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2010 The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
Terence Tao, Tamar Ziegler
Anal. PDE 3(1): 1-20 (2010). DOI: 10.2140/apde.2010.3.1


The inverse conjecture for the Gowers norms Ud(V) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm fUd(V) if and only if it correlates with a phase polynomial ϕ=eF(P) of degree at most d1, thus P:VF is a polynomial of degree at most d1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case  charFd 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.


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Terence Tao. Tamar Ziegler. "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle." Anal. PDE 3 (1) 1 - 20, 2010.


Received: 30 October 2008; Revised: 29 October 2009; Accepted: 10 December 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1252.11012
MathSciNet: MR2663409
Digital Object Identifier: 10.2140/apde.2010.3.1

Primary: 11T06, 37A15

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.3 • No. 1 • 2010
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