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Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\\ldots,s_k$, there is a real number $t$ such that $\\|s_i t\\|\\geq \\frac{1}{k+1}$ for all $i$.\n  We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\\mathbf d=(d_1,\\ldots,d_k)\\in(0,1/2]^k$, let $\\mathsf{MLPS}_k$ be the set of vectors "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.27941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T04:27:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9ba7e1637bb7cd2a8aa38adfb5a9e05efed5b99ff1f5105810d2617c0137e22d","abstract_canon_sha256":"a4995e3ae933f80be26143ae654e6de7b1417189abbff2fa78a8899326409411"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T01:04:53.391648Z","signature_b64":"fm0p+JdR+HosPmfB0S1j/oht4HoCW7xY3m5HSR00E8GrPB+RwqEbHn5ADXJq3OzjFaZrVUwH+VR/YjCt902iAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab18592bdc56ec78df15364f460440b3d3728504b26deffc8bee54264f861984","last_reissued_at":"2026-05-28T01:04:53.391188Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T01:04:53.391188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mixed thresholds in the Lonely Runner Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Alathea Jensen","submitted_at":"2026-05-27T04:27:10Z","abstract_excerpt":"The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\\ldots,s_k$, there is a real number $t$ such that $\\|s_i t\\|\\geq \\frac{1}{k+1}$ for all $i$.\n  We introduce and study a version of the conjecture in which the required distances may vary with $i$. 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