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arXiv · 2304.03633

On the generalized Hausdorff dimension of Besicovitch sets

Xianghong Chen·Lixin Yan·Yue Zhong
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Abstract

Keich (1999) showed that the sharp gauge function for the generalized Hausdorff dimension of Besicovitch sets in $\mathbb R^2$ is between $r^2\log 1/r$ and $r^2(\log 1/r) (\log\log 1/r)^{2+\varepsilon}$ 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 $r^2(\log 1/r) (\log\log 1/r)^{\varepsilon}$. 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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Xianghong Chen, Lixin Yan, Yue Zhong. 2023-04-07. On the generalized Hausdorff dimension of Besicovitch sets. https://arxiv.org/abs/2304.03633

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