{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:12:25Z","timestamp":1772136745799,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\lVert x \\rVert$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integer speeds $v_1, \\ldots, v_n$, we define its maximum loneliness, also known as the gap $\\delta$, to be$$\\text{ML}(v_1,\\ldots,v_n) = \\delta(v_1,\\ldots,v_n) = \\max_{t \\in \\mathbb{R}}\\min_{1 \\leq i \\leq n} \\lVert tv_i \\rVert.$$The Loneliness Spectrum Conjecture, recently proposed by Kravitz, asserts that $$\\exists s \\in \\mathbb{N}, \\text{ML}(v_1,\\ldots,v_n) = \\frac{s} {sn + 1} \\text{ or } \\text{ML}(v_1,\\ldots,v_n) \\geq \\frac{1}{n}.$$We disprove the Loneliness Spectrum Conjecture for $n = 4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n = 4$ whenever there exists a pair of speeds with a common factor of at least $3$ and prove some related results.<\/jats:p>","DOI":"10.37236\/13840","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:31Z","timestamp":1772135011000},"source":"Crossref","is-referenced-by-count":0,"title":["Amending the Lonely Runner Spectrum Conjecture"],"prefix":"10.37236","volume":"33","author":[{"given":"Ho Tin","family":"Fan","sequence":"first","affiliation":[]},{"given":"Alec","family":"Sun","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p38\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:32Z","timestamp":1772135012000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13840","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,27]]},"article-number":"P1.38"}}