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

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 (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.