We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f: V → ℂ on a finite-dimensional vector space V over a finite field F has large Gowers uniformity norm ∥f∥U S+1(V), then there exists a (non-classical) polynomial P: V → T of degree at most s such that f correlates with the phase e(P) = e 2πiP. This conjecture had already been established in the "high characteristic case", when the characteristic of F is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28]. © 2011 Springer Basel AG.
CITATION STYLE
Tao, T., & Ziegler, T. (2012). The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic. Annals of Combinatorics, 16(1), 121–188. https://doi.org/10.1007/s00026-011-0124-3
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