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The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic
Terence Tao1 and Tamar Ziegler2
1UCLA Department of Mathematics, Los Angeles, CA 90095-1596, USA
2Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Annals of Combinatorics 16 (1) pp.121-188 March, 2012
AMS Subject Classification: 11B30, 11T06
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 ||Us+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) = e2πiP. 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].
Keywords: finite fields, polynomials, Gowers uniformity norms


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