On variants of the Furstenberg set problem
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914953267511296 |
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| author | Fraser, Jonathan M. |
| author_facet | Fraser, Jonathan M. |
| contents | Given an integer $d \geq 2$, $s \in (0,1]$, and $t \in [0,2(d-1)]$, suppose a set $X$ in $\mathbb{R}^d$ has the following property: there is a collection of lines of packing dimension $t$ such that every line from the collection intersects $X$ in a set of packing dimension at least $s$. We show that such sets must have packing dimension at least $\max\{s,t/2\}$ and that this bound is sharp. In particular, the special case $d=2$ solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is $\max\{s,t+1-d\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On variants of the Furstenberg set problem Fraser, Jonathan M. Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78 Given an integer $d \geq 2$, $s \in (0,1]$, and $t \in [0,2(d-1)]$, suppose a set $X$ in $\mathbb{R}^d$ has the following property: there is a collection of lines of packing dimension $t$ such that every line from the collection intersects $X$ in a set of packing dimension at least $s$. We show that such sets must have packing dimension at least $\max\{s,t/2\}$ and that this bound is sharp. In particular, the special case $d=2$ solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is $\max\{s,t+1-d\}$. |
| title | On variants of the Furstenberg set problem |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A80, 28A78 |
| url | https://arxiv.org/abs/2409.03678 |