发表机构
Southern University of Science and Technology; Northwestern Polytechnical University（南方科技大学; 西北工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了中心点属于给定不可数Borel集的零测度紧致Kakeya集，证明可数性是唯一障碍。
AI 中文摘要
本文考虑具有附加限制的Kakeya集，即单位线段的中心属于给定集合。特别地，对于每个不可数Borel集$A\subset\mathbb{R}^d$，$d\geq 2$，我们构造了$\mathbb{R}^d$的一个Lebesgue测度为零的紧致子集，该子集在每个方向上包含一个中心位于$A$中的单位线段。注意到，如果$A$是可数的，那么每个这样的Kakeya集（甚至不必是紧致的）都必须具有正的Lebesgue测度。因此，我们的结果表明，可数性实际上是唯一的障碍。
英文摘要
In this paper, we consider Kakeya sets with the additional restriction that centers of the unit line segments belong to a given set. In particular, for every uncountable Borel set $A\subset\mathbb{R}^d$, $d\geq 2$, we construct a compact subset of $\mathbb{R}^d$ of Lebesgue measure zero that contains, in every direction, a unit line segment whose center lies in $A$. Notice that every such Kakeya set (not even necessarily compact) must have positive Lebesgue measure if $A$ is countable. So our result shows that the countability is in fact the only obstruction.
Comments10 pages. 2 figures, results from the 2026 SUSTech Undergraduate Summer Research Program in Real Analysis
