The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

TL;DR

This paper proves the inverse conjecture for Gowers norms over finite fields in large characteristic by linking it to ergodic theory, with partial results in low characteristic and a discussion of open problems.

Contribution

It develops a new correspondence principle to connect finite field Gowers norms with ergodic theory, proving the inverse conjecture in large characteristic cases.

Findings

Established the inverse conjecture for Gowers norms in large characteristic.

Provided partial results for low characteristic cases.

Linked finite field problems to ergodic theory via a novel correspondence principle.

Abstract

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$, thus $P: V \to \F$ is a polynomial of degree at most $d-1$. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case $\charac(F) \geq d$ from an ergodic theory counterpart, which was recently established by Bergelson and the authors. In low characteristic we obtain a partial result, in which the phase polynomial $\phi$ is allowed to be of some larger degree $C(d)$. The full inverse conjecture remains open in low characteristic; the counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture.