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

作者：Green Ben^*; Tao Terence

来源：Mathematical Proceedings of the Cambridge Philosophical Society, 2010, 149: 1-19.

DOI：10.1017/S0305004110000186

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.

• 出版日期2010-7