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

TL;DR

This paper proves a conjecture relating uniformity norms and linear forms over finite fields, using inverse theorems, and establishes a decomposition of functions into structured and uniform parts.

Contribution

It confirms a conjecture connecting uniformity of degree k-1 with linear independence of kth powers of linear forms in finite fields, utilizing recent inverse theorems.

Findings

Proved the conjecture for sufficiently large prime p.

Established a function decomposition into polynomial phases and uniform parts.

Applied inverse theorems for the U^k norm in finite fields.

Abstract

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.