Israel Journal of Mathematics

, Volume 54, Issue 3, pp 307–316

A szemerédi type theorem for sets of positive density inRk

  • J. Bourgain
Article

Abstract

Letk≧2 andA a subset ofRk of positive upper density. LetV be the set of vertices of a (non-degenerate) (k−1)-dimensional simplex. It is shown that there existsl=l(A, V) such thatA contains an isometric image ofl′. V wheneverl′>l. The casek=2 yields a new proof of a result of Katznelson and Weiss [4]. Using related ideas, a proof is given of Roth’s theorem on the existence of arithmetic progressions of length 3 in sets of positive density.

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References

  1. 1.
    J. Bourgain,On the spherical maximal function in the plane, preprint, IHES.Google Scholar
  2. 2.
    H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math.31 (1977), 204–256.MATHMathSciNetGoogle Scholar
  3. 3.
    H. Furstenberg and Y. Katznelson,An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math.34 (1978), 275–291.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Y. Katznelson and B. Weiss, preprint.Google Scholar
  5. 5.
    K. Roth,On certain sets of integers, J. London Math. Soc.28 (1953), 104–109.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Salem,Oeuvres Mathematiques, Hermann, Paris, 1967.MATHGoogle Scholar
  7. 7.
    E. Szemerédi,On sets of integers containing no k elements in arithmetic progression, Acta Arith.27 (1975), 199–245.MATHMathSciNetGoogle Scholar
  8. 8.
    Séminaire d’Analyse Fonctionnelle, Ecole Polytechnique, 1978–79, Exp. 9.Google Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • J. Bourgain
    • 1
  1. 1.Department of MathematicsI.H.E.S.Bures-sur-YvetteFrance

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