An improved bound on the Minkowski dimension of Besicovitch sets in R^3

Katz, Nets Hawk; Łaba, Izabella; Tao, Terence

An improved bound on the Minkowski dimension of Besicovitch sets in R^3

Authors

Katz, Nets Hawk

Łaba, Izabella

Tao, Terence

Description

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 + εfor some absolute constant ε> 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.
64 pages, published version