Terence Tao and Tamar Ziegler. The inverse conjecture for the Gowers norm over finite fields via the correspondence principle.
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 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 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 by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture.