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 ${\mathrm{S}\mathrm{L}}_2$ example are essentially disjoint. We also establish an $L^p$ inequality regarding Nikodym-type maximal function in the first Heisenberg group.