Abstract
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function \({f : V \rightarrow \mathbb{C}}\) on a finite-dimensional vector space V over a finite field \({\mathbb{F}}\) has large Gowers uniformity norm \({{\parallel{f}\parallel_{U^{s+1}(V)}}}\) , then there exists a (non-classical) polynomial \({P: V \rightarrow \mathbb{T}}\) 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 \({\mathbb{F}}\) 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].
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The first author is supported by a grant from the MacArthur Foundation, and by NSF grant CCF-0649473.
The second author is supported by ISF grant 557/08, and by an Alon fellowship.
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Tao, T., Ziegler, T. The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic. Ann. Comb. 16, 121–188 (2012). https://doi.org/10.1007/s00026-011-0124-3
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DOI: https://doi.org/10.1007/s00026-011-0124-3