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

By Terence Tao, T. Ziegler. 2008

108 citations recorded by the source corpus.

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 normk fkU d.V/ if and only if it correlates with a phase polynomial D eF.P/ of degree at most d 1, thus PV 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 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8c3abb67530d07e5279db246b5480d44a899647a

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Citations in the corpus

The true complexity of a system of linear equations (2007, background): “For example, it was shown by Host and Kra [12] and Ziegler [18] independently that when the linear forms L1, . . . , Lm describe an arithmetic progression of length m, then there exists a characteristic factor for the average, which is isomorphic to an inverse limit of a sequence of (m− 2)-step nils”
The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic (2011, unknown): “The purpose of this appendix is to prove this proposition, which explains why we were unable to use the ergodic theory method from [3, 43] to establish the main results in this paper.”