{
  "abstract": "For distinct positive integers v_1, ..., v_n let ML(v_1, ..., v_n) be the largest number L such that at some time t every t vi is at distance at least L from the nearest integer; the Lonely Runner Conjecture asserts that ML >= 1/(n+1). Write ML = p/q in lowest terms. Kravitz conjectured that whenever ML < 1/n one has q = np + 1; Fan and Sun found counterexamples for n = 4, conjectured that q = np + k with 1 <= k <= n always holds, and observed that in their data for n = 6 only k = 1 and k = 3 occur. We explain this observation. For six speeds we show that all but finitely many tuples with ML < 1/6 satisfy ML = (P-1)/(6P) for an integer P congruent to 1 or 5 modulo 6; in particular k is 1 or 3 and the denominator q is odd. The proof determines the three infinite two-parameter families of tuples on which such values concentrate, computes ML exactly on each family, and describes exactly where k = 3 occurs. An exhaustive search over the 2 x 10^9 sextuples with speeds at most 110 finds no exception. For five speeds the same method, together with Chen's classification of the tuples attaining ML = 1/5, shows that all but finitely many tuples with ML < 1/5 satisfy Kravitz's original conjecture. The computations are exact and the code is provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03444",
  "author_total": 1,
  "authors": [
    {
      "name": "Francesco Cordella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0amadad365ra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03444",
  "fulltext_url": null,
  "id": "e0amadad365ra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03444",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7208731451",
  "pdf_url": "https://arxiv.org/pdf/2609.03444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0amadad365ra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.03444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Odd denominators in the Lonely Runner spectrum for six speeds",
  "title_text": "Odd denominators in the Lonely Runner spectrum for six speeds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/e0amadad365ra",
  "venue": "arXiv",
  "versions": []
}