Revista Matemática Iberoamericana


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Volume 31, Issue 3, 2015, pp. 799–810
DOI: 10.4171/RMI/854

Published online: 2015-10-29

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

Allan Greenleaf[1], Alex Iosevich[2], Bochen Liu[3] and Eyvindur Palsson[4]

(1) University of Rochester, USA
(2) University of Rochester, USA
(3) University of Rochester, USA
(4) Williams College, Williamstown, USA

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.

Keywords: Erdős–Falconer problems, Mattila integral

Greenleaf Allan, Iosevich Alex, Liu Bochen, Palsson Eyvindur: A group-theoretic viewpoint on Erdős–Falconer problems and the Mattila integral. Rev. Mat. Iberoamericana 31 (2015), 799-810. doi: 10.4171/RMI/854