On the generalized Hausdorff dimension of Besicovitch sets

2023/04/07 by Xianghong Chen, Lixin Yan, Chen, Xianghong +3
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2304.03633

openalex publication_date 2023/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Keich (1999) showed that the sharp gauge function for the generalized Hausdorff dimension of Besicovitch sets in \mathbb R² is between r²log 1/r and r²(log 1/r) (loglog 1/r)^2+ε by refining an argument of Bourgain (1991). It is not known whether the iterated logarithms in Keich's bound are necessary. In this paper we construct a family of Besicovitch line sets whose sharp gauge function is smaller than r²(log 1/r) (loglog 1/r)^ε. Moreover, these Besicovitch sets are minimal in the sense that there is essentially only one line in the set pointing in each direction.

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