On the packing dimension of Furstenberg sets
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866910566928351232 |
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| author | Shmerkin, Pablo |
| author_facet | Shmerkin, Pablo |
| contents | We prove that if $α\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge α$ is at least $1/2+α+c(α)$ for some $c(α)>0$. In particular, this holds for $α$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\geα$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $α$-Furstenberg sets over the "trivial" estimate for all values of $α\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $α$-Furstenberg set resembles a set of dimension close to $1/2+α$, if they exist, are rather sparse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_15569 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the packing dimension of Furstenberg sets Shmerkin, Pablo Classical Analysis and ODEs Metric Geometry Primary: 28A78, 28A80 We prove that if $α\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge α$ is at least $1/2+α+c(α)$ for some $c(α)>0$. In particular, this holds for $α$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\geα$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $α$-Furstenberg sets over the "trivial" estimate for all values of $α\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $α$-Furstenberg set resembles a set of dimension close to $1/2+α$, if they exist, are rather sparse. |
| title | On the packing dimension of Furstenberg sets |
| topic | Classical Analysis and ODEs Metric Geometry Primary: 28A78, 28A80 |
| url | https://arxiv.org/abs/2006.15569 |