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

Katz, Nets; Zahl, Joshua

doi:10.1090/jams/907

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

J. Amer. Math. Soc. 32 (2019), 195-259

DOI: https://doi.org/10.1090/jams/907

Published electronically: August 29, 2018

HTML | PDF | 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