We obtain nontrivial exponents for Erdős–Falconer type point configuration problems. Let denote the set of distinct congruent -dimensional simplices determined by -tuples of points from . For , we prove that there exists a such that, if , , with , then the -imensional Lebesgue measure of is positive. Results of this type were previously obtained for triangles in the plane in  and for higher and in . We improve upon those exponents, using a group action perspective, which also sheds light on the classical approach to the Falconer distance problem.
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Allan Greenleaf, Alex Iosevich, Bochen Liu, Eyvindur Palsson, A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral. Rev. Mat. Iberoam. 31 (2015), no. 3, pp. 799–810DOI 10.4171/RMI/854