Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields

TL;DR

This paper establishes the first quantitative bounds for the inverse Gowers $U^4$-norm theorem over low characteristic finite fields, specifically for $p=2,3$, advancing understanding in additive combinatorics.

Contribution

It provides the first quantitative bounds for the inverse theorem over $ extbf{F}_p^n$ for $p=2,3$, extending previous results for larger primes.

Findings

Solved the integration problem for all $k$-linear forms

Partially solved the symmetrization problem for trilinear forms

Identified open problems for symmetrization of low-characteristic forms

Abstract

This paper gives the first quantitative bounds for the inverse theorem for the Gowers $U^4$-norm over $\mathbb{F}_p^n$ when $p=2,3$. We build upon earlier work of Gowers and Mili\'cevi\'c who solved the corresponding problem for $p\geq 5$. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all $k$-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of low-characteristic $k$-linear forms whose resolution, combined with recent work of Gowers and Mili\'cevi\'c, would give quantitative bounds for the inverse theorem for the Gowers $U^{k+1}$-norm over $\mathbb{F}_p^n$ for all $k,p$.