Israel Journal of Mathematics

, Volume 87, Issue 1–3, pp 193–201 | Cite as

Hausdorff dimension and distance sets

  • Jean Bourgain


According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, yA} of distances for a Souslin setA of ℝ n has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject.


Maximal Function HAUSDORFF Dimension Image Measure Fine Theory Spherical Average 


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Copyright information

© Hebrew University 1994

Authors and Affiliations

  • Jean Bourgain
    • 1
  1. 1.Département de MathématiquesI.H.E.S.Bures-sur-YvetteFrance

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