Published January 2019
| Submitted
Journal Article
Open
An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ^3
- Creators
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Katz, Nets Hawk
- Zahl, Joshua
Abstract
We prove that every Besicovitch set in ℝ^3 must have Hausdorff dimension at least 5/2 + ϵ_0 for some small constant ϵ_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 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.
Additional Information
© 2018 American Mathematical Society. Received by the editors May 20, 2017, and, in revised form, September 16, 2017, and May 21, 2018. Published electronically: August 29, 2018. The first author was supported by NSF grants DMS 1266104 and DMS 1565904. The second author was supported by an NSERC Discovery grant. The authors would like to thank Terry Tao, Daniel Di Benedetto, and the anonymous referee for helpful comments and suggestions to an earlier version of this manuscript.Attached Files
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Additional details
- Eprint ID
- 90740
- DOI
- 10.1090/jams/907
- Resolver ID
- CaltechAUTHORS:20181108-083518614
- DMS-1266104
- NSF
- DMS-1565904
- NSF
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- Created
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2018-11-08Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field