Vol. 3, No. 1, 2010

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The inverse conjecture for the Gowers norm over finite fields via the correspondence principle

Terence Tao and Tamar Ziegler

Vol. 3 (2010), No. 1, 1–20
Abstract

The inverse conjecture for the Gowers norms ${U}^{d}\left(V\right)$ for finite-dimensional vector spaces $V$ over a finite field $\mathbb{F}$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\parallel f{\parallel }_{{U}^{d}\left(V\right)}$ if and only if it correlates with a phase polynomial $\varphi ={e}_{\mathbb{F}}\left(P\right)$ of degree at most $d-1$, thus $P:V\to \mathbb{F}$ 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 from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial $\varphi$ is allowed to be of some larger degree $C\left(d\right)$. The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.

Keywords
Gowers uniformity norm, Furstenberg correspondence principle, characteristic factor, polynomials over finite fields
Mathematical Subject Classification 2000
Primary: 11T06, 37A15
Milestones
Received: 30 October 2008
Revised: 29 October 2009
Accepted: 10 December 2009
Published: 4 March 2010
Authors
 Terence Tao University of California Mathematics Department Los Angeles, CA 90095-1555 United States http://www.math.ucla.edu/~tao/ Tamar Ziegler Department of Mathematics Technion 32000 Haifa Israel