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{
    "pk": 64813,
    "title": "Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem",
    "subtitle": null,
    "abstract": "We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number $x$, let $\\Vert x \\Vert$ denote the distance from $x$ to the nearest integer. For each set of positive integer speeds $v_1, \\dots, v_n$, we define the associated maximum loneliness to be $$\\operatorname{ML}(v_1, \\dots, v_n)=\\max_{t \\in \\mathbb{R}}\\min_{1 \\leq i \\leq n} \\Vert tv_i \\Vert.$$\nThe Lonely Runner Conjecture asserts that $\\operatorname{ML}(v_1, \\dots, v_n) \\geq 1/(n+1)$ for all choices of $v_1, \\dots, v_n$. We make the stronger conjecture that for each choice of $v_1, \\dots, v_n$, we have either $\\operatorname{ML}(v_1, \\dots, v_n)=s/(ns+1)$ for some $s \\in \\mathbb{N}$ or $\\operatorname{ML}(v_1, \\dots, v_n) \\geq 1/n$. This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for $n \\leq 3$; and confirming it for $n=4$ and $n=6$ in the case where one speed is much faster than the rest.\nMathematics Subject Classifications: 11K60 (primary), 11J13, 11J71, 52C07",
    "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": [],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3wx931fh",
    "frozenauthors": [
        {
            "first_name": "Noah",
            "middle_name": "",
            "last_name": "Kravitz",
            "name_suffix": "",
            "institution": "Grace Hopper College, Yale University, New Haven, CT 06510, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2021-11-12T21:56:58Z",
    "date_accepted": "2021-11-12T21:56:58Z",
    "date_published": "2021-12-15T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64813/galley/49623/download/"
        }
    ]
}