Israel Journal of Mathematics

, Volume 54, Issue 3, pp 307-316

First online:

A szemerédi type theorem for sets of positive density inR k

  • J. BourgainAffiliated withDepartment of Mathematics, I.H.E.S.

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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.