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

TL;DR

This paper establishes a quasipolynomial inverse theorem for the Gowers U^4 norm over finite vector spaces, introducing new techniques that improve bounds and combine algebraic and combinatorial methods.

Contribution

It presents a novel quasipolynomial inverse theorem for the U^4 norm, utilizing a new abstract Balog-Szemerédi-Gowers theorem and combining multiple advanced techniques.

Findings

Proves a quasipolynomial bound for the inverse theorem.

Introduces the abstract Balog-Szemerédi-Gowers theorem.

Combines algebraic regularity, bilinear Bogolyubov, and dependent random choice methods.

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.