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 $\Vert f{\Vert }_{U^{s+1}(V)}$, then there exists a (non-classical) polynomial $P: V \to \T$ of degree at most $s$ such that $f$ correlates with the phase $e(P)=e^{2\pi iP}$. 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.

68 pages, no figures, to appear, Annals of Combinatorics. This is the final version, incorporating the referee's suggestions

References in corpus (5)

• Nilspaces, nilmanifolds and their morphisms
• On the norm convergence of nonconventional ergodic averages
• Gowers norms, regularization and limits of functions on abelian groups
• Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
• Multiple recurence and convergence for sequences related to the prime numbers

Cited by in corpus (4)

• New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
• An inverse theorem for Gowers norms of trace functions over prime fields
• Higher-order Fourier analysis of ${\mathbb{F}}_p^n$ and the complexity of systems of linear forms
• Correlation Testing for Affine Invariant Properties on ${\mathbb{F}}_p^n$ in the High Error Regime