Erdős distinct distances problem

In what follows let g(n) denote the minimal number of distinct distances between n points in the plane, or equivalently the smallest possible cardinality of their distance set. In his 1946 paper, Erdős proved the estimates

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

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

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)

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

• Falconer's conjecture
• The Erdős unit distance problem

• Chung, Fan (1984), "The number of different distances determined by n points in the plane" (PDF), Journal of Combinatorial Theory, Series A, 36 (3): 342–354, doi:10.1016/0097-3165(84)90041-4, MR 0744082CS1 maint: ref=harv (link).
• 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 1134448CS1 maint: ref=harv (link).
• Erdős, Paul (1946), "On sets of distances of n points" (PDF), American Mathematical Monthly, 53 (5): 248–250, doi:10.2307/2305092, JSTOR 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 1310.52019CS1 maint: ref=harv (link). 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 (ed.), Towards a theory of geometric graphs, Contemporary Mathematics, 342, Providence, RI: American Mathematical Society, pp. 119–126, doi:10.1090/conm/342/06136, ISBN 978-0-8218-3484-8, MR 2065258
• Moser, Leo (1952), "On the different distances determined by n points", American Mathematical Monthly, 59 (2): 85–91, doi:10.2307/2307105, JSTOR 2307105, MR 0046663CS1 maint: ref=harv (link).
• Solymosi, József; Tóth, Csaba D. (2001), "Distinct Distances in the Plane", Discrete and Computational Geometry, 25 (4): 629–634, doi:10.1007/s00454-001-0009-z, MR 1838423CS1 maint: ref=harv (link).
• 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 2399013CS1 maint: ref=harv (link).
• Székely, László A. (1993), "Crossing numbers and hard Erdös problems in discrete geometry", Combinatorics, Probability and Computing, 11 (3): 1–10, doi:10.1017/S0963548397002976, MR 1464571CS1 maint: ref=harv (link).
• 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.

• 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.