2025-01-1920232397312910.19086/da.842682-s2.0-85173093047https://doi.org/10.19086/da.84268https://hdl.handle.net/20.500.14288/26298We 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 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 Zω-actions of order 2 on probability spaces established recently by Shalom and the authors. © 2023 Asgar Jamneshan, and Terence TaoMathematicsJournal Article1138810100001Q150733