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arXiv · 2112.13759

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

Asgar Jamneshan·Terence Tao
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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.

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Asgar Jamneshan, Terence Tao. 2021-12-27. The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches. https://arxiv.org/abs/2112.13759

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