Generalizations of Fourier analysis, and how to apply them

TL;DR

This paper surveys the extension of Fourier analysis techniques in additive combinatorics, highlighting existing generalizations, their limitations, and the potential for discovering more comprehensive theories.

Contribution

It provides a comprehensive overview of current Fourier analysis generalizations in additive combinatorics and discusses open problems and future directions.

Findings

Some generalizations are very effective in specific contexts.

Certain theories retain some Fourier properties but not all.

There are promising hints of more complete theories yet to be developed.

Abstract

This is a survey of the use of Fourier analysis in additive combinatorics, with a particular focus on situations where it cannot be straightforwardly applied, but needs to be generalized first. Sometimes very satisfactory generalizations exist, while sometimes we have to make do with theories that have some of the desirable properties of Fourier analysis but not all of them. In the latter case, there are intriguing hints that there may be more satisfactory theories yet to be discovered. This article grew out of the Colloquium Lectures at the Joint Meeting of the AMS and the MAA, given in Seattle in January 2016.