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


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 $\mathbb{R}^3$. In this paper we show that the Minkowski dimension must in fact be greater than $5/2 + \varepsilon$ for some absolute constant $\varepsilon > 0$. 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.



Nets Hawk Katz

Izabella Łaba

Terence Tao