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

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

1999-03-29

arxiv.org/abs/math/9903166v2

Ann. of Math. (2) 152 (2000), no. 2, 383-446

Mathematics

Classical Analysis and ODEs

64 pages, published version

Scientific paper

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.

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