The Inverse Conjecture for the Gowers Norm over Finite Fields via the Correspondence Principle Terence Tao and Tamar Ziegler

The inverse conjecture for the Gowers norms U(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 φ = eF(P ) 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 and the authors in [2]. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples in [15], [13] in this setting can be avoided by a slight reformulation of the conjecture.