## Abstract

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 language | American English |
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Pages (from-to) | 121-188 |

Number of pages | 68 |

Journal | Annals of Combinatorics |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2012 |

Externally published | Yes |

### 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.

## Keywords

- Gowers uniformity norms
- finite fields
- polynomials