Sets with few distinct distances do not have heavy lines

Orit E. Raz; Oliver Roche-Newton; Micha Sharir

doi:10.1016/j.disc.2015.02.009

Sets with few distinct distances do not have heavy lines

Orit E. Raz^*, Oliver Roche-Newton, Micha Sharir

^*Corresponding author for this work

School of Computer Science

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n43^/52polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

Original language	English
Pages (from-to)	1484-1492
Number of pages	9
Journal	Discrete Mathematics
Volume	338
Issue number	8
DOIs	https://doi.org/10.1016/j.disc.2015.02.009
State	Published - 6 Aug 2015

Funding

Funders	Funder number
Austrian Science Fund	F5511-N26
National Science Foundation
Israeli Centers for Research Excellence	4/11
United States-Israel Binational Science Foundation
Israel Science Foundation	2012/229
Hermann Minkowski-MINERVA Center for Geometry
Tel Aviv University

Keywords

Distinct distances
Incidences

Access to Document

10.1016/j.disc.2015.02.009

Cite this

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title = "Sets with few distinct distances do not have heavy lines",

abstract = "Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n43/52polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.",

keywords = "Distinct distances, Incidences",

author = "Raz, {Orit E.} and Oliver Roche-Newton and Micha Sharir",

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AU - Raz, Orit E.

AU - Roche-Newton, Oliver

AU - Sharir, Micha

PY - 2015/8/6

Y1 - 2015/8/6

N2 - Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n43/52polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

AB - Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n43/52polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

KW - Distinct distances

KW - Incidences

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JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

ER -

Sets with few distinct distances do not have heavy lines

Abstract

Funding

Keywords

Access to Document

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