{
  "abstract": "For positive integers $v_1,\\dots,v_n$ let $\\mathrm{ML}(v_1,\\dots,v_n)=\\max_t\\min_i\\|v_it\\|$, where $\\|x\\|$ is the distance to the nearest integer. Kravitz asked which values below $1/n$ occur. For $n=4$, Jain and Kravitz showed that the values below $1/4$ are $m/(4m+2)$, $m\\ge2$, up to finitely many undetermined exceptions. We settle the four-runner case completely. A primitive quadruple has $\\mathrm{ML}<1/4$ if and only if it is, up to order, $\\{1,2,3,4k\\}$, or $\\{x,y,x+y,x+2y\\}$ with $\\gcd(x,y)=1$ and $y\\not\\equiv x,2x\\pmod 4$, or $\\{1,3,4,14\\}$. The values below $1/4$ are exactly $m/(4m+2)$ with $m\\ge2$, $m\\ne3$, so $3/14$ is the only exception, and Conjecture 4.1 of Fan and Sun holds. A density lemma bounds the length of a quadruple on neither critical subtorus by $512$, and an exact search over 800,816,099 quadruples completes the proof. The same method shows that the two-dimensional subtori of $(\\mathbb{R}/\\mathbb{Z})^4$ with loneliness below $2/5$ take exactly the three-runner values in that range, which between $1/3$ and $2/5$ are the fractions $p/q$ with $1\\le 2q-5p\\le 4$. Hence the accumulation points of the four-runner spectrum in $(0,2/5]$ are the three-runner values there, as Giri and Kravitz conjectured, and $9/25$ is not the loneliness of a two-dimensional subtorus. For every $n$, assuming the conjecture for $n-1$ and $n-2$ speeds, we prove $\\mathrm{ML}(v)>1/n-(n(n-1))^{n-2}/(2|v|)$, which is sharp in $|v|$, and that every $n$-tuple with $\\mathrm{ML}<1/n$ and $|v|\\ge n(n(n-1))^{2n-4}$ lies on a two-dimensional subtorus of loneliness $1/n$. This answers a question of Cordella. For five speeds, a two-dimensional subtorus of $(\\mathbb{R}/\\mathbb{Z})^5$ with loneliness below $2/9$ is the product of a coordinate circle with the circle of $\\{1,2,3,4\\}$ or $\\{1,3,4,7\\}$. We also give infinitely many subtori whose maximizers lie off every circle with a repeated speed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40694",
  "author_total": 3,
  "authors": [
    {
      "name": "Hyunjun Yi"
    },
    {
      "name": "Karam Gill"
    },
    {
      "name": "Bryan Bu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ce3zspc0a9w65.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40694",
  "fulltext_url": null,
  "id": "ce3zspc0a9w65",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40694",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ce3zspc0a9w65.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40694"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Four runners that are barely lonely",
  "title_text": "Four runners that are barely lonely",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/ce3zspc0a9w65",
  "venue": "arXiv",
  "versions": []
}