Divisibility of spheres with measurable pieces
Clinton T. Conley, Jan Grebík, Oleg Pikhurko · L’Enseignement Mathématique · 2024
For an r -tuple (\gamma_1,\ldots,\gamma_r) of special orthogonal d\times d matrices, we say that the Euclidean (d-1) -dimensional sphere \mathbb S^{d-1} is (\gamma_1,\ldots,\gamma_r) -divisible if there is a subset A\subseteq \mathbb S^{d-1} such that its translations by the rotations \gamma_1,\ldots,\gamma_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.