An application of ergodic theory to a problem in geometric Ramsey theory

Let E be a measurable subset of ℝ^k, k > 2, with D̄(E) > 0. Let V = {0, v₁,..., v_k+1} ∈ ℝ^k, where v₁,..., v_k+1 are affinely independent. 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 [FKW] showing a similar property for ℝ², V = {0, v₁, v₂}.