An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$

Abstract

We prove that any Besicovitch set in $\mathbb{R}^3$ must have Hausdorff dimension at least $5/2+ε_0$ for some small constant $ε_0>0$. This follows from a more general result about the volume of unions of tubes that satisfy the Wolff axioms. Our proof grapples with a new "almost counter example" to the Kakeya conjecture, which we call the $SL_2$ example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension $5/2$. We believe this example may be an interesting object for future study.