  We prove that any Besicovitch set in $\mathbb{R}^3$ must have Hausdorff
dimension at least $5/2+\epsilon_0$ for some small constant $\epsilon_0>0$.
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.
