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Erdős’s Unit Distance Problem

  • Endre Szemerédi
Chapter

Abstract

We survey some problems and results around one of Paul Erdős’s favorite questions, first published 70 years ago: What is the maximum number of times that the unit distance can occur among n points in the plane? This simple and beautiful question has generated a lot of important research in discrete geometry, in extremal combinatorics, in additive number theory, in Fourier analysis, in algebra, and in other fields, but we still do not seem to be close to a satisfactory answer.

Keywords

Unit Circle Unit Distance Geometric Graph Combinatorial Geometry Diameter Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I would like to thank János Pach and József Solymosi for many interesting discussions on the topic of this survey, and for their help in preparing the manuscript. Work was supported by ERC-AdG. 321104, and OTKA NK 104183 grants.

References

  1. 1.
    M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi: Crossing free graphs, Ann. Discrete Math. 12 (1982) 9–12.Google Scholar
  2. 2.
    F. Amoroso and E. Viada: Small points on subvarieties of a torus, Duke Mathematical Journal 150 (2009), No. 3, 407–442.Google Scholar
  3. 3.
    S. Avital and H. Hanani: Graphs, continuation, Gilyonot Le’matematika 3, issue 2 (1966), 2–8.Google Scholar
  4. 4.
    J. Beck and J. Spencer: Unit distances, J. Combin. Theory Ser. A 37 (1984), no. 3, 231–238.Google Scholar
  5. 5.
    B. Bollobás: Extremal Graph Theory. London Mathematical Society Monographs, 11, Academic Press, London-New York, 1978.Google Scholar
  6. 6.
    K. Borsuk: Drei Sätze iiber die n-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177–190.Google Scholar
  7. 7.
    J. Bourgain, N. Katz, and T. Tao: A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), no. 1, 27–57.Google Scholar
  8. 8.
    J. Bourgain, A. Glibichuk, and S. Konyagin: Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), no. 2, 380–398.Google Scholar
  9. 9.
    P. Brass: On lattice polyhedra and pseudocircle arrangements, in: Charlemagne and his Heritage—1200 Years of Civilization and Science in Europe, Vol. 2: Mathematical Arts, P. L. Butzer et al., eds., Brepols Verlag, 1988, 297–302.Google Scholar
  10. 10.
    P. Brass: On the maximum number of unit distances among n points in dimension four, in: Intuitive Geometry (I. Bárány et al., eds.), Bolyai Soc. Math. Studies 4, Springer, Berlin, 1997, 277–290.Google Scholar
  11. 11.
    P. Brass: On point sets with many unit distances in few directions, Discrete Comput. Geom. 19 (1998), no. 3, 355–366.Google Scholar
  12. 12.
    P. Brass, W. Moser, and J. Pach: Research Problems in Discrete Geometry, Springer, New York, 2005.Google Scholar
  13. 13.
    P. Brass and J. Pach: The maximum number of times the same distance can occur among the vertices of a convex n-gon, is \(O(n\log n)\), J. Combinatorial Theory Ser. A 94 (2001), 178–179.Google Scholar
  14. 14.
    F. R. K. Chung: On the number of different distances determined by n points in the plane, J. Combin. Theory, Ser. A 36 (1984), 342–354.Google Scholar
  15. 15.
    F. R. K. Chung, E. Szemerédi, and W. T. Trotter: The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7 (1992), 1–11.Google Scholar
  16. 16.
    K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), 99–160.Google Scholar
  17. 17.
    H. Edelsbrunner and P. Hajnal: A lower bound on the number of unit distances between the points of a convex polygon, J. Combinatorial Theory Ser. A 56 (1991), 312–316.Google Scholar
  18. 18.
    G. Elekes: On the number of sums and products, Acta Arith. 81 (1997), 365–367.Google Scholar
  19. 19.
    P. Erdős: On sequences of integers none of which divides the product of two others, Mitteilungen des Forschungsinstituts für Mathematik und Mechanik, Tomsk 2 (1938), 74–82.Google Scholar
  20. 20.
    P. Erdős: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.Google Scholar
  21. 21.
