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Abstract

This dissertation is comprised of four articles, each related to a discrete extremal problem. Several topics appear in more than one chapter. These include polynomial methods, Fourier analysis, and combinatorial number theory. In Chapter 2 we prove that the number of directions contained in a set of the form A x B ⊂ AG(2,p) where p is prime, is at least |A||B| - min{|A|,|B|} + 2. Here A and B are subsets of GF(p) each with at least two elements and |A||B|