Israel Journal of Mathematics

, Volume 230, Issue 2, pp 949–972 | Cite as

On the Hausdorff dimension of pinned distance sets

  • Pablo ShmerkinEmail author


We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|xy| : yA} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension 1 (in particular, for many xA). This verifies a strong variant of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint s = 1.


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  1. [1]
    J. Bourgain, On the Erdős–Volkmann and Katz–Tao ring conjectures, Geometric and Functional Analysis 13 (2003), 334–365.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    C. A. Cabrelli, K. E. Hare and U. M. Molter, Sums of Cantor sets yielding an interval, Journal of the Australian Mathematical Society 73 (2002), 405–418.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.Google Scholar
  4. [4]
    K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), 206–212.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 2014.Google Scholar
  6. [6]
    A. Ferguson, J. M. Fraser and T. Sahlsten, Scaling scenery of (×m,×n) invariant measures, Advances in Mathematics 268 (2015), 564–602.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    K. Hambrook, A. Iosevich and A. Rice, Group actions and a multi-parameter falconer distance problem, preprint, arXiv:1705.03871.Google Scholar
  8. [8]
    M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Annals of Mathematics 180 (2014), 773–822.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Annals of Mathematics 175 (2012), 1001–1059.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Iosevich and B. Liu, Pinned distance problem, slicing measures and local smoothing estimates, preprint, arXiv:1706.09851, 2017.zbMATHGoogle Scholar
  11. [11]
    N. H. Katz and T. Tao, Some connections between Falconer’s distance set conjecture and sets of Furstenburg type, New York Journal of Mathematics 7:149–187, 2001.MathSciNetzbMATHGoogle Scholar
  12. [12]
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, Cambridge, 1995.Google Scholar
  13. [13]
    P. Mattila, Hausdorff dimension, projections, and the fourier transform, Publicacions Matemàtiques 48 (2004), 3–48.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    P. Mattila and T. Orponen, Hausdorff dimension, intersections of projections and exceptional plane sections, Proceedings of the American Mathematical Society 144 (2016), 3419–3430.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    T. Orponen, On the distance sets of self-similar sets, Nonlinearity 25 (2012), 1919–1929.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    T. Orponen, On the distance sets of Ahlfors–David regular sets, Advances inMathematics 307 (2017), 1029–1045.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    R. Pemantle and Y. Peres, Galton–Watson trees with the same mean have the same polar sets, Annals of Probability 23 (1995), 1102–1124.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Y. Peres and W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Mathematical Journal 102 (2000), 193–251.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Y. Peres and P. Shmerkin, Resonance between Cantor sets, Ergodic Theory and Dynamical Systems 29 (2009), 201–221.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    P. Shmerkin, On distance sets, box-counting and Ahlfors-regular sets, Discrete Analysis 2017 (2017), Paper No. 9.Google Scholar
  21. [21]
    T. Wolff, Decay of circular means of Fourier transforms of measures, International Mathematics Research Notices 10 (1999), 547–567.MathSciNetCrossRefzbMATHGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsTorcuato Di Tella UniversityBuenos AiresArgentina
  2. 2.CONICET, Godoy CruzBuenos AiresArgentina

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