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

Tao, Terence; Ziegler, Tamar

doi:10.2140/apde.2010.3.1

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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

DOI: 10.2140/apde.2010.3.1

Abstract

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 $ϕ = e_{F} (P)$ 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 $char F \geq d$ 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 $ϕ$ is allowed to be of some larger degree $C (d)$ . 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

Vol. 3, No. 1, 2010