Linear forms and higher-degree uniformity for functions on Fnp

W. T. Gowers, J. Wolf

Research output: Contribution to journalArticle (Academic Journal)peer-review

26 Citations (Scopus)

Abstract

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.
Original languageEnglish
Pages (from-to)36-69
Number of pages34
JournalGeometric and Functional Analysis
Volume21
Issue number1
DOIs
Publication statusPublished - 1 Feb 2011

Bibliographical note

40 pages

Keywords

  • math.NT
  • math.CO
  • 11B30

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