SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2502-2509, January 2020.
We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical Kakeya sets and center spherical Kakeya sets, respectively. For $n\ge 4$ we prove a general lower bound on the size of any set containing $q-1$ different spheres which applies to both kinds of spherical Kakeya sets. We provide constructions which meet the main terms of this lower bound. We also give a construction showing that we cannot get a lower bound of order of magnitude $q^n$ if we take lower-dimensional objects, such as circles in $\mathbb{F}_q^3$ instead of spheres, showing that there are significant differences to the line Kakeya problem. Finally, we study the case of dimension $n=1$, which is different and equivalent to the study of sum and difference sets that cover $\mathbb{F}_q$.

中文翻译：

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有限域中的球形Kakeya问题

SIAM离散数学杂志，第34卷，第4期，第2502-2509页，2020年1月。
我们研究有限域$ \ mathbb {F} _q $上$ n $维向量空间的子集，其中奇数$ q $包含每个半径的球体或中心的第一个坐标的球体。我们分别把这样的集合称为半径球面Kakeya集和中心球面Kakeya集。对于$ n \ ge 4 $，我们证明了包含$ q-1 $不同球体的任何集合的大小的一般下界，这适用于两种球形Kakeya集。我们提供满足此下限主要条款的结构。我们还给出了一个结构，该结构表明，如果我们采用较低维的对象（例如$ \ mathbb {F} _q ^ 3 $中的圆而不是球形），则无法获得数量级$ q ^ n $的下界，表明Kakeya问题线有显着差异。最后，我们研究维数$ n = 1 $的情况，