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 $5/2+{\epsilon }_0$ 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 $S{L}_2$ 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.

65 pages, 11 figures. v3: Incorporates referee suggestions

References in corpus (1)

• Polynomial Wolff axioms and Kakeya-type estimates in ${\mathbb{R}}^4$

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