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 D̄(E) > 0. Let V = {0, v₁,..., v_k} ⊆ ℝ_m. We show that for t large enough, we can find an isometric copy of tV 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.