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

Abstract

We consider unions of S L (2) lines in R3. These are lines of the form L=(a,b,0)+span(c,d,1) where ab-cd=1. We show that if L is a Kakeya set of S L (2) lines, then the union U L has Hausdorff dimension 3. This answers a question of Wang and Zahl. The S L (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 R3, 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.