Kakeya Configurations in Lie Groups and Homogeneous Spaces

TL;DR

This paper investigates continuous Kakeya configurations in Lie groups and homogeneous spaces, showing they must contain open neighborhoods and have positive measure, with special results in nilpotent groups.

Contribution

It extends Kakeya set concepts to Lie groups, proving that such configurations contain neighborhoods and characterizing their structure in nilpotent groups.

Findings

Kakeya configurations contain open neighborhoods of the identity in connected Lie groups.

In nilpotent Lie groups, the only subspace containing such configurations is the entire group.

Results are related to finite field Kakeya problems and extend to homogeneous spaces.

Abstract

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.