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On Falconer’s distance set problem in the plane

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

Abstract

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.

Notes

Acknowledgements

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).

References

  1. 1.
    Bourgain, J.: Hausdorff dimension and distance sets. Israle J. Math. 87(1–3), 193–201 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bourgain, J.: On the Erdős-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13(2), 334–365 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bourgain, J., Demeter, C.: The proof of the \(l^2\) decoupling conjecture. Ann. Math. (2) 182(1), 351–389 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Du, X., Guth, L., Li, X.: A sharp Schrödinger maximal estimate in \(\mathbb{R}^2\). Ann. Math. 186, 607–640 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Du, X., Guth, L., Li, X., Zhang, R.: Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimate (2018). arXiv:1803.01720
  6. 6.
    Du, X., Guth, L., Ou, Y., Wang, H., Wilson, B., Zhang, R.: Weighted restriction estimates and application to Falconer distance set problem (2018). arXiv:1802.10186
  7. 7.
    Du, X., Zhang, R.: Sharp \(L^2\) estimate of Schrödinger maximal function in higher dimensions (2018). arXiv:1805.02775
  8. 8.
    Eswarathasan, S., Iosevich, A., Taylor, K.: Fourier integral operators, fractal sets and the regular value theorem. Adv. Math. 228, 2385–2402 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Erdoğan, B.: A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not. 2005(23), 1411–1425 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Erdös, P.: On sets of distances of n points. Am. Math. Mon. 53, 248–250 (1946)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Falconer, K.J.: On the Hausdorff dimensions of distance sets. Mathematika 32(2), 206–212 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Garibaldi, J.: Erdös distance problem for convex metrics. Thesis (Ph.D.), University of California, Los Angeles (2004)Google Scholar
  13. 13.
    Guth, L.: Restriction estimates using polynomial partitioning II (2016). arXiv:1603.04250
  14. 14.
    Guth, L., Katz, N.H.: On the Erdős distinct distance problem in the plane. Ann. Math. (2) 181(1), 155–190 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Herz, C.S.: Fourier transforms related to convex sets. Ann. Math. (2) 75(1), 81–92 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hofmann, S., Iosevich, A.: Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics. Proc. Am. Mat. Soc. 133, 133–144 (2005)zbMATHCrossRefGoogle Scholar
  17. 17.
    Iosevich, A., Rudnev, M., Uriarte-Tuero, I.: Theory of dimension for large discrete sets and applications. Math. Model. Nat. Phenom. 9(5), 148–169 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Iosevich, A., Łaba, I.: K-distance sets, Falconer conjecture, and discrete analogs. Integers Electron. J. Comb. Number Theory 5 (2005). #A08 (In: Topics in Combinatorial Number Theory: Proceedings of the Integers Conference 2003 in Honor of Tom Brown, DIMATIA, ITI Series, vol. 261)Google Scholar
  19. 19.
    Iosevich, A., Liu, B.: Pinned distance problem, slicing measures and local smoothing estimates (2017). arXiv:1706.09851
  20. 20.
    Iosevich, A., Rudnev, M.: Distance measures for well-distributed sets. Discret. Comput. Geom. 38, 61–80 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Iosevich, A., Senger, S.: Sharpness of Falconer’s \(\frac{d+1}{2}\) estimate. Ann. Acad. Sci. Fenn. Math. 41(2), 713–720 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Katz, N., Tao, T.: Some connections between Falconer’s distance set conjecture and sets of Furstenburg type. N. Y. J. Math. 7, 149–187 (2001)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Katz, N., Tardos, G.: A new entropy inequality for the Erdos distance problem. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs, vol. 342, pp. 119–126. Contemporary Mathematics Publication, American Mathematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Keleti, T., Shmerkin, P.: New bounds on the dimensions of planar distance sets (2018). arXiv:1801.08745
  25. 25.
    Liu, B.: An \(L^2\)-identity and pinned distance problem (2018). arXiv:1802.00350
  26. 26.
    Mattila, P.: On the Hausdorff dimension and capacities of intersections. Mathematika 32, 213–217 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mattila, P.: Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets. Mathematika 34, 207–228 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Moser, L.: On the different distances determined by n points. Am. Math. Mon. 59(2), 85–91 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Orponen, T.: On the distance sets of Ahlfors-David regular sets. Adv. Math. 307, 1029–1045 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Orponen, T.: On the dimension and smoothness of radial projections (2017). arXiv:1710.11053v2 (preprint)
  31. 31.
    Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions and the dimension of exceptions. Duke Math J. 102, 193–251 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Shmerkin, P.: On distance sets, box-counting and Ahlfors regular sets. Discret. Anal. 9, 22 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Shmerkin, P.: On the Hausdorff dimension of pinned distance sets (2017). arXiv:1706.00131
  34. 34.
    Solymosi, J., Vu, V.: Near optimal bounds for the Erdős distinct distances problem in high dimensions. Combinatorica 28(1), 113–125 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Stein, E.: Harmonic Analysis. Princeton University Press, Princeton (1993)Google Scholar
  36. 36.
    Wolff, T.: Decay of circular means of Fourier transforms of measures. Int. Math. Res. Not. 1999(10), 547–567 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Wolff, T.: Lectures on Harmonic Analysis. University Lecture Series, vol. 29. American Mathematican Society, Providence (2003)zbMATHGoogle Scholar

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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