Relative Lonely Runner spectra
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912149807300608 |
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| author | Jain, Vanshika Kravitz, Noah |
| author_facet | Jain, Vanshika Kravitz, Noah |
| contents | For a subtorus $T \subseteq (\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the $L^\infty$-distance from $T$ to the point $(1/2, \ldots, 1/2)$. For a subtorus $U \subseteq (\mathbb{R}/\mathbb{Z})^n$, define $\mathcal{S}_1(U)$, the Lonely Runner spectrum relative to $U$, to be the set of all values of $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $U$ not contained in the union of the coordinate hyperplanes of $(\mathbb{R}/\mathbb{Z})^n$. The relative spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$ is the ordinary Lonely Runner spectrum that has been studied previously.
Giri and the second author recently showed that the relative spectra $\mathcal{S}_1(U)$ for $2$-dimensional subtori $U \subseteq (\mathbb{R}/\mathbb{Z})^n$ essentially govern the accumulation points of the Lonely Runner spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$. In the present work, we prove that such relative spectra $\mathcal{S}_1(U)$ have a very rigid arithmetic structure, and that one can explicitly find a complete characterization of each such relative spectrum with a finite calculation; carrying out this calculation for a few specific examples sheds light on previous constructions in the literature on the Lonely Runner Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative Lonely Runner spectra Jain, Vanshika Kravitz, Noah Number Theory Combinatorics For a subtorus $T \subseteq (\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the $L^\infty$-distance from $T$ to the point $(1/2, \ldots, 1/2)$. For a subtorus $U \subseteq (\mathbb{R}/\mathbb{Z})^n$, define $\mathcal{S}_1(U)$, the Lonely Runner spectrum relative to $U$, to be the set of all values of $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $U$ not contained in the union of the coordinate hyperplanes of $(\mathbb{R}/\mathbb{Z})^n$. The relative spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$ is the ordinary Lonely Runner spectrum that has been studied previously. Giri and the second author recently showed that the relative spectra $\mathcal{S}_1(U)$ for $2$-dimensional subtori $U \subseteq (\mathbb{R}/\mathbb{Z})^n$ essentially govern the accumulation points of the Lonely Runner spectrum $\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)$. In the present work, we prove that such relative spectra $\mathcal{S}_1(U)$ have a very rigid arithmetic structure, and that one can explicitly find a complete characterization of each such relative spectrum with a finite calculation; carrying out this calculation for a few specific examples sheds light on previous constructions in the literature on the Lonely Runner Problem. |
| title | Relative Lonely Runner spectra |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2411.12684 |