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Combinatorica

, Volume 36, Issue 3, pp 349–364 | Cite as

Few distinct distances implies no heavy lines or circles

  • Adam ShefferEmail author
  • Joshua Zahl
  • Frank de Zeeuw
Original Paper

Abstract

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set \(\mathcal{P}\) of n points determines o(n) distinct distances, then no line contains Ω(n 7/8) points of \(\mathcal{P}\) and no circle contains Ω(n 5/6) points of \(\mathcal{P}\).

We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].

A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.

Mathematics Subject Classication (2000)

52C10 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsMITCambridgeUSA
  3. 3.EPFLLausanneSwitzerland

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