Nilfactors of Rm-actions and configurations in sets of positive upper density in Rm
Abstract
We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of Rm, with positive upper density. Let V=0,v1,...,vk be a subset of Rm. 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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