A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral

We obtain nontrivial exponents for Erdos-Falconer type point configuration problems. Let T_k(E) denote the set of distinct congruent k-dimensional simplices determined by (k + 1)-tuples of points from E. For 1 ≤ k ≤ d, we prove that there exists a t_k,d < d such that, if E ⊂ R^d, d ≥ 2, with dim_H(E) > t_k,d, then the (₂^k+1)-dimensional Lebesgue measure of T_k(E) is positive. Results of this type were previously obtained for triangles in the plane (k = d = 2) in [8] and for higher k and d in [7]. We improve upon those exponents, using a group action perspective, which also sheds light on the classical approach to the Falconer distance problem.