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

Abstract

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 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$-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.

11 pages. Many improvements based on referee comments. To appear, Revista Matemática Iberoamericana

References in corpus (1)

• On the number of non-congruent lattice tetrahedra

Cited by in corpus (2)

• On the number of non-congruent lattice tetrahedra
• Dimensions of triangle sets