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

Terence Tao*, Tamar Ziegler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


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].

Original languageAmerican English
Pages (from-to)121-188
Number of pages68
JournalAnnals of Combinatorics
Issue number1
StatePublished - Mar 2012
Externally publishedYes

Bibliographical note

Funding Information:
∗ 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.


  • Gowers uniformity norms
  • finite fields
  • polynomials


Dive into the research topics of 'The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic'. Together they form a unique fingerprint.

Cite this