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

Abstract

We consider unions of lines in . These are lines of the form where . We show that if is a Kakeya set of lines, then the union has Hausdorff dimension 3. This answers a question of Wang and Zahl. The 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 , 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.