Tamar Ziegler. An application of ergodic theory to a problem in geometric ramsey theory
Let E be a measurable subset of Rk ,k>2, with d(E)>0. Let V = {u1,...,uk+1} ⊆ Rk , where u1,...,uk+1 are affinely independent. We show that forr large enough, we can find an isometric copy of V 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,u1,u2}.