JournalsrmiVol. 31, No. 3pp. 799–810

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

We obtain nontrivial exponents for Erdős–Falconer type point configuration problems. Let Tk(E)T_k(E) denote the set of distinct congruent kk-dimensional simplices determined by (k+1)(k+1)-tuples of points from EE. For 1kd1 \le k \le d, we prove that there exists a tk,d<dt_{k,d} < d such that, if ERdE \subset {\mathbb R}^d, d2d \ge 2, with dimH(E)>tk,d\mathrm {dim}_{{\mathcal H}}(E)>t_{k,d}, then the (k+12){k+1 \choose 2}-imensional Lebesgue measure of Tk(E)T_k(E) is positive. Results of this type were previously obtained for triangles in the plane (k=d=2)(k=d=2) in [8] and for higher kk and dd 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