AI 中文总结
该研究针对第一Heisenberg群的CC测地线Kakeya集，修正了Kakeya猜想的自然预测，证明其精确下界为4，构造了满维零测集及不同曲率的低维零测Kakeya集，完善了相关维数理论。
AI 中文摘要
我们研究第一Heisenberg群中的CC测地线Kakeya集，其定义为包含从单位元出发的所有单位速CC测地线段的左平移。Kakeya猜想的自然类似物预测这类集合的Heisenberg Hausdorff维数为4，但我们证明该预测不成立：它们的精确下界为4，且在紧CC测地线Kakeya集中仍保持精确。我们还构造了一个Heisenberg Hausdorff维数为4且Lebesgue测度为零的CC测地线Kakeya集。最后，对每个κ∈(0,2π]，我们构造了一个曲率为κ的紧Kakeya集，其Euclidean和Heisenberg Hausdorff维数均为1且Lebesgue测度为零。
英文摘要
We study CC-geodesic Kakeya sets in the first Heisenberg group, namely Borel sets $E$ such that, for every unit-speed CC-geodesic segment of length $1$ issuing from the identity, some left translate of the segment is contained in $E$. The natural analogue of the Kakeya conjecture would predict full Heisenberg Hausdorff dimension 4 for such sets. We show that this prediction fails: the sharp lower bound for their Heisenberg Hausdorff dimension is 3, and it remains sharp even among compact CC-geodesic Kakeya sets. By adjoining a Lebesgue-null set of full Heisenberg Hausdorff dimension, we obtain a CC-geodesic Kakeya set of full Heisenberg Hausdorff dimension 4 and zero Lebesgue measure. Finally, when one prescribes only geodesic segments with one fixed nonzero curvature parameter $κ$, rather than segments of all curvatures, the condition is weaker. For every $κ\in(0,2π]$, we construct a compact curvature-$κ$ Kakeya set of zero Lebesgue measure whose Euclidean and Heisenberg Hausdorff dimensions are both equal to $1$.
Comments19 pages
