## An improved bound on the Minkowski dimension of Besicovitch sets in $$\mathbb{R}^3$$.(English)Zbl 0980.42014

A Besicovitch set (or Kakeya set) $$E\subset \mathbb R^n$$ is a set which contains a unit line segment in each direction. The Kakeya conjecture is: a Besicovitch set in $$\mathbb R^n$$ must have Hausdorff dimension $$n$$? This has been verified for $$n=2$$ but is open otherwise. The authors treat the Minkowski dimension. If $$E\subset \mathbb R^n$$, one defines $$\delta$$-entropy $$\mathcal E_\delta(E)$$ to be the maximal possible cardinality for a $$\delta$$-separated subset of $$E$$, and the Minkowski dimension $$\overline{\dim} E$$ is defined as $$\overline{\dim} E=\limsup_{\delta\to 0} \log_{1/\delta}\mathcal E_\delta (E)$$. T. Wolff showed $$\overline{\dim} E\geq {n}/{2}+1$$, and J. Bourgain showed $$\overline{\dim} E={13n}/{25}+{12}/{25}$$. By combining the ideas of Wolff and Bourgain with some observations on the structure of extremal counterexamples to the Kakeya problem, the authors get the following improvement in the dimension 3: There exists an $$\varepsilon >0$$ such that $$\overline{\dim} E\geq {5}/{2}+\varepsilon$$ for all Besicovitch sets $$E$$ in $$\mathbb R^3$$. As a byproduct of the argument, they obtain that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which they call “stickiness,” “planiness,” and “graininess.”.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 28A78 Hausdorff and packing measures
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