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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.03633 |
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| _version_ | 1866913303154917376 |
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| author | Chen, Xianghong Yan, Lixin Zhong, Yue |
| author_facet | Chen, Xianghong Yan, Lixin Zhong, Yue |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03633 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the generalized Hausdorff dimension of Besicovitch sets Chen, Xianghong Yan, Lixin Zhong, Yue Classical Analysis and ODEs 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. |
| title | On the generalized Hausdorff dimension of Besicovitch sets |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2304.03633 |