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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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