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Unit Distances in Three Dimensions


We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

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[28]Zahl, J. (2011) An improved bound on the number of point-surface incidences in three dimensions. arXiv:1104.4987. First posted (v1) 26 April 2011; revised and corrected 22 September 2011.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
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