More unit distances in arbitrary norms
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912621369753600 |
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| author | Greilhuber, Josef Schildkraut, Carl Tidor, Jonathan |
| author_facet | Greilhuber, Josef Schildkraut, Carl Tidor, Jonathan |
| contents | For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\geq 3$ and a typical norm on $\mathbb R^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | More unit distances in arbitrary norms Greilhuber, Josef Schildkraut, Carl Tidor, Jonathan Combinatorics Metric Geometry 52C10 (Primary) 52A20, 05C62 (Secondary) For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\geq 3$ and a typical norm on $\mathbb R^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$. |
| title | More unit distances in arbitrary norms |
| topic | Combinatorics Metric Geometry 52C10 (Primary) 52A20, 05C62 (Secondary) |
| url | https://arxiv.org/abs/2410.07557 |