论文标题
有限领域的球形Kakeya问题
The Spherical Kakeya Problem in Finite Fields
论文作者
论文摘要
我们研究有限字段$ \ mathbb {f} _q $的$ n $二维矢量空间的子集，用于奇数$ q $，该$ q $包含每个半径的球体或中心每个第一个坐标的球体。我们分别称之为Radii球形Kakeya套件和中心球形Kakeya套件。 对于$ n \ ge 4 $，我们证明了一般的下限，其大小包含$ q-1 $不同的球体的大小，适用于两种球形凯基亚套件。我们提供符合该下限的主要术语的结构。 我们还提供了一个施工，表明如果我们采用较低的尺寸对象，例如$ \ mathbb {f} _q^3 $而不是球体，则无法获得〜$ q^n $的下限〜$ q^n $，这表明与Line Kakeya问题存在显着差异。 最后，我们研究了尺寸$ n = 1 $的情况，该情况与覆盖$ \ m athbb {f} _Q $的总和和差异集的研究不同。
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 $\mathbb{F}_q$.
