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Nilfactors of ℝ^m-actions and configurations in sets of positive upper density in ℝ^m

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Tamar Ziegler. Nilfactors of ℝ^m-actions and configurations in sets of positive upper density in ℝ^m

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Abstract

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of ℝ^m, with positive upper density. Let V={0,v₁,...,v_k} be a subset of ℝ^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2.

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