Quasipolynomial inverse theorem for the $\mathsf{U}^4(\mathbb{F}_p^n)$ norm

Abstract

The inverse theory for Gowers uniformity norms is one of the central topics in additive combinatorics and one of the most important aspects of the theory is the question of bounds. In this paper, we prove a quasipolynomial inverse theorem for the $\mathsf{U}^4$ norm in finite vector spaces. The proof follows a different strategy compared to the existing quantitative inverse theorems. In particular, the argument relies on a novel argument, which we call the abstract Balog-Szemer\'edi-Gowers theorem, and combines several other ingredients such as algebraic regularity method, bilinear Bogolyubov argument and algebraic dependent random choice.