A Kakeya maximal estimate for regulus strips

We prove Kakeya-type estimates for regulus strips. As a result, we obtain another epsilon improvement over the Kakeya conjecture in \mathbb{R}^{3} , by showing that the regulus strips in the \textup{SL}_{2} example are essentially disjoint. We also establish an L^{p} inequality regarding a Nikodym-type maximal function in the first Heisenberg group.