Annals of Combinatorics

, Volume 16, Issue 1, pp 121–188 | Cite as

The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

  • Terence Tao
  • Tamar Ziegler


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 iP . 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].

Mathematics Subject Classification

11B30 11T06 


finite fields polynomials Gowers uniformity norms 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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