An improved bound on the Minkowski dimension of Besicovitch sets in R3

• 成果类型: Article
• 署名作者: Katz, NH; Laba, I; Tao, T
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661389
• 发表日期: 2000
• 页码: 383-446
• 关键词: restriction

摘要

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.