The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

TL;DR

This paper establishes a quantitative inverse theorem for the Gowers $U^3$ norm on any finite abelian group, combining Fourier-analytic and ergodic methods to extend previous results to all such groups.

Contribution

It provides a unified proof of the inverse theorem for the $U^3$ norm on arbitrary finite abelian groups, using both Fourier-analytic and ergodic approaches.

Findings

Quantitative inverse theorem for $U^3(G)$ on all finite abelian groups.

Extension of previous results from specific groups to general finite abelian groups.

Integration of Fourier-analytic and ergodic methods in the proof.

Abstract

We state and prove a quantitative inverse theorem for the Gowers uniformity norm $U^3(G)$ on an arbitrary finite abelian group $G$; the cases when $G$ was of odd order or a vector space over ${\mathbf F}_2$ had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host--Kra type for ergodic ${\mathbf Z}^\omega$-actions of order $2$ on probability spaces established recently by Shalom and the authors.