Kakeya books and projections of Kakeya sets

2017/04/14 by Han Yu, Yu, Han
Mathematics · #28A78 #28A80 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1704.04488

openalex publication_date 2017/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Here we show some results related with Kakeya conjecture which says that for any integer n≥ 2, a set containing line segments in every dimension in ℝⁿ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set K⊂ℝⁿ, the Hausdorff dimension of orthogonal projections on k≤ n subspaces is independent of directions then the Kakeya conjecture is true. Moreover, the converse is also true.

Citations

• Fractal Geometry: Mathematical Foundations and Applications.

Related

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• Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements
• Dimension estimates for Kakeya sets defined in an axiomatic setting
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• Restricting directions for Kakeya sets