紧李群中的Kakeya集与维数压缩
Kakeya sets and dimension compression in compact Lie groups
AI总结：
本文针对紧半单李群中的两类Kakeya集问题，确定了其最优Minkowski维数的上下界，推测局部Kakeya集问题的上界为精确值，结合李理论构造、关联几何与数的几何完成相关证明。
AI中文摘要：
我们研究紧半单李群中的两个Kakeya集问题。第一个问题受群结构启发，要求Kakeya集包含每个闭一维环面的左陪集。对于维数为d、秩为r的紧半单李群G，我们确定了该问题的最优Minkowski维数，证明其恰好为(d+r)/2。上界通过与Chevalley对合相关的李理论构造得到，下界则结合了关联几何与数的几何。我们还考虑了更接近经典调和分析表述的局部Kakeya集问题，该问题要求包含每个方向上固定长度的单参数弧。对于此问题，我们得到奇数秩r时的下界为(d+1)/2、偶数秩r时的下界为(d+2)/2，以及上界(d+r)/2，并推测该上界应是精确的。
英文摘要：
We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group $G$ of dimension $d$ and rank $r$, we determine the optimal Minkowski dimension for this problem, proving that it is exactly $(d+r)/2$. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound $(d+1)/2$ and $(d+2)/2$ for odd and even $r$, and upper bound $(d+r)/2$. We conjecture that the upper bound should be sharp.
