On the packing dimension of distance sets with respect to $C^1$ and polyhedral norms
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| Format: | Preprint |
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2025
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| _version_ | 1866908321890435072 |
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| author | Altaf, Iqra Bushling, Ryan Wilson, Bobby |
| author_facet | Altaf, Iqra Bushling, Ryan Wilson, Bobby |
| contents | We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$. One of the main tools is a nonlinear projection theorem extending a result of M. Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the packing dimension of distance sets with respect to $C^1$ and polyhedral norms Altaf, Iqra Bushling, Ryan Wilson, Bobby Classical Analysis and ODEs 28A80 We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$. One of the main tools is a nonlinear projection theorem extending a result of M. Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms. |
| title | On the packing dimension of distance sets with respect to $C^1$ and polyhedral norms |
| topic | Classical Analysis and ODEs 28A80 |
| url | https://arxiv.org/abs/2504.11660 |