Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 3
On the Multiplicity of Arrangements of Congruent Zones on the Sphere
Authors:
András Bezdek Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA

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Ferenc Fodor Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary

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Viktor Vígh Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary

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Tamás Zarnócz Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary

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Pages:
262–273
Online Publication Date:
01 Oct 2024
Publication Date:
21 Nov 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2024.04319
Keywords:
Covering multiplicity; density; Euclidean plank; spherical zone
Restricted access

Consider an arrangement of 𝑛 congruent zones on the 𝑑-dimensional unit sphere 𝑆𝑑−1, where a zone is the intersection of an origin symmetric Euclidean plank with 𝑆𝑑−1. We prove that, for sufficiently large 𝑛, it is possible to arrange 𝑛 congruent zones of suitable width on 𝑆𝑑−1 such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover 𝑆𝑑−1 by 𝑛 congruent zones such that each point of 𝑆𝑑−1 belongs to at most 𝐴𝑑 ln 𝑛 zones, where the 𝐴𝑑 is a constant that depends only on 𝑑. This extends the corresponding 3-dimensional result of Frankl, Nagy and Naszódi [8]. Moreover, we also examine coverings of 𝑆𝑑−1 with congruent zones under the condition that each point of the sphere belongs to the interior of at most 𝑑 − 1 zones.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

Editorial Board

  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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