%0 Journal Article
%T The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
%A Terence Tao
%A Tamar Ziegler
%J Mathematics
%D 2008
%I arXiv
%R 10.2140/apde.2010.3.1
%X The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$, thus $P: V \to \F$ 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 $\charac(F) \geq d$ 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 $\phi$ 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.
%U http://arxiv.org/abs/0810.5527v2