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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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