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

Allan Greenleaf, Alex Iosevich, Bochen Liu, Eyvindur Palsson

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

Abstract

We obtain nontrivial exponents for Erdos-Falconer type point configuration problems. Let Tk(E) denote the set of distinct congruent k-dimensional simplices determined by (k + 1)-tuples of points from E. For 1 ≤ k ≤ d, we prove that there exists a tk,d < d such that, if E ⊂ Rd, d ≥ 2, with dimH(E) > tk,d, then the (2k+1)-dimensional Lebesgue measure of Tk(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.

Original languageEnglish
Pages (from-to)799-810
Number of pages12
JournalRevista Matematica Iberoamericana
Volume31
Issue number3
DOIs
StatePublished - 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© European Mathematical Society.

Keywords

  • Erdos-Falconer problems
  • Mattila integral

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