挂屋书籍和挂屋集的投影
Kakeya books and projections of Kakeya sets
摘要：
本文给出了与Kakeya猜想有关的一些结果，该猜想认为对任意整数$n\geq 2$，在$\mathbb{R}^n$的每一维中包含线段的集合都具有全豪斯多夫维数和盒维数.本文证明了对线段的位置有一定限制的Kakeya集的Kakeya书具有满盒维数。本文还证明了Kakeya集的投影性质与Kakeya猜想之间的关系。如果对任何Kakeya集$K\subset\mathbb{R}^n$，$k\leq n$子空间上正交投影的豪斯多夫维数与方向无关，则Kakeya猜想成立.此外，反过来也是如此。
英文摘要：
Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent of directions then the Kakeya conjecture is true. Moreover, the converse is also true.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
