A note on Kakeya sets of horizontal and $SL(2)$ lines

Abstract

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.

7 pages