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 finitedimensional vector space $V$ over a finite field $\F$ has large Gowers uniformity norm $\f\_{U^{s+1}(V)}$, then there exists a (nonclassical) 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 nonclassical polynomials, in the spirit of the work of Green and the first author and of Kaufman and Lovett.
 Publication:

arXiv eprints
 Pub Date:
 January 2011
 arXiv:
 arXiv:1101.1469
 Bibcode:
 2011arXiv1101.1469T
 Keywords:

 Mathematics  Combinatorics;
 11B30;
 11T06
 EPrint:
 68 pages, no figures, to appear, Annals of Combinatorics. This is the final version, incorporating the referee's suggestions