The structure of Lonely Runner spectra
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917200002023424 |
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| author | Giri, Vikram Kravitz, Noah |
| author_facet | Giri, Vikram Kravitz, Noah |
| contents | For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture, we study the structure of the sets $\mathcal{S}(n)$. The main purpose of this note is to show that the set of accumulation points of $\mathcal{S}(n)$ is precisely $\mathcal{S}(n-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01462 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The structure of Lonely Runner spectra Giri, Vikram Kravitz, Noah Combinatorics Number Theory For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture, we study the structure of the sets $\mathcal{S}(n)$. The main purpose of this note is to show that the set of accumulation points of $\mathcal{S}(n)$ is precisely $\mathcal{S}(n-1)$. |
| title | The structure of Lonely Runner spectra |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2304.01462 |