# A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral

### Allan Greenleaf

University of Rochester, USA### Alex Iosevich

University of Rochester, USA### Bochen Liu

University of Rochester, USA### Eyvindur Palsson

Williams College, Williamstown, USA

## 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 $\mathrm {dim}_{{\mathcal H}}(E)>t_{k,d}$, then the ${k+1 \choose 2}$-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.

## Cite this article

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. 799–810

DOI 10.4171/RMI/854