Linear forms and quadratic uniformity for functions on ℤ N

W. T. Gowers, J. Wolf

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

21 Citations (Scopus)


We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\mathbb{F}_p^n$. While in [GW09] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].
Original languageEnglish
Pages (from-to)121-186
Number of pages66
JournalJournal d'Analyse Mathématique
Issue number1
Publication statusPublished - Jun 2011

Bibliographical note

26 pages


  • math.NT
  • math.CO
  • 11B30


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