The inverse theorem for the U³ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

2021/12/27 by Asgar Jamneshan, Terence Tao, Jamneshan, Asgar +1 · 1 citation
Mathematics · #11B30 #28E05 #37A15 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2112.13759

openalex publication_date 2021/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We state and prove a quantitative inverse theorem for the Gowers uniformity norm U³(G) on an arbitrary finite abelian group G; the cases when G was of odd order or a vector space over \mathbf F₂ 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^ω-actions of order 2 on probability spaces established recently by Shalom and the authors.

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• An inverse theorem for all finite abelian groups via nilmanifolds

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