On higher-order Fourier analysis in characteristic $p$

TL;DR

This paper develops a nilspace framework for higher-order Fourier analysis over finite fields, providing structural insights and applications in ergodic theory, including new proofs of inverse theorems and analysis of Host-Kra factors.

Contribution

It introduces and characterizes $p$-homogeneous nilspaces, advancing the algebraic understanding of higher-order Fourier analysis in characteristic $p$ and applying this to ergodic theory.

Findings

Characterization of $p$-homogeneous nilspaces via algebraic properties

Structure theorem for finite $p$-homogeneous nilspaces

New proof of Tao-Ziegler inverse theorem for Gowers norms

Abstract

In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb{F}_p$, with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call $p$-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite $p$-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian $p$-groups. The applications include a description of the Host-Kra factors of ergodic $\mathbb{F}_p^\omega$-systems as $p$-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the $k$-th Host-Kra factor is an Abramov system of order $\leq k$, extending a result of Bergelson-Tao-Ziegler that holds for $k< p$. We illustrate the utility of $p$-homogeneous nilspaces also by showing that the above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on $\mathbb{F}_p^n$.