    P. Erdős: On sets of distances of n points in Euclidean space, Magyar Tudom. Akad. Matem. Kut. Int. Közl. (Publ. Math. Inst. Hung. Acad. Sci.) 5 (1960), 165–169.Google Scholar
  22. 22.
    P. Erdős, D. Hickerson, and J. Pach: A problem of Leo Moser about repeated distances on the sphere, Amer. Math. Monthly 96 (1989), 569–575.Google Scholar
  23. 23.
    P. Erdős and L. Moser: Problem 11, Canad. Math. Bulletin 2 (1959), 43.Google Scholar
  24. 24.
    P. Erdős and A. H. Stone: On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087–1091.Google Scholar
  25. 25.
    P. Erdős and E. Szemerédi: On sums and products of integers, in: Studies in Pure Mathematics. To the Memory of Paul Turán (Erdős et al., eds.), Akadémiai Kiadó–Birkhäuser, Budapest–Basel, 1983, 213–218.Google Scholar
  26. 26.
    P. Fishburn and J. A. Reeds: Unit distances between vertices of a convex polygon, Comput. Geom. Theory Appl. 2 (1992), 81–91.Google Scholar
  27. 27.
    Z. Füredi: The maximum number of unit distances in a convex n-gon, J. Combinatorial Theory Ser. A 55 (1990) 316–320.Google Scholar
  28. 28.
    B. Grünbaum: A proof of Vázsonyi’s conjecture, Bull. Res. Council Israel, Sect. A 6 (1956), 77–78.Google Scholar
  29. 29.
    L. Guth and N. H. Katz: Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. 225 (2010), no. 5, 2828–2839.Google Scholar
  30. 30.
    L. Guth and N. H. Katz: On the Erdős distinct distance problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155–190.Google Scholar
  31. 31.
    F. Harary and H. Harborth: Extremal animals, J. Combinatorics, Information & System Sciences 1 (1976), 1–8.Google Scholar
  32. 32.
    A. Heppes: Beweis einer Vermutung von A. Vázsonyi, Acta Math. Acad. Sci. Hungar. 7 (1956), 463–466.Google Scholar
  33. 33.
    H. Hopf and E. Pannwitz: Aufgabe Nr. 167, Jahresbericht d. Deutsch. Math.-Verein. 43 (1934), 114.Google Scholar
  34. 34.
    T. Jenrich: A 64-dimensional two-distance counterexample to Borsuk’s conjecture, arXiv:1308.0206.Google Scholar
  35. 35.
    S. Józsa and E. Szemerédi: The number of unit distance on the plane, in: Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Colloq. Math. Soc. János Bolyai 10, North-Holland, Amsterdam, 1975, 939–950.Google Scholar
  36. 36.
    J. Kahn and G. Kalai: A counterexample to Borsuk’s conjecture, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 60–62.Google Scholar
  37. 37.
    H. Kaplan, J. Matouěk, Z. Safernová, and M. Sharir: Unit distances in three dimensions, Combin. Probab. Comput. 21 (2012), no. 4, 597–610.Google Scholar
  38. 38.
    H. Kaplan, J. Matoušek, and M. Sharir: Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique, Discrete Comput. Geom. 48 (2012), no. 3, 499–517.Google Scholar
  39. 39.
    N. H. Katz: On arithmetic combinatorics and finite groups, Illinois J. Math. 49 (2005), 33–43.Google Scholar
  40. 40.
    N. H. Katz and G. Tardos: A new entropy inequality for the Erdős distance problem, in: Towards a Theory of Geometric Graphs, Contemp. Math. 342, Amer. Math. Soc., Providence, 2004, 119–126.Google Scholar
  41. 41.
    S. Konyagin and M. Rudnev: On new sum-product-type estimates, SIAM J. Discrete Math. 27 (2013), no. 2, 973–990.Google Scholar
  42. 42.
    A. Kupavskii: Diameter graphs in R 4, Discrete Comput. Geom. 51 (2014), no. 4, 842–858.Google Scholar
  43. 43.
    A. B. Kupavskii and A. Polyanskii: Proof of Schur’s conjecture in R d, arXiv:1402.3694v1.Google Scholar
  44. 44.
    Y. S. Kupitz: Extremal Problems of Combinatorial Geometry, Lecture Notes Series 53, Aarhus University, Denmark, 1979.Google Scholar
  45. 45.
    T. Leighton; Complexity Issues in VLSI. Foundations of Computing Series, MIT Press, Cambridge, MA, 1983.Google Scholar
  46. 46.
    H. B. Mann: On linear relations between roots of unity, Mathematika 12 (1965), 107–117.Google Scholar
  47. 47.
    J. Matoušek: The number of unit distances is almost linear for most norms, Adv. Math. 226 (2011), no. 3, 2618–2628.Google Scholar
  48. 48.
