## Abstract

Let F a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm U^{k}(X) for an ergodic action of the infinite abelian group on a probability space is generated by phase polynomials of degree less than C(k) on X, where C(k) depends only on k. In the case where we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case with a partial result in low characteristic.

Original language | American English |
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Pages (from-to) | 1539-1596 |

Number of pages | 58 |

Journal | Geometric and Functional Analysis |

Volume | 19 |

Issue number | 6 |

DOIs | |

State | Published - Mar 2010 |

Externally published | Yes |

### Bibliographical note

Funding Information:1.6 Acknowledgements. The authors would like to thank Tim Austin, Ben Green, Bernard Host, Bryna Kra, and Trevor Wooley for many enlightening conversations and suggestions. The first author is supported by NSF grant DMS-0600042. The second author is supported by a grant from the MacArthur Foundation, and by NSF grant DMS-0649473. The first and third authors are supported by BSF grant No. 2006094. The third author is supported by a Landau fellowship of the Taub foundations, and by an Alon fellowship. The authors also thank the anonymous referees for many useful suggestions and corrections.

## Keywords

- Characteristic factors
- Gowers uniformity norms
- Polynomials over finite fields

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