Incidences between Points and Lines in Three Dimensions

Micha Sharir, Noam Solomon

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R3, so that no plane contains more than s lines, is (eqution presented) (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n). This bound, originally obtained by Guth and Katz [8] as a major step in their solution of Erdos s distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth [6]. The present paper presents a different and simpler derivation, with better bounds than those in [6], and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.

Original languageEnglish
Title of host publication31st International Symposium on Computational Geometry, SoCG 2015
EditorsJanos Pach, Janos Pach, Lars Arge
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages16
ISBN (Electronic)9783939897835
StatePublished - 1 Jun 2015
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: 22 Jun 201525 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference31st International Symposium on Computational Geometry, SoCG 2015


  • Algebraic Geometry
  • Combinatorial Geometry
  • Incidences
  • The Polynomial Method


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