An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ^3

Abstract

We prove that every Besicovitch set in 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 satisfies the Wolff axioms. Our proof grapples with a new "almost counterexample" 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.

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