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 group G; the cases when G was of odd order or a vector space over 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 Z ω -actions of order 2 on probability spaces established recently by Shalom and the authors.

An uncountable Furstenberg–Zimmer structure theory

“…Beyond the natural (intrinsic) motivation of generalizing ergodic theory to uncountable systems, we hope that an uncountable Host-Kra-Ziegler structure theory (such as for the action of hyperfinite abelian groups on Loeb probability spaces) may help to clarify the relationship between the ergodic-theoretical and analytic (higher-order Fourier analysis) approaches to Szemerédi's theorem. See [29,30] for some recent progress.…”