论文标题
球体与可测量的零件的分裂性
Divisibility of Spheres with Measurable Pieces
论文作者
论文摘要
对于特殊正交$ d \ times d $矩阵的$ r $ -tuple $（γ_1，\ ldots，γ_r）$，我们说euclidean $（d-1）$ dimensional $ s $ s^{d-1} s^{d-1} $，使其翻译由旋转$γ_1，\ ldots，γ_R$分区球形。由Mycielski和Wagon的一些古老的开放问题激励，我们研究了该概念的版本，在该概念上，对于球形测量，必须可衡量$ a $。我们的主要结果表明，对于“通用”（各种含义）$ r $ r $ tuple的旋转，不可能可衡量的可划分性。这与Conley，Marks和Unger的最新结果形成鲜明对比，这意味着，对于每个具有Baire属性的零件，对于每个“通用” $ r $ tuple来说，可以分裂。
For an $r$-tuple $(γ_1,\ldots,γ_r)$ of special orthogonal $d\times d$ matrices, we say that the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is $(γ_1,\ldots,γ_r)$-divisible if there is a subset $A\subseteq S^{d-1}$ such that its translations by the rotations $γ_1,\ldots,γ_r$ 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.
