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We establish a correspondence between inverse sumset theorems (which can beviewed as classifications of approximate (abelian) groups) and inverse theoremsfor the Gowers norms (which can be viewed as classifications of approximatepolynomials). In particular, we show that the inverse sumset theorems ofFreiman type are equivalent to the known inverse results for the Gowers U^3norms, and moreover that the conjectured polynomial strengthening of the formeris also equivalent to the polynomial strengthening of the latter. We establishthis equivalence in two model settings, namely that of the finite field vectorspaces F_2^n, and of the cyclic groups Z/NZ.  In both cases the argument involves clarifying the structure of certain typesof approximate homomorphism.

An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm