Nilfactors of R-actions and Configurations in Sets of Positive Upper Density in R

  title={Nilfactors of R-actions and Configurations in Sets of Positive Upper Density in R},
  author={T. Ziegler},
We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R, with D̄(E) > 0. Let V = {0, v1, . . . , vk} ⊂ R . 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 [FuKaW] showing a similar property for m = k = 2. 

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