Projection Theorems and Linear Hashing

Abstract

This thesis gathers several different ways to prove projection theorems, which have applications in problems with max collisions in hashing, getting good bounds on the frequency of exceptional sets in geometry, or finding bounds on the sizes of Kakeya and related sets. We examine several proofs of theorems, many of which are combinatorial or geometric in nature, while providing meta commentary on why the techniques in the proofs work so well in these settings, compare the different strengths of each theorem, and suggest some improvements that can be made.