On the Hausdorff dimension of circular Furstenberg sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910751575244800 |
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| author | Fässler, Katrin Liu, Jiayin Orponen, Tuomas |
| author_facet | Fässler, Katrin Liu, Jiayin Orponen, Tuomas |
| contents | For $0 \leq s \leq 1$ and $0 \leq t \leq 3$, a set $F \subset \mathbb{R}^{2}$ is called a circular $(s,t)$-Furstenberg set if there exists a family of circles $\mathcal{S}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{S} \geq t$ such that $$\dim_{\mathrm{H}} (F \cap S) \geq s, \qquad S \in \mathcal{S}.$$ We prove that if $0 \leq t \leq s \leq 1$, then every circular $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $\dim_{\mathrm{H}} F \geq s + t$. The case $s = 1$ follows from earlier work of Wolff on circular Kakeya sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_11587 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff dimension of circular Furstenberg sets Fässler, Katrin Liu, Jiayin Orponen, Tuomas Classical Analysis and ODEs Metric Geometry 28A80, 28A78 For $0 \leq s \leq 1$ and $0 \leq t \leq 3$, a set $F \subset \mathbb{R}^{2}$ is called a circular $(s,t)$-Furstenberg set if there exists a family of circles $\mathcal{S}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{S} \geq t$ such that $$\dim_{\mathrm{H}} (F \cap S) \geq s, \qquad S \in \mathcal{S}.$$ We prove that if $0 \leq t \leq s \leq 1$, then every circular $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $\dim_{\mathrm{H}} F \geq s + t$. The case $s = 1$ follows from earlier work of Wolff on circular Kakeya sets. |
| title | On the Hausdorff dimension of circular Furstenberg sets |
| topic | Classical Analysis and ODEs Metric Geometry 28A80, 28A78 |
| url | https://arxiv.org/abs/2305.11587 |