Israel Journal of Mathematics

, Volume 114, Issue 1, pp 271–288 | Cite as

An application of ergodic theory to a problem in geometric ramsey theory



LetE be a measurable subset of ℝ k ,k>2, with XXX(E)>0. LetV = {0,υ 1, …,υ k+1} ε ℝ k , whereυ 1, …,υ k+1 are affinely independent. We show that forr large enough, we can find an isometric copy ofrV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FKW] showing a similar property for ℝ2,V = {0,υ 1,υ 2}.


Ergodic Theory Haar Measure Ergodic Theorem Measurable Subset Compact Abelian Group 
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  1. [Bo]
    J. Bourgain,A Szemerédi type theorem for sets of positive density ink, Israel Journal of Mathematics54 (1986), 307–316.MATHCrossRefMathSciNetGoogle Scholar
  2. [FM]
    K.J. Falconer and J.M. Marstrand,Plane sets with positive density at infinity contain all large distances, The Bulletin of the London Mathematical Society18 (1986), 471–474.MATHCrossRefMathSciNetGoogle Scholar
  3. [Fu1]
    H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, 1981.MATHGoogle Scholar
  4. [Fu2]
    H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Mathématique31 (1977), 204–256.MATHMathSciNetCrossRefGoogle Scholar
  5. [FKW]
    H. Furstenberg, Y. Katznelson and B. Weiss,Ergodic theory and configurations in sets of positive density, inMathematics of Ramsey Theory (J. Neśetŕil and V. Rődl, eds.), Springer-Verlag, Berlin, 1990.Google Scholar
  6. [Gr]
    R. L. Graham,Recent trends in Euclidean Ramsey theory, Discrete Mathematics136 (1994), 119–127.MATHCrossRefMathSciNetGoogle Scholar
  7. [Pe]
    K. Petersen,Ergodic Theory, Cambridge University Press, 1983.Google Scholar
  8. [PS]
    C. Pugh and M. Shub,Ergodic Elements of Ergodic Actions, Compositio Mathematica23 (1971), 115–122.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1999

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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