On higher order Fourier analysis

TL;DR

This paper develops a comprehensive higher order Fourier analysis framework for compact abelian groups, establishing inverse theorems, regularity lemmas, and a new limit theory using algebraic and ultraproduct methods.

Contribution

It introduces an algebraic interpretation of higher order Fourier analysis via compact nilspaces and proves general inverse theorems and regularity lemmas within this framework.

Findings

Established a new inverse theorem for Gowers's uniformity norms.

Developed a regularity lemma for functions on abelian groups.

Formulated a limit theory analogous to graph limits for abelian groups.

Abstract

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact $k$-step nilspaces. As a byproduct of our results we obtain a new type of limit theory for functions on abelian groups in the spirit of the so-called graph limit theory. Our proofs are based on an exact (non-approximative) version of higher order Fourier analysis which appears on ultra product groups.