Israel Journal of Mathematics

, Volume 54, Issue 3, pp 307–316

A szemerédi type theorem for sets of positive density inRk

  • J. Bourgain

DOI: 10.1007/BF02764959

Cite this article as:
Bourgain, J. Israel J. Math. (1986) 54: 307. doi:10.1007/BF02764959


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.

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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