Informace o publikaci
Divisibility of spheres with measurable pieces
| Autoři | |
|---|---|
| Rok publikování | 2024 |
| Druh | Článek v odborném periodiku |
| Časopis / Zdroj | ENSEIGNEMENT MATHEMATIQUE |
| Fakulta / Pracoviště MU | |
| Citace | |
| www | https://ems.press/journals/lem/articles/14255106 |
| Doi | https://doi.org/10.4171/LEM/1058 |
| Klíčová slova | Euclidean sphere; divisibility under a group action; measurable set; special orthogonal group |
| Popis | 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. |