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

Authors: Nets Hawk Katz and Joshua Zahl
Journal: J. Amer. Math. Soc. 32 (2019), 195-259
MSC (2010): Primary 42B25
Published electronically: August 29, 2018
MathSciNet review: 3868003
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Abstract: 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.

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Additional Information

Nets Hawk Katz
Affiliation: California Institute of Technology, Pasadena, California 91125
MR Author ID: 610432

Joshua Zahl
Affiliation: University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
MR Author ID: 849921
ORCID: 0000-0001-5129-8300

Received by editor(s): May 20, 2017
Received by editor(s) in revised form: September 16, 2017, and May 21, 2018
Published electronically: August 29, 2018
Additional Notes: The first author was supported by NSF grants DMS 1266104 and DMS 1565904
The second author was supported by an NSERC Discovery grant.
Article copyright: © Copyright 2018 American Mathematical Society