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

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,v_1,…,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.