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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.03633 |
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Table of 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.