Erdős distinct distances problem

In discrete geometry, the Erdős distinct distances problem states that between n distinct points on a plane there are at least n^{1 − o(1)} distinct distances. It was posed by Paul Erdős in 1946 and proven by Guth & Katz (2015).

The conjecture

In what follows let g(n) denote the minimal number of distinct distances between n points in the plane. In his 1946 paper, Erdős proved the estimates

${\sqrt {n-3/4}}-1/2\leq g(n)\leq cn/{\sqrt {\log n}}$

for some constant $c$. The lower bound was given by an easy argument. The upper bound is given by a ${\sqrt {n}}\times {\sqrt {n}}$ square grid. For such a grid, there are $O(n/{\sqrt {\log n}})$ numbers below n which are sums of two squares; see Landau–Ramanujan constant. Erdős conjectured that the upper bound was closer to the true value of g(n), and specifically that $g(n)=\Omega (n^{c})$ holds for every c < 1.

Partial results

Paul Erdős' 1946 lower bound of g(n) = Ω(n^1/2) was successively improved to:

• g(n) = Ω(n^2/3) (Moser 1952)
• g(n) = Ω(n^5/7) (Chung 1984)
• g(n) = Ω(n^4/5/log n) (Chung, Szemerédi & Trotter 1992)
• g(n) = Ω(n^4/5) (Székely 1993)
• g(n) = Ω(n^6/7) (Solymosi & Tóth 2001)
• g(n) = Ω(n^{(4e/(5e − 1)) − ɛ}) (Tardos 2003)
• g(n) = Ω(n^{((48 − 14e)/(55 − 16e)) − ɛ}) (Katz & Tardos 2004)
• g(n) = Ω(n/log n) (Guth & Katz 2015)

Higher dimensions

Erdős also considered the higher-dimensional variant of the problem: for d≥3 let g_d(n) denote the minimal possible number of distinct distances among n point in the d-dimensional Euclidean space. He proved that g_d(n) = Ω(n^1/d) and g_d(n) = O(n^2/d) and conjectured that the upper bound is in fact sharp, i.e., g_d(n) = Θ(n^2/d) . Solymosi & Vu (2008) obtained the lower bound g_d(n) = Ω(n^{2/d - 2/d(d+2)}).

See also

• Falconer's conjecture
• The Erdős unit distance problem

References

• Chung, Fan (1984), "The number of different distances determined by n points in the plane" (PDF), Journal of Combinatorial Theory, (A), 36: 342–354, doi:10.1016/0097-3165(84)90041-4, MR 0744082.
• Chung, Fan; Szemerédi, Endre; Trotter, William T. (1992), "The number of different distances determined by a set of points in the Euclidean plane" (PDF), Discrete and Computational Geometry, 7: 342–354, doi:10.1007/BF02187820, MR 1134448.
• Erdős, Paul (1946), "On sets of distances of n points" (PDF), American Mathematical Monthly, 53: 248–250, doi:10.2307/2305092.
• Garibaldi, Julia; Iosevich, Alex; Senger, Steven (2011), The Erdős Distance Problem, Student Mathematical Library, 56, Providence, RI: American Mathematical Society, ISBN 978-0-8218-5281-1, MR 2721878.
• Guth, Larry; Katz, Nets Hawk (2015), "On the Erdős distinct distances problem in the plane", Annals of Mathematics, 181 (1): 155–190, arXiv:1011.4105, doi:10.4007/annals.2015.181.1.2, MR 3272924, Zbl 06383662. See also The Guth-Katz bound on the Erdős distance problem by Terry Tao and Guth and Katz’s Solution of Erdős’s Distinct Distances Problem by János Pach.
• Katz, Nets Hawk; Tardos, Gábor (2004), "A new entropy inequality for the Erdős distance problem", in Pach, János, Towards a theory of geometric graphs, Contemporary Mathematics, 342, Providence, RI: American Mathematical Society, pp. 119–126, doi:10.1090/conm/342/06136, ISBN 0-8218-3484-3, MR 2065258
• Moser, Leo (1952), "On the different distances determined by n points", American Mathematical Monthly, 59: 85–91, doi:10.2307/2307105, MR 0046663.
• Solymosi, József; Tóth, Csaba D. (2001), "Distinct Distances in the Plane", Discrete and Computational Geometry, 25: 629–634, doi:10.1007/s00454-001-0009-z, MR 1838423.
• Solymosi, József; Vu, Van H. (2008), "Near optimal bounds for the Erdős distinct distances problem in high dimensions", Combinatorica, 28: 113–125, doi:10.1007/s00493-008-2099-1, MR 2399013.
• Székely, László A. (1993), "Crossing numbers and hard Erdös problems in discrete geometry", Combinatorics, Probability and Computing, 11: 1–10, doi:10.1017/S0963548397002976, MR 1464571.
• Tardos, Gábor (2003), "On distinct sums and distinct distances", Advances in Mathematics, 180: 275–289, doi:10.1016/s0001-8708(03)00004-5, MR 2019225.

External links

• William Gasarch's page on the problem
• János Pach's guest post on Gil Kalai's blog
• Terry Tao's blog entry on the Guth-Katz proof, gives a detailed exposition of the proof.