Equivalence of polynomial conjectures in additive combinatorics

TL;DR

This paper proves that two major conjectures in additive combinatorics, the polynomial Freiman-Ruzsa and inverse Gowers conjectures for $U^3$, are logically equivalent, linking set structure and quadratic functions.

Contribution

It establishes the equivalence between the polynomial Freiman-Ruzsa and inverse Gowers conjectures for $U^3$, unifying two central problems in additive combinatorics.

Findings

Proves the equivalence of the two conjectures.

Shows that a strong form of one conjecture implies the strong form of the other.

Connects the structure of small doubling sets with quadratic function behavior.

Abstract

We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.