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

TL;DR

This paper demonstrates a fundamental equivalence between inverse sumset theorems and inverse Gowers norm theorems, linking classifications of approximate groups and polynomials in finite groups and vector spaces.

Contribution

It establishes a formal equivalence between inverse sumset theorems and inverse Gowers norm theorems, including their polynomial conjectures, in two key algebraic settings.

Findings

Inverse sumset theorems are equivalent to inverse Gowers U^3 norm results.

Polynomial strengthening conjectures are also equivalent in both frameworks.

The structure of approximate homomorphisms is clarified in the equivalence proof.

Abstract

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.