Montreal Lecture Notes on Quadratic Fourier Analysis

Abstract

These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand it at the present time. Specifically, we will describe ``quadratic objects'' of various types and their relation to additive structures, particularly four-term arithmetic progressions. I will focus on qualitative results, referring the reader to the literature for the many interesting quantitative questions in this theory. Thus these lectures have a distinctly ``soft'' flavour in many places. Some of the notes cover unpublished work which is joint with Terence Tao. This will be published more formally at some future juncture.