Open Collections

Abstract

We discuss four different problems. The first, a joint work with Malabika Pramanik, concerns large subsets of ℝn that do not contain various types of configurations. We show that a collection of v points satisfying a continuously differentiable v-variate equation in ℝ can be avoided by a set of Hausdorff dimension 1/(v-1) and Minkowski dimension 1. The second problem concerns large subsets of vector spaces over non-archimedean local fields that do not contain configurations. Results analogous to the real-variable cases are obtained in this setting. The third problem is the construction of measure-zero Besicovitch-type sets in Kn for non-archimedean local fields K. This construction is based on a Euclidean construction of Wisewell and an earlier construction of Sawyer. The fourth problem, a joint work with Kyle Hambrook, is the construction of an explicit Salem set in ℚp. This set is based on a Euclidean construction of Kaufman. [An errata to this thesis/dissertation was made available on 2019-04-26.]