# A Kakeya maximal function estimate in four dimensions using planebrushes

## TL;DR

This paper introduces the planebrush, a new geometric tool, to improve Kakeya maximal function and Hausdorff dimension estimates in four-dimensional space, advancing understanding of Besicovitch sets.

## Contribution

The paper develops the planebrush method, a novel geometric approach, to obtain sharper Kakeya maximal function and Hausdorff dimension bounds in D.

## Key findings

- Maximal function estimate in D at dimension 3.049
- Hausdorff dimension of Besicovitch sets in D at least 3.059
- Introduction of the planebrush as a new geometric tool

## Abstract

We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in $\mathbb{R}^4$ at dimension 3.049, and we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least 3.059.