Divisibility of spheres with measurable pieces

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F24%3A00138608" target="_blank" >RIV/00216224:14330/24:00138608 - isvavai.cz</a>
Result on the web
<a href="https://ems.press/journals/lem/articles/14255106" target="_blank" >https://ems.press/journals/lem/articles/14255106</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4171/LEM/1058" target="_blank" >10.4171/LEM/1058</a>

Result language
angličtina
Original language name
Divisibility of spheres with measurable pieces
Original language description
For an r-tuple (y 1 , ... , y r ) of special orthogonal d x d matrices, we say that the Euclidean (d - 1) -dimensional sphere S d-1 is (y 1 , ... , y r ) -divisible if there is a subset A c S d-1 such that its translations by the rotations y 1 , ... , y 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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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