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{
    "pk": 65031,
    "title": "Relative Lonely Runner spectra",
    "subtitle": null,
    "abstract": "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 two-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.\n \nMathematics Subject Classifications: 11J13, 52C07, 11J06, 11B75\n \nKeywords: Lonely Runner conjecture, Diophantine approximation, spectra, combinatorial number theory",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Lonely Runner conjecture"
        },
        {
            "word": "Diophantine approximation"
        },
        {
            "word": "spectra"
        },
        {
            "word": "combinatorial number theory"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3mx8w3js",
    "frozenauthors": [
        {
            "first_name": "Vanshika",
            "middle_name": "",
            "last_name": "Jain",
            "name_suffix": "",
            "institution": "Department of Mathematics, Princeton University, Princeton, NJ 08540, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Noah",
            "middle_name": "",
            "last_name": "Kravitz",
            "name_suffix": "",
            "institution": "Department of Mathematics, Princeton University, Princeton, NJ 08540, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2026-04-20T10:59:16+03:00",
    "date_accepted": "2026-04-20T10:59:16+03:00",
    "date_published": "2026-04-20T10:00:00+03:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65031/galley/49841/download/"
        }
    ]
}