The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f: V → ℂ on a finite-dimensional vector space V over a finite field F has large Gowers uniformity norm ∥f∥U ^S+1(V), then there exists a (non-classical) polynomial P: V → T of degree at most s such that f correlates with the phase e(P) = e ^2πiP. This conjecture had already been established in the "high characteristic case", when the characteristic of F is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28].