The true complexity of a system of linear equations

W. T. Gowers, J. Wolf

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


It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called "quadratic Fourier analysis", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.
Original languageUndefined/Unknown
JournalProceedings of the London Mathematical Society
Publication statusPublished - 1 Nov 2007

Bibliographical note

30 pages


  • math.NT
  • math.CO

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