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

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

11 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	American 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
Externally published	Yes

Bibliographical note

Funding Information:
Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation . Work by Micha Sharir was also supported by Grant 2012/229 from the US–Israel Binational Science Foundation , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Oliver Roche-Newton was supported by the Austrian Science Fund (FWF), Project F5511-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation.

Publisher Copyright:
© 2015 Elsevier B.V.

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",

note = "Funding Information: Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation . Work by Micha Sharir was also supported by Grant 2012/229 from the US–Israel Binational Science Foundation , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Oliver Roche-Newton was supported by the Austrian Science Fund (FWF), Project F5511-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. Publisher Copyright: {\textcopyright} 2015 Elsevier B.V.",

year = "2015",

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doi = "10.1016/j.disc.2015.02.009",

language = "American English",

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AU - Roche-Newton, Oliver

AU - Sharir, Micha

N1 - Funding Information: Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation . Work by Micha Sharir was also supported by Grant 2012/229 from the US–Israel Binational Science Foundation , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Oliver Roche-Newton was supported by the Austrian Science Fund (FWF), Project F5511-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. Publisher Copyright: © 2015 Elsevier B.V.

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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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U2 - 10.1016/j.disc.2015.02.009

DO - 10.1016/j.disc.2015.02.009

M3 - Article

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SN - 0012-365X

VL - 338

SP - 1484

EP - 1492

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

ER -

Sets with few distinct distances do not have heavy lines

Abstract

Bibliographical note

Keywords

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