A Kakeya maximal estimate for regulus strips

TL;DR

This paper advances Kakeya-type estimates for regulus strips, leading to improved bounds in the Kakeya conjecture in three dimensions and establishing new inequalities in the Heisenberg group.

Contribution

It introduces new Kakeya estimates for regulus strips and demonstrates their implications for the Kakeya conjecture and Nikodym maximal functions in the Heisenberg group.

Findings

Achieved epsilon improvement over the Kakeya conjecture in R^3

Showed regulus strips in the SL_2 example are essentially disjoint

Established an L^p inequality for Nikodym maximal function in the Heisenberg group

Abstract

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