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

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We obtain nontrivial exponents for Erd os-Falconer type problems. Let Tₖ(E) denote the set of distinct congruent k-dimensional simplexes determined by (k+1)-tuples of points from E. We prove that there exists s₀(d)<d such that, if E ⊂ Rᵈ, d ≥ 2, with dimₕ(E)>s₀(d), then the k+1 2-dimensional Lebesgue measure of Tₖ(E) is positive. Results were previously obtained for triangles in the plane GI12 and in higher dimensions GGIP12. In this paper, we improve upon those exponents, using a group-theoretic method that sheds new light on the classical approach to these problems. The key to our approach is a group action perspective which leads to natural and effective formulae related to the classical Mattila integral.