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

Abstract

The inverse conjecture for the Gowers norms $U^d\left(V\right)$ for finite-dimensional vector spaces $V$ over a finite field $\mathbb{F}$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\parallel f{\parallel }_{U^d\left(V\right)}$ if and only if it correlates with a phase polynomial $\varphi =e_{\mathbb{F}}\left(P\right)$ of degree at most $d-1$, thus $P:V\to \mathbb{F}$ 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 $\text{ char}\mathbb{F}\ge d$ from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial $\varphi$ is allowed to be of some larger degree $C\left(d\right)$. The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.