On the Number of Rich Lines in Truly High Dimensional Sets

Zeev Dvir, Sivakanth Gopi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a truly d-dimensional configuration of points v1, . . . , vn ε ℂd. More formally, we show that, if the number of r-rich lines is significantly larger than n2/rd then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor rd can be replaced with a tight rd+1. If true, this would generalize the classic Szemerédi-Trotter theorem which gives a bound of n2/r3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R3 in the seminal work of Guth and Katz [7] and was also recently proved over R4 (under some additional restrictions) [14]. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r - 2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of [1] to convert these local linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.

Original languageEnglish (US)
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
Pages584-598
Number of pages15
ISBN (Electronic)9783939897835
DOIs
StatePublished - Jun 1 2015
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: Jun 22 2015Jun 25 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume34
ISSN (Print)1868-8969

Other

Other31st International Symposium on Computational Geometry, SoCG 2015
CountryNetherlands
CityEindhoven
Period6/22/156/25/15

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Additive Combinatorics
  • Combinatorial Geometry
  • Designs
  • Incidences
  • Polynomial Method

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