The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle

Abstract

The inverse conjecture for the Gowers norms Ud(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 ∥Ud(V)if and only if it correlates with a phase polynomial (Equation Presented) of degree at most d-1, thus P:V ⌜ 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 char F > 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 ∅ is allowed to be of some larger degree C(d) 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