An inverse theorem for the uniformity seminorms associated with the action of $F^\omega$

TL;DR

This paper characterizes the universal characteristic factor for Gowers-Host-Kra seminorms in finite field actions, showing it is generated by low-degree phase polynomials, extending results from integers to finite fields.

Contribution

It establishes an inverse theorem for uniformity seminorms in finite fields, identifying the structure of characteristic factors via phase polynomials of bounded degree.

Findings

Characteristic factor generated by phase polynomials of degree less than C(k)

Sharp result C(k)=k when k ≤ characteristic of the field

Extension of Host-Kra results from integers to finite fields

Abstract

Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^\omega}$ of the infinite abelian group $\F^\omega$ on a probability space $X = (X,\B,\mu)$ is generated by phase polynomials $\phi: X \to S^1$ of degree less than $C(k)$ on $X$, where $C(k)$ depends only on $k$. In the case where $k \leq \charac(\F)$ we obtain the sharp result $C(k)=k$. This is a finite field counterpart of an analogous result for $\Z$ by Host and Kra. In a companion paper to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case $k \leq \charac(\F)$, with a partial result in low characteristic.