{
  "abstract": "For a set $V$ of $n-1$ distinct positive integers let $\\mathrm{LR}(V)=\\max_{t}\\min_{v\\in V}\\|vt\\|$, where $\\|x\\|$ is the distance from $x$ to the nearest integer. The Lonely Runner Conjecture asserts $\\mathrm{LR}(V)\\ge 1/n$, and $V$ is called tight if equality holds. Goddyn and Wong determined which of the sets obtained from $[n-1]=\\{1,\\dots,n-1\\}$ by multiplying a single entry by an integer are tight. We complete the classification of arbitrary one-entry modifications $([n-1]\\setminus\\{r\\})\\cup\\{w\\}$, $w\\ge n$: for $n\\ge4$ such a set is tight if and only if it is of Goddyn--Wong type or is one of the two sporadic sets $\\{1,3,4,7\\}$ and $\\{1,3,4,5,9\\}$ found by Wills. The proof is elementary and rests on an exact description of the region left uncovered by the retained speeds. We then show that the largest speed $M(n)$ occurring in a tight one-entry modification satisfies $M(n)<0.60\\,n\\log n+52\\,n$ and $\\limsup_{n\\to\\infty}M(n)/(n\\log n)=1/2$; in particular, tight sets are not confined to speeds $O(n)$. Along the way we give a short proof of Liu's closed formula for $\\mathrm{LR}([n-1]\\setminus\\{2\\})$ and show that $\\mathrm{LR}([n-1]\\setminus\\{r\\})>1/(n-1)$ for every $r\\le n-2-2\\sqrt n$, extending the range $r\\le(n-1)/2$ covered by Liu's theorem. For two-entry modifications we show that no tight instance removes a speed $q$ with $2\\le q\\le n/10$ once $n\\ge\\max(40,10q)$, whatever the second removed speed and the inserted speeds are; an exact-arithmetic census for $8\\le n\\le140$ with inserted speeds at most $8n$ finds only two tight two-entry modifications, both already listed by Goddyn and Wong.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13599",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuhan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c46q7497g0s3j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13599",
  "fulltext_url": null,
  "id": "c46q7497g0s3j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13599",
  "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"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7203651984",
  "pdf_url": "https://arxiv.org/pdf/2608.13599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c46q7497g0s3j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.13599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds",
  "title_text": "Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/c46q7497g0s3j",
  "venue": "arXiv",
  "versions": []
}