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

Katz, Nets¹ ; Zahl, Joshua²

¹ California Institute of Technology, Pasadena, California 91125
² University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4

Journal of the American Mathematical Society, Tome 32 (2019) no. 1, pp. 195-259

Résumé

We prove that every 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 satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the $\operatorname {SL}_2$ 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.

Détail

DOI : 10.1090/jams/907

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