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

Abstract

We obtain nontrivial exponents for Erdős–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 \le k \le d , we prove that there exists a t_{k,d} < d such that, if E \subset {\mathbb R}^d , d \ge 2 , with \mathrm {dim}_{{\mathcal H}}(E)>t_{k,d} , then the {k+1 \choose 2} -imensional 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.