李群与齐性空间中的Kakeya算子
Kakeya Configurations in Lie Groups and Homogeneous Spaces
摘要：
本文研究了连通李群及相应的齐性空间中有向和无向的连续Kakeya线和针构型。这些是Kakeya线（针）集（$\mathbb{R}^n$的子集，其中可以将线（分别为单位长度的间隔）连续地转向所有方向{ \bf，而不重复“方向”}）的类似物。在一般假设下，证明了连通李群中的连续Kakeya线配置集必包含单位元的开邻域，因而必有正Haar测度.在连通幂零李群$G$中，$G$中唯一包含这样一个无向线配置的子空间被证明是$G$本身。最后讨论了齐性空间中的一些类似问题。这些问题是由Z.在有限域设置中的Dvir。
英文摘要：
In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line (respectively an interval of unit length) through all directions {\bf continuously, without repeating a "direction"}.) We show under some general assumptions that any such continuous Kakeya line configuration set in a connected Lie group must contain an open neighborhood of the identity, and hence must have positive Haar measure. In connected nilpotent Lie groups $G$, the only subspace of $G$ that contains such an unoriented line configuration is shown to be $G$ itself. Finally some similar questions in homogeneous spaces are addressed. These questions were motivated by work of Z. Dvir in the finite field setting.
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