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The inverse conjecture
for the Gowers norm over finite fields via the correspondence
principle
Terence Tao and Tamar Ziegler |
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Vol. 3 (2010), No. 1, 1–20
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Abstract |
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The inverse conjecture for the Gowers norms for finite-dimensional vector spaces over a finite field asserts, roughly speaking, that a bounded function has large Gowers norm if and only if it correlates with a phase polynomial of degree at most , thus is a polynomial of degree at most . 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 is allowed to be of some larger degree . 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
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Mathematical Subject Classification 2000
Primary: 11T06, 37A15
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Milestones
Received: 30 October 2008
Revised: 29 October 2009
Accepted: 10 December 2009
Published: 4 March 2010
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Authors |
| Terence Tao | |
| University of California Mathematics Department Los Angeles, CA 90095-1555 United States |
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| http://www.math.ucla.edu/~tao/ | |
| Tamar Ziegler | |
| Department of Mathematics Technion 32000 Haifa Israel |
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