Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10363799" target="_blank" >RIV/00216208:11320/17:10363799 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.12" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.12</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.12" target="_blank" >10.4230/LIPIcs.SoCG.2017.12</a>

Result language

angličtina

Original language name

Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Original language description

Let d and k be integers with 1 &lt;= k &lt;= d-1. Let Lambda be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in the intersection of Lambda with K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n * ... * n grid is at least Omega(n^(d(d-k)/(d-1)-epsilon)) and at most O(n^(d(d-k)/(d-1))), where epsilon &gt; 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K. We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d &gt; =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA14-14179S" target="_blank" >GA14-14179S: Algorithmic, structural and complexity aspects of configurations in the plane</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2017

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

33rd International Symposium on Computational Geometry (SoCG 2017)

ISBN

978-3-95977-038-5

ISSN

1868-8969

e-ISSN

neuvedeno

Number of pages

16

Pages from-to

1-16

Publisher name

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Place of publication

Dagstuhl, Germany

Event location

Brisbane

Event date

Jul 4, 2017

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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