Fourier Analytic Methods in Additive Number Theory

In recent years, analytic methods have become prominent in additive number theory. In particular, finite Fourier analysis is well-suited to solve some problems that are too difficult for purely combinatorial techniques. Among these is Szemerédi’s Theorem, a statement regarding the density of integral sets and the existence of arithmetic progressions in those sets. In this thesis, we give a general introduction to classical Fourier analysis over R and discrete Fourier analysis over the group of integers modulo N. We then give a complete explanation of Timothy Gowers’s 1998 proof of Szemerédi’s Theorem for arithmetic progressions of length four. This proof relies entirely on finite Fourier analytic methods. As a result, our explanation of it provides readers with a thorough demonstration of how these techniques are useful in additive number theory. We also give a short description of other problems in this field for which analytic methods are helpful.