Exponential odd-distance sets under the Manhattan metric
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909361730748416 |
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| author | Díaz, Alberto Espuny Hogan, Emma Illingworth, Freddie Michel, Lukas Portier, Julien Yan, Jun |
| author_facet | Díaz, Alberto Espuny Hogan, Emma Illingworth, Freddie Michel, Lukas Portier, Julien Yan, Jun |
| contents | We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential odd-distance sets under the Manhattan metric Díaz, Alberto Espuny Hogan, Emma Illingworth, Freddie Michel, Lukas Portier, Julien Yan, Jun Combinatorics Metric Geometry We construct a set of $2^n$ points in $\mathbb{R}^n$ such that all pairwise Manhattan distances are odd integers, which improves the recent linear lower bound of Golovanov, Kupavskii and Sagdeev. In contrast to the Euclidean and maximum metrics, this shows that the odd-distance set problem behaves very differently to the equilateral set problem under the Manhattan metric. Moreover, all coordinates of the points in our construction are integers or half-integers, and we show that our construction is optimal under this additional restriction. |
| title | Exponential odd-distance sets under the Manhattan metric |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2410.18281 |