On Falconer’s distance set problem in the plane

  • Larry Guth
  • Alex Iosevich
  • Yumeng OuEmail author
  • Hong Wang


If \(E \subset \mathbb {R}^2\) is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point \(x \in E\) so that the set of distances \(\{ |x-y| \}_{y \in E}\) has positive Lebesgue measure.



The authors are grateful to the anonymous referees for helpful comments and suggestions. Larry Guth is supported by a Simons Investigator grant. Alex Iosevich is supported in part by the NSA Grant H98230-15-0319. Yumeng Ou is supported in part by NSF-DMS #1854148 (previously #1764454).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Larry Guth
    • 1
  • Alex Iosevich
    • 2
  • Yumeng Ou
    • 3
    Email author
  • Hong Wang
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of MathematicsUniversity of RochesterRochesterUSA
  3. 3.Department of MathematicsCity University of New York, Baruch CollegeNew YorkUSA

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