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Divisibility of spheres with measurable pieces
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Conley, Clinton T., Grebík, Jan and Pikhurko, Oleg (2024) Divisibility of spheres with measurable pieces. L'Enseignement Mathematique . doi:10.4171/LEM/1058 ISSN 0013-8584. (In Press)
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Official URL: https://doi.org/10.4171/LEM/1058
Abstract
For an r-tuple (γ 1 ,…,γ r ) of special orthogonal d×d matrices, we say that the Euclidean (d−1)-dimensional sphere S d−1 is (γ 1 ,…,γ r )-divisible if there is a subset A⊆Sd−1 such that its translations by the rotations γ
1 ,…,γ r partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set A has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) r-tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" r-tuple, divisibility is possible with parts that have the property of Baire.
Item Type: | Journal Article | |||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Journal or Publication Title: | L'Enseignement Mathematique | |||||||||
Publisher: | EMS Press | |||||||||
ISSN: | 0013-8584 | |||||||||
Official Date: | 9 February 2024 | |||||||||
Dates: |
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DOI: | 10.4171/LEM/1058 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | In Press | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Copyright Holders: | © 2024 EMS Press EMS Press is an imprint of the European Mathematical Society - EMS - Publishing House GmbH, a subsidiary of the European Mathematical Society. | |||||||||
Date of first compliant deposit: | 22 July 2022 | |||||||||
Date of first compliant Open Access: | 13 March 2024 | |||||||||
Funder: | Jan Grebík was supported by Leverhulme Research Project Grant RPG-2018-424. Oleg Pikhurko was supported by ERC Advanced Grant 101020255 and Leverhulme Research Project Grant RPG-2018-424. | |||||||||
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