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 ${\dim }_{\mathcal{H}}(E)>t_{k,d}$, then the $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$-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.