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

By Nets Hawk Katz, I. Łaba, Terence Tao. 2000

72 citations recorded by the source corpus.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ed4b3a02b66c70a2e7ca57403df22f41694dbcd7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

On the multilinear restriction and Kakeya conjectures (2005, result): “However, our multilinear results do suggest (in some non-rigorous sense) that if there were some counterexamples to either the linear restriction or Kakeya conjectures, then they would have to be somewhat “non-transverse” (or “plany”, in the terminology of [ 18 ]).”
Recent progress on the Kakeya conjecture (2000, methodology): “See [ 14 ] for a rigorous version of these statements in the n = 3 case, and [16] for the higher-dimensional case.”
A polynomial bound in Freiman's theorem (2002, background): “For d = 2, this is a known result, but the question for d > 2 appears to be very hard (for d = 3, the best result so far is Hausdorff-dim A 5/2, for Minkowski-dim A 5/2+ (see [ KLT ])).”
The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture (2008, result): “The phenomenon of “planiness” was discovered by Katz, Laba and Tao in [ 10 ], and the estimate below is similar to some estimates from that paper.”
Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type (2001, background): “Very informally, the point is that the arguments in [ 10 ] can be pushed a bit further to conclude that a Besicovitch set of dimension exactly 5/2 must essentially be a “Heisenberg group” over a ring of dimension 1/2.”
A sum-product estimate in finite fields, and applications (2003, result): “It was observed in [ KLT2000 ], [MT] that if one replaced the finite field Z/qZ by a finite field F which contained a subfield G of index 2, then one could obtain a “Heisenberg group” counterexample which showed that Theorem 8.1 must fail for that field.”