Kakeya sets from lines in $SL_2$
Kakeya sets from lines in $SL_2$
We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was also recently proved by F\"assler and Orponen using different techniques. Our method combines …