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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb {R}^3$
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by Nets Hawk Katz and Joshua Zahl HTML | PDF
J. Amer. Math. Soc. 32 (2019), 195-259 Request permission

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.
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 195-259
  • MSC (2010): Primary 42B25
  • DOI: https://doi.org/10.1090/jams/907
  • MathSciNet review: 3868003