to these spaces and all of its main properties carry over. In particular, we will show that we can reconstruct information about the geometry of a subset from their associated distance zeta function through fractal tube formulas. We also provide a selection of examples in Ahlfors spaces, as well as hints that the theory can be expanded to a more general setting.