    F. Morić and J. Pach: Remarks on Schur’s conjecture, Comput. Geom. 48 (2015), no. 7, 520–527.Google Scholar
  49. 49.
    J. Pach: Geometric graph theory, in: Surveys in Combinatorics, 1999 (J. D. Lamb and D. A. Preece, eds.), London Mathematical Society Lecture Notes 267, Cambridge University Press, Cambridge, 1999, 167–200.Google Scholar
  50. 50.
    J. Pach: Geometric intersection patterns and the theory of geometric graphs, in: Proceedings of the International Congress of Mathematicians 2014 (ICM 2014, Seoul, Korea), 455–474.Google Scholar
  51. 51.
    J. Pach and M. Sharir: On the number of incidences between points and curves, Combin. Probab. Comput. 7 (1998), 121–127.Google Scholar
  52. 52.
    J. Pach and G. Tardos: Forbidden paths and cycles in ordered graphs and matrices, Israel J. Math. 155 (2006), 359–380.Google Scholar
  53. 53.
    Z. Schur, M. A. Perles, H. Martini, and Y. S. Kupitz: On the number of maximal regular simplices determined by n points in \(\mathbb{R}^{d}\), in: Discrete and Computational Geometry, The Goodman-Pollack Festschrift (Aronov et al., eds.), Algorithms Combin. 25, Springer, Berlin, 2003, 767–787.Google Scholar
  54. 54.
    R. Schwartz: Using the subspace theorem to bound unit distances, Moscow Journal of Combinatorics and Number Theory 3 (2013), No. 1, 108–117.Google Scholar
  55. 55.
    R. Schwartz, J. Solymosi, and F. de Zeeuw, Rational distances with rational angles, Mathematika 58 (2012), no. 2, 409–418.Google Scholar
  56. 56.
    J. Solymosi: On the number of sums and products, Bull. London Math. Soc. 37 (2005), no. 4, 491–494.Google Scholar
  57. 57.
    J. Solymosi and T. Tao: An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), no. 2, 255–280.Google Scholar
  58. 58.
    J. Solymosi and Cs. Tóth: Distinct distances in the plane, Discrete Comput. Geom. 25 (2001), 629–634.Google Scholar
  59. 59.
    J. Spencer, E. Szemerédi, and W. T. Trotter: Unit distances in the Euclidean plane, in: Graph Theory and Combinatorics (B. Bollobás, ed.), Academic Press, London, 1984, 293–303.Google Scholar
  60. 60.
    S. Straszewicz: Sur un problème géométrique de P. Erdős, Bull. Acad. Pol. Sci., Cl. III 5 (1957), 39–40.Google Scholar
  61. 61.
    K. J. Swanepoel: Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), 1–27.Google Scholar
  62. 62.
    K, J. Swanepoel and P. Valtr: The unit distance problem on spheres, in: Towards a Theory of Geometric Graphs, J. Pach, ed., Contemporary Mathematics 342, American Mathematical Society, Providence, 2004, 273–279.Google Scholar
  63. 63.
    L. A. Székely: Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6 (1997), 353–358.Google Scholar
  64. 64.
    E. Szemerédi and W. T. Trotter, Jr.: A combinatorial distinction between the Euclidean and projective planes, European J. Combin. 4 (1983), no. 4, 385–394.Google Scholar
  65. 65.
    E. Szemerédi and W. T. Trotter, Jr.: Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381–392.Google Scholar
  66. 66.
    T. Tao and V. H. Vu: Additive Combinatorics. Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, Cambridge, 2010.Google Scholar
  67. 67.
    C. D. Tóth: The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), no. 1, 95–126.Google Scholar
  68. 68.
    P. Turán: Egy gráfelméleti szélsőértékfeladatról, Matematikai és Fizikai Lapok 48 (1941), 436–452.Google Scholar
  69. 69.
    P. Valtr, Strictly convex norms allowing many unit distances and related touching questions, manuscript, Charles University, Prague, 2005.Google Scholar
  70. 70.
    P. van Wamelen: The maximum number of unit distances among n points in dimension four, Beiträge Algebra Geom. 40 (1999), no. 2, 475–477.Google Scholar
  71. 71.
    J. Zahl: An improved bound on the number of point-surface incidences in three dimensions, Contrib. Discrete Math. 8 (2013), no. 1, 100–121.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Renyi Alfred Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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