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  1. Free, publicly-accessible full text available July 1, 2026
  2. 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 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. 
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