On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913762843295744 |
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| author | Orponen, Tuomas Shmerkin, Pablo |
| author_facet | Orponen, Tuomas Shmerkin, Pablo |
| contents | Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t \in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\mathbb{R}^{2}$ is no smaller than $2s + ε$, where $ε> 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $ε$-improvement over a result of Wolff from 1999.
The same method also yields an $ε$-improvement to Kaufman's projection theorem from 1968. We show that if $s \in (0,1)$, $t \in (s,2]$ and $K \subset \mathbb{R}^{2}$ is an analytic set with $\dim_{\mathrm{H}} K = t$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} π_{e}(K) \leq s\} \leq s - ε,$$ where $ε> 0$ only depends on $s$ and $t$. Here $π_{e}$ is the orthogonal projection to $\mathrm{span}(e)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_03338 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane Orponen, Tuomas Shmerkin, Pablo Classical Analysis and ODEs Combinatorics Metric Geometry 28A80 (Primary) 28A75, 28A78 (Secondary) Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t \in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in $\mathbb{R}^{2}$ is no smaller than $2s + ε$, where $ε> 0$ depends only on $s$ and $t$. For $s>1/2$ and $t = 1$, this is an $ε$-improvement over a result of Wolff from 1999. The same method also yields an $ε$-improvement to Kaufman's projection theorem from 1968. We show that if $s \in (0,1)$, $t \in (s,2]$ and $K \subset \mathbb{R}^{2}$ is an analytic set with $\dim_{\mathrm{H}} K = t$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} π_{e}(K) \leq s\} \leq s - ε,$$ where $ε> 0$ only depends on $s$ and $t$. Here $π_{e}$ is the orthogonal projection to $\mathrm{span}(e)$. |
| title | On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A80 (Primary) 28A75, 28A78 (Secondary) |
| url | https://arxiv.org/abs/2106.03338 |