{
  "abstract": "We study the Lonely Runner Conjecture (LRC), conceived by Jörg M. Wills in the 1960's: Given positive integers $n_1, n_2, \\dots, n_k$, there exists a positive real number $t$ such that for all $1 \\le j \\le k$ the distance of $t \\,n_j$ to the nearest integer is at least $\\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\\mathbf{n}$ of LRC, $\\mathbf{m} \\cdot \\mathbf{n} = 0$ for some $\\mathbf{m} \\in \\mathbb{Z}^k$ with $0 < \\| \\mathbf{m} \\|_1 \\le \\min(2k+3, \\ \\frac{ k+1 }{ k-1 } \\mathrm{flt}(k))$ where $\\mathrm{flt}(k)$ denotes Khinchin's (1948) flatness constant limiting the lattice width of a $k$-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwiński (2012) that when we choose $\\mathbf{n}$ at random then, with probability tending to 1, the measure of loneliness $\\frac{1}{ k+1 }$ can be replaced by $\\frac 1 2 - ε$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06259",
  "author_total": 2,
  "authors": [
    {
      "name": "Matthias Beck"
    },
    {
      "name": "Samuel Everett"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fqhnz5mj9eqtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06259",
  "fulltext_url": null,
  "id": "fqhnz5mj9eqtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06259",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212010233",
  "pdf_url": "https://arxiv.org/pdf/2609.06259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fqhnz5mj9eqtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.06259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06259"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Lonely Runner Relations",
  "title_text": "Lonely Runner Relations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/fqhnz5mj9eqtn",
  "venue": "arXiv",
  "versions": []
}