论文标题
关于Kakeya的注释，水平和$ SL（2）$ lines
A note on Kakeya sets of horizontal and $SL(2)$ lines
论文作者
论文摘要
我们考虑$ \ mathbb {r}^{3} $的$ sl（2）$行的工会。这些是$$ l =（a，b，0） + \ mathrm {span}（c，d，1），$ ad -ad -ad -bc = 1 $的行。我们表明，如果$ \ Mathcal {l} $是$ sl（2）$行的Kakeya集，则Union $ \ CUP \ MATHCAL {L} $具有Hausdorff Dimension $ 3 $。这回答了王和扎尔的问题。 $ sl（2）$线可以在第一个海森伯格组中用水平线识别，我们获得了有关水平线工会的更一般性语句的主要结果。该语句是通过$ \ mathbb {r}^{3} $之间的点线二元性原理建立的，结合了由于GAN，Guo，Guth，Guth，Harris，Maldague和Wang的限制性预测家庭的最新工作。 我们的结果还具有与水平线相关的Nikodym集的推论，该集合回答了Kim问题的特殊情况。
We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal and conical lines in $\mathbb{R}^{3}$, combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim.
