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).
Similar content being viewed by others
References
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.
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.
B. Chazelle and J. Friedman: A deterministic view of random sampling and its use in geometry, Combinatorica 10(3) (1990), 229–249.
F. Chung: The number of different distances determined by n points in the plane, J. Combin. Theory Ser. A 36(3) (1984), 342–354.
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.
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.
P. Erdős: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.
A. Iosevich: Curvature, Combinatorics, and the Fourier Transform; Notices of the American Mathematical Society 48 (2001), 577–583.
A. Iosevich: Szemerédi-Trotter incidence theorem, related results, and amusing consequences; in Proceedings of Minicorsi di Analisi Matematica, Padova (to appear).
J. Matousek: Lectures on Discrete Geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, 2002, xvi+481 pp.
L. Moser: On the different distances determined by n points, Amer. Math. Monthly 59 (1952), 85–91.
J. Solymosi and Cs. D. Tóth: Distinct distances in the plane, Discrete Comput. Geom. 25(4) (The Micha Sharir birthday issue) (2001), 629–634.
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.
L. Székely: Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6(3) (1997), 353–358.
G. Tardos: On distinct sums and distinct distances, Advances in Mathematics 180(1) (2003), 275–289.
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-008-2099-1