{"paper":{"title":"Problems on Track Runners","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Adrian Dumitrescu, Csaba D. T\\'oth","submitted_at":"2015-08-28T17:43:58Z","abstract_excerpt":"Consider the circle $C$ of length 1 and a circular arc $A$ of length $\\ell\\in (0,1)$.\n  It is shown that there exists $k=k(\\ell) \\in \\mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds so that at any time $t \\geq 0$, at least one of the $k$ runners is not in $A$.\n  On the other hand, we show the following:\n  Assume that $k$ runners $1,2,\\ldots,k$, with constant rationally independent (thus distinct) speeds $\\xi_1,\\xi_2,\\ldots,\\xi_k$, run clockwise along a circle of length $1$, starting from arbitrary points. For every circular arc $A\\subs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07289","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}