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)
$g (n) = Ω (n (4 e /(5 e - 1)) - ɛ)$ (Tardos 2003)
$g (n) = Ω (n ((48 - 14 e)/(55 - 16 e)) - ɛ)$ (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 $g d (n) = Ω (n 2/ d - 2/ d (d +2))$ .

References

Chung, Fan (1984), "The number of different distances determined by $n$ points in the plane" (PDF), Journal of Combinatorial Theory, Series A, 36: 342&ndash, 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&ndash, 354, doi:10.1007/BF02187820, MR 1134448 .
Erdős, Paul (1946), "On sets of distances of $n$ points" (PDF), American Mathematical Monthly, 53: 248&ndash, 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 1310.52019 . 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&ndash, 91, doi:10.2307/2307105, MR 0046663 .
Solymosi, József; Tóth, Csaba D. (2001), "Distinct Distances in the Plane", Discrete and Computational Geometry, 25: 629&ndash, 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&ndash, 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&ndash, 10, doi:10.1017/S0963548397002976, MR 1464571 .
Tardos, Gábor (2003), "On distinct sums and distinct distances", Advances in Mathematics, 180: 275&ndash, 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.

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