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- Title
A note on distinct distances.
- Authors
Raz, Orit E.
- Abstract
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 (|S| \\ 2) 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.
- Subjects
DISTANCES; POINT set theory; ALGEBRAIC curves; ALGEBRAIC varieties
- Publication
Combinatorics, Probability & Computing, 2020, Vol 29, Issue 5, p650
- ISSN
0963-5483
- Publication type
Article
- DOI
10.1017/S096354832000022X