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