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

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