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7009 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 76 of 351

  • A note on distinct distances

  • Part of
  • Orit E. Raz
  • Journal:
    Combinatorics, Probability and Computing / Volume 29 / Issue 5 / September 2020
    Published online by Cambridge University Press:
    16 July 2020, pp. 650-663

  • We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2].

    We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S ⊂ P of size at least Ω(n4/(9+12(d−1))), such that S spans $\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$ distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].

    Both results are consequences of a common technical tool.

Page 76 of 351