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

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d)<d$ such that, if $E\subset {\mathbb{R}}^d,d\ge 2$, with $di{m}_{\mathcal{H}}(E)>s_0(d)$, then the $\left(\genfrac{}{}{0px}{}{k+1}{2}\right)$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{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.