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

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 d 1, thus P : V 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  charF 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.

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