On the Mattila-Sjölin theorem for distance sets

Alex Iosevich, Mihalis Mourgoglou, Krystal Taylor

Research output: Contribution to journalArticlepeer-review

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Abstract

We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set E⊂ Rd, d ≥2, is greater than d+1/2, then the distance set Δ(E) = {|x-y|:x,y ∈ E} contains an interval. We prove this result for distance sets ΔB(E) = {||x-y||B: x,y ∈ E}, where ||·||B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.

Original languageEnglish (US)
Pages (from-to)557-562
Number of pages6
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume37
Issue number1
DOIs
StatePublished - Feb 2012

Keywords

  • Arithmetic of the lattice
  • Bilinear operators
  • Distribution of angles
  • Erdos problems
  • Falconer distance problem

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