Higher-order Fourier analysis of $\mathbb{F}_p^n$ and the complexity of systems of linear forms

TL;DR

This paper extends higher-order Fourier analysis to subsets of finite fields, identifying the minimal uniformity degree needed to approximate indicator functions in linear form averages, generalizing prior results.

Contribution

It determines the smallest degree of uniformity for which a function can be approximated by a bounded component in linear form averages, generalizing Gowers and Wolf's work.

Findings

Identifies the minimal uniformity degree for approximation.

Extends analysis to multiple subsets in linear forms.

Solves an open problem from Gowers and Wolf's 2011 paper.

Abstract

Consider a subset $A$ of $\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming that $\|f_2\|_{U^k}$ is sufficiently small, it is possible to discard $f_2$ and replace $1_A$ with $f_1$ in the average over this family of linear forms, affecting it only negligibly. Previously, Gowers and Wolf solved this problem for the case where $f_1$ is a constant function. Furthermore, our main result solves Problem 7.6 in [W. T. Gowers and J. Wolf. Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$. Geom. Funct. Anal., 21(1):36--69, 2011] regarding the analytic averages that involve more than one subset of $\mathbb{F}_p^n$.] regarding the analytic averages that involve more than one subset of $\mathbb{F}_p^n$.