Abstract
Letk≧2 andA a subset ofR k 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
J. Bourgain,On the spherical maximal function in the plane, preprint, IHES.
H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math.31 (1977), 204–256.
H. Furstenberg and Y. Katznelson,An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math.34 (1978), 275–291.
Y. Katznelson and B. Weiss, preprint.
K. Roth,On certain sets of integers, J. London Math. Soc.28 (1953), 104–109.
R. Salem,Oeuvres Mathematiques, Hermann, Paris, 1967.
E. Szemerédi,On sets of integers containing no k elements in arithmetic progression, Acta Arith.27 (1975), 199–245.
Séminaire d’Analyse Fonctionnelle, Ecole Polytechnique, 1978–79, Exp. 9.
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Bourgain, J. A szemerédi type theorem for sets of positive density inR k . Israel J. Math. 54, 307–316 (1986). https://doi.org/10.1007/BF02764959
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DOI: https://doi.org/10.1007/BF02764959