Linear forms and quadratic uniformity for functions on $\mathbb{F}_p^n$

TL;DR

This paper improves bounds on the number of solutions to linear systems in finite fields, using a new decomposition method that enhances the application of the $U^3$ inverse theorem, resulting in a doubly exponential relation between uniformity and error.

Contribution

It introduces a novel decomposition approach leveraging the Hahn-Banach theorem, replacing the structure theorem, to achieve better bounds in quadratic uniformity problems.

Findings

Achieved doubly exponential bounds between uniformity and error.

Developed a new decomposition method using Hahn-Banach theorem.

Enhanced the application of the $U^3$ inverse theorem.

Abstract

We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\mathbb{F}_p^n$. While in [GW09] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].