An Improved Bound on the Minkowski Dimension of Besicovitch Sets in ℝ 3

Nets Hawk Katz, Izabella Łaba, Terence Tao

Type: Article

Publication Date: 2000-09-01

Citations: 73

DOI: https://doi.org/10.2307/2661389

Abstract

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . In this paper we show that the Minkowski dimension must in fact be greater than 5= 2+ for some absolute constant > 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call \stickiness, \planiness, and \graininess.

Locations

Annals of Mathematics - View
arXiv (Cornell University) - PDF
Annals of Mathematics - View
arXiv (Cornell University) - PDF

Similar Works

Action	Title	Year	Authors
+	An improved bound on the Minkowski dimension of Besicovitch sets in R^3	1999	Nets Hawk Katz Izabella Łaba Terence Tao
+	An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension	2000	Izabella Łaba Terence Tao
+ PDF Chat	An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension	2001	Izabella Łaba Terence Tao
+	On the generalized Hausdorff dimension of Besicovitch sets	2023	Xianghong Chen Lixin Yan Yue Zhong
+	An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$	2017	Nets Hawk Katz Joshua Zahl
+	An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$	2017	Nets Hawk Katz Joshua Zahl
+	An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³	2018	Nets Hawk Katz Joshua Zahl
+	Quantitative Besicovitch projection theorem for irregular sets of directions	2022	Damian Dąbrowski
+	The Kakeya Needle Problem and Besikovitch Set ( With Idea of Exhibit for Mathematics Of The Planet Earth ) TIFR VSRP Project Report	2013	Debashis Chatterjee
+	Dimensions of modified Besicovitch-Eggleston sets	2006	Yuqiong Xie Zhi‐Xiong Wen
+ PDF	A Kakeya maximal function estimate in four dimensions using planebrushes	2020	Nets Hawk Katz Joshua Zahl
+	A note on Borsuk's problem in Minkowski spaces	2023	А. М. Райгородский Арсений Сагдеев
+	A Kakeya maximal function estimate in four dimensions using planebrushes	2019	Nets Hawk Katz Joshua Zahl
+	A Kakeya maximal function estimate in four dimensions using planebrushes	2019	Nets Hawk Katz Joshua Zahl
+ PDF	GEOMETRIC PROBABILITY IN THE MINKOWSKI PLANE	2003	Weon-Bae Kim Yong-I Kim
+	Minkowski’s theorem on lattice points in convex sets	1968	K. Chandrasekharan
+	Continuously parametrized Besicovitch sets in R^n	2011	Esa Järvenpää Maarit Järvenpää Tamás Keleti András Máthé
+ PDF	On the best constant for the Besicovitch covering theorem	1994	Zoltán Füredi Peter A. Loeb
+	On minimal Besicovitch arrangements	2018	Stéphane Blondeau da Silva
+	An $L^{4/3}$ $SL_2$ Kakeya maximal inequality	2023	Terence L. J. Harris

Works That Cite This (68)

Action	Title	Year	Authors
+	Some results in support of the Kakeya Conjecture	2014	Jonathan M. Fraser Eric J. Olson James C. Robinson
+	An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³	2018	Nets Hawk Katz Joshua Zahl
+	Factorization in generalized arithmetic progressions and application to the Erd?s-Szemer�di sum-product problems	2003	Mei-Chu Chang
+ PDF Chat	An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension	2001	Izabella Łaba Terence Tao
+	An x-ray estimate in $R^n$	1999	Izabella Łaba Terence Tao
+	An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$	2017	Nets Hawk Katz Joshua Zahl
+	A Kakeya maximal function estimate in four dimensions using planebrushes	2019	Nets Hawk Katz Joshua Zahl
+ PDF	Some Results in Support of the Kakeya Conjecture	2017	Jonathan M. Fraser Olson Robinson
+	Algebraic Methods in Discrete Analogs of the Kakeya Problem	2008	Larry Guth Nets Hawk Katz
+	Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem	2006	Joshua Cooper

Works Cited by This (17)

Action	Title	Year	Authors
+ PDF	Recent work connected with the Kakeya problem	2003	Thomas Wolff
+	Sums of Finite Sets	1996	Imre Z. Ruzsa
+	An x-ray estimate in $R^n$	1999	Izabella Łaba Terence Tao
+	A bilinear approach to cone multipliers I. Restriction estimates	2000	Terence Tao Ana Vargas
+ PDF	An improved bound for Kakeya type maximal functions	1995	Thomas Wolff
+	The Bochner-Riesz conjecture implies the restriction conjecture	1999	Terence Tao
+	A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions	1998	Wilhelm Schlag
+	Besicovitch type maximal operators and applications to fourier analysis	1991	Jean Bourgain
+ PDF	An inequality related to the isoperimetric inequality	1949	Lynn H. Loomis Hassler Whitney
+ PDF	A mixed norm estimate for the X-ray transform	1998	Thomas Wolff