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

HTML articles powered by AMS MathViewer

- 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.## References

- Jonathan Bennett, Anthony Carbery, and Terence Tao,
*On the multilinear restriction and Kakeya conjectures*, Acta Math.**196**(2006), no. 2, 261–302. MR**2275834**, DOI 10.1007/s11511-006-0006-4 - Jean Bourgain,
*The discretized sum-product and projection theorems*, J. Anal. Math.**112**(2010), 193–236. MR**2763000**, DOI 10.1007/s11854-010-0028-x - J. Bourgain,
*Besicovitch type maximal operators and applications to Fourier analysis*, Geom. Funct. Anal.**1**(1991), no. 2, 147–187. MR**1097257**, DOI 10.1007/BF01896376 - Jean Bourgain and Larry Guth,
*Bounds on oscillatory integral operators based on multilinear estimates*, Geom. Funct. Anal.**21**(2011), no. 6, 1239–1295. MR**2860188**, DOI 10.1007/s00039-011-0140-9 - J. Bourgain, N. Katz, and T. Tao,
*A sum-product estimate in finite fields, and applications*, Geom. Funct. Anal.**14**(2004), no. 1, 27–57. MR**2053599**, DOI 10.1007/s00039-004-0451-1 - Roy O. Davies,
*Some remarks on the Kakeya problem*, Proc. Cambridge Philos. Soc.**69**(1971), 417–421. MR**272988**, DOI 10.1017/s0305004100046867 - Larry Guth,
*Degree reduction and graininess for Kakeya-type sets in $\Bbb {R}^3$*, Rev. Mat. Iberoam.**32**(2016), no. 2, 447–494. MR**3512423**, DOI 10.4171/RMI/891 - Larry Guth and Nets Hawk Katz,
*On the Erdős distinct distances problem in the plane*, Ann. of Math. (2)**181**(2015), no. 1, 155–190. MR**3272924**, DOI 10.4007/annals.2015.181.1.2 - L. Guth, J. Zahl,
*Polynomial Wolff axioms and Kakeya-type estimates in ${\mathbb {R}}^4$*, Proc. London Math. Soc.**117**(2018), no. 1, 192–220. - D. Hilbert and S. Cohn-Vossen,
*Geometry and the imagination*, Chelsea Publishing Co., New York, N. Y., 1952. Translated by P. Neményi. MR**0046650** - Nets Hawk Katz, Izabella Łaba, and Terence Tao,
*An improved bound on the Minkowski dimension of Besicovitch sets in $\textbf {R}^3$*, Ann. of Math. (2)**152**(2000), no. 2, 383–446. MR**1804528**, DOI 10.2307/2661389 - Nets Katz and Terence Tao,
*Recent progress on the Kakeya conjecture*, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 161–179. MR**1964819**, DOI 10.5565/PUBLMAT_{E}sco02_{0}7 - T. Tao,
*Stickiness, Graininess, Planiness, and a Sum-Product approach to the Kakeya Problem*, blog post: https://terrytao.wordpress.com/2014/05/07/stickiness-graininess-planiness-and-a-sum-product-approach-to-the-kakeya-problem (2014). - T. Tao,
*The two-ends reduction for the Kakeya maximal conjecture*, blog post: https://terrytao.wordpress.com/2009/05/15/the-two-ends-reduction-for-the-kakeya-maximal- conjecture (2009). - Terence Tao and Van Vu,
*Additive combinatorics*, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR**2289012**, DOI 10.1017/CBO9780511755149 - Thomas Wolff,
*An improved bound for Kakeya type maximal functions*, Rev. Mat. Iberoamericana**11**(1995), no. 3, 651–674. MR**1363209**, DOI 10.4171/RMI/188 - Thomas Wolff,
*A mixed norm estimate for the X-ray transform*, Rev. Mat. Iberoamericana**14**(1998), no. 3, 561–600. MR**1681585**, DOI 10.4171/RMI/245 - Thomas Wolff,
*Recent work connected with the Kakeya problem*, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. MR**1660476** - Richard Wongkew,
*Volumes of tubular neighbourhoods of real algebraic varieties*, Pacific J. Math.**159**(1993), no. 1, 177–184. MR**1211391**, DOI 10.2140/pjm.1993.159.177

## 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