2008-2009

Kyle Kinneberg

**Monday, May 4
Millikan 213
(Pomona College), 2:30 PM
**

**Title: **Some applications
of harmonic analysis to problems in additive number theory

Abstract: In recent years, analytic methods have become prominent in additive number theory. In particular, discrete Fourier analysis is well-suited to solve some problems that are too difficult for purely combinatorial techniques. Among these is Szemeredi's Theorem, which states that for any positive density d and positive integer k, we can find an interval {1,...,N} long enough so that every subset of this interval with size at least dN contains an arithmetic progression of length k. In this talk, we will discuss discrete Fourier analytic approaches to Szemeredi's Theorem. We will focus on Roth's work from the early 1950s that first established the result for k=3 and Gowers's work from the late 1990s that gave a new proof of the result for k=4. We will also make some remarks regarding Gowers's proof for arbitrary k.