The inverse conjecture for the Gowers norm over finite fields in low characteristic

Abstract

We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ 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 $P: V \to \T$ of degree at most $s$ such that $f$ correlates with the phase $e(P) = e^{2\pi i P}$. 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, 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 and of Kaufman and Lovett.