Distance sets bounds for polyhedral norms via effective dimension
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912100904861696 |
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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 norm on $\mathbb{R}^d$ and every $E \subseteq \mathbb{R}^d$, the Hausdorff dimension of the distance set of $E$ with respect to that norm is at least $\dim_{\mathrm{H}} E - (d-1)$. An explicit construction follows, demonstrating that this bound is sharp for every polyhedral norm on $\mathbb{R}^d$. The techniques of algorithmic complexity theory underlie both the computations and the construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Distance sets bounds for polyhedral norms via effective dimension Altaf, Iqra Bushling, Ryan Wilson, Bobby Classical Analysis and ODEs 28A80, 03D32 We prove that, for every norm on $\mathbb{R}^d$ and every $E \subseteq \mathbb{R}^d$, the Hausdorff dimension of the distance set of $E$ with respect to that norm is at least $\dim_{\mathrm{H}} E - (d-1)$. An explicit construction follows, demonstrating that this bound is sharp for every polyhedral norm on $\mathbb{R}^d$. The techniques of algorithmic complexity theory underlie both the computations and the construction. |
| title | Distance sets bounds for polyhedral norms via effective dimension |
| topic | Classical Analysis and ODEs 28A80, 03D32 |
| url | https://arxiv.org/abs/2305.06937 |