Near optimal bounds for the Erdős distinct distances problem in high dimensions

Abstract

We show that the number of distinct distances in a set of n points in ℝd is Ω(n 2/d − 2 / d(d + 2)), d ≥ 3. Erdős’ conjecture is Ω(n 2/d).

This is a preview of subscription content, access via your institution.

References

  1. [1]

    J. Pach and P. Agarwal: Combinatorial geometry, Wiley-Interscience Series in Discrete Mathematics and Optimization, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995, xiv+354 pp.

    Google Scholar 

  2. [2]

    B. Aronov, J. Pach, M. Sharir and G. Tardos: Distinct Distances in Three and Higher Dimensions, Combinatorics, Probability and Computing 13(3) (2004), 283–293.

    Article  MathSciNet  MATH  Google Scholar 

  3. [3]

    B. Chazelle and J. Friedman: A deterministic view of random sampling and its use in geometry, Combinatorica 10(3) (1990), 229–249.

    Article  MathSciNet  MATH  Google Scholar 

  4. [4]

    F. Chung: The number of different distances determined by n points in the plane, J. Combin. Theory Ser. A 36(3) (1984), 342–354.

    Article  MathSciNet  MATH  Google Scholar 

  5. [5]

    F. Chung, E. Szemerédi and W. Trotter: The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7(1) (1992), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  6. [6]

    K. Clarkson, H. Edelsbrunner, L. Gubias, M. Sharir and E. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), 99–160.

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    P. Erdős: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

    Article  MathSciNet  Google Scholar 

  8. [8]

    A. Iosevich: Curvature, Combinatorics, and the Fourier Transform; Notices of the American Mathematical Society 48 (2001), 577–583.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    A. Iosevich: Szemerédi-Trotter incidence theorem, related results, and amusing consequences; in Proceedings of Minicorsi di Analisi Matematica, Padova (to appear).

  10. [10]

    J. Matousek: Lectures on Discrete Geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, 2002, xvi+481 pp.

    Google Scholar 

  11. [11]

    L. Moser: On the different distances determined by n points, Amer. Math. Monthly 59 (1952), 85–91.

    Article  MathSciNet  MATH  Google Scholar 

  12. [12]

    J. Solymosi and Cs. D. Tóth: Distinct distances in the plane, Discrete Comput. Geom. 25(4) (The Micha Sharir birthday issue) (2001), 629–634.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    J. Solymosi and V. H. Vu: Distinct distances in high dimensional homogeneous sets, in Towards a Theory of Geometric Graphs (J. Pach, ed.), pp. 259–268, Contemporary Mathematics, vol. 342, Amer. Math. Soc., 2004.

  14. [14]

    L. Székely: Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6(3) (1997), 353–358.

    Article  MathSciNet  MATH  Google Scholar 

  15. [15]

    G. Tardos: On distinct sums and distinct distances, Advances in Mathematics 180(1) (2003), 275–289.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to József Solymosi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Solymosi, J., Vu, V.H. Near optimal bounds for the Erdős distinct distances problem in high dimensions. Combinatorica 28, 113–125 (2008). https://doi.org/10.1007/s00493-008-2099-1

Download citation

Mathematics Subject Classification (2000)

  • 52C10