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arXiv · 1704.07210

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

Nets Hawk Katz·Joshua Zahl
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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 $SL_2$ 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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Nets Hawk Katz, Joshua Zahl. 2017-04-24. An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$. https://doi.org/10.1090/jams/907

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