An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³

We prove that every Besicovitch set in R 3 \mathbb {R}^3 must have Hausdorff dimension at least 5 / 2 + ϵ 0 5/2+\epsilon _0 for some small constant ϵ 0 > 0 \epsilon _0>0 . This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the SL 2 \operatorname {SL}_2 example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension 5 / 2 5/2 . We believe this example may be an interesting object for future study.