An improved bound on the Minkowski dimension of Besicovitch sets in R^3
Abstract
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + \epsilon for some absolute constant \epsilon > 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.'' The purpose of this paper is to improve upon the known bounds for the Minkowski dimension of Besicovitch sets in three dimensions. As a by-product of the argument we obtain some strong conclusions on the structure of Besicovitch sets with almost-minimal Minkowski dimension.
Keywords
Cite
@article{arxiv.math/9903166,
title = {An improved bound on the Minkowski dimension of Besicovitch sets in R^3},
author = {Nets Hawk Katz and Izabella Łaba and Terence Tao},
journal= {arXiv preprint arXiv:math/9903166},
year = {2007}
}
Comments
64 pages, published version