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

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.

48 pages, no figures. This is the published version

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