The Inverse Theorem for the U3 Gowers Uniformity Norm on Arbitrary Finite Abelian Groups: Fourier-analytic and Ergodic Approaches

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 V omega-actions of order 2 on probability spaces established recently by Shalom and the authors.