A point set everywhere dense in the plane
type: Item
instance of: scholarly article
title: A point set everywhere dense in the plane
zbMATH Open document ID: 0573.52003
author: Károly Bezdek
author: János Pach
published in: Elemente der Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141360
review text: Let \(0_ 1\), \(0_ 2\) be two points in the plane having mutual distance at most 2. Put \(S_ 1=\{0_ 1,0_ 2\}\). If \(S_ i\), \(i\geq 1\), has already been determined, let \(S_{i+1}\) be the union of \(S_ i\) and the set of all intersection points of unit circles whose centers are in \(S_ i.\) The authors prove the following conjecture of L. Fejes-Tóth: The set \(\cup^{\infty}_{i=1}S_ i\) is either identical with the set of vertices of a regular triangle-lattice of side 1, or it is everywhere dense in the plane. It is shown, that this result is valid for Minkowski planes. Similar assertions can be proved for hyperbolic planes and spheres.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 3915104
zbMATH Keywords: point sets everywhere dense in the plane
reviewed by: Valeriu Soltan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3915104