{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:UWHSM4TTMXVBPXFX34NLW7JDRU","short_pith_number":"pith:UWHSM4TT","schema_version":"1.0","canonical_sha256":"a58f26727365ea17dcb7df1abb7d238d3734603d5d61252598856be1e3790519","source":{"kind":"arxiv","id":"1212.5063","version":2},"attestation_state":"computed","paper":{"title":"On constant-multiple-free sets contained in a random set of integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sang June Lee","submitted_at":"2012-12-20T14:46:29Z","abstract_excerpt":"For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$.\n  The extremal problem on estimating the maximum possible size of $r$-multiple-free sets contained in $[n]:={1,2,...,n}$ has been studied for its own interest in combinatorial number theory and application to coding theory. Let $a$, $b$ be positive integers such that $a<b$ and the greatest common divisor of $a$ and $b$ is 1. Wakeham and Wood showed that the maximum size of $(b/a)$-multiple-free sets contained in $[n]$ is $\\frac{b}{b+1}n+O"},"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":"1212.5063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-20T14:46:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6a10b8ace274d7702a8d3287ce7037b2c4ea62c229378ec9d8105a1ff0c542f2","abstract_canon_sha256":"3a731d187687243255ad0b510482723b3d654b743eb96635e14e38f45f5e56f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:35.682836Z","signature_b64":"5vJ89bQ8UPbA7LC6eU1W3BlzztgGjJuECdIGYe0eSZrVmGe8EMQBwXCyfUEiBdW+31rlaEtNnpRZS+kaQxIVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a58f26727365ea17dcb7df1abb7d238d3734603d5d61252598856be1e3790519","last_reissued_at":"2026-05-18T02:23:35.682033Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:35.682033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On constant-multiple-free sets contained in a random set of integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sang June Lee","submitted_at":"2012-12-20T14:46:29Z","abstract_excerpt":"For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$.\n  The extremal problem on estimating the maximum possible size of $r$-multiple-free sets contained in $[n]:={1,2,...,n}$ has been studied for its own interest in combinatorial number theory and application to coding theory. Let $a$, $b$ be positive integers such that $a<b$ and the greatest common divisor of $a$ and $b$ is 1. Wakeham and Wood showed that the maximum size of $(b/a)$-multiple-free sets contained in $[n]$ is $\\frac{b}{b+1}n+O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5063","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1212.5063","created_at":"2026-05-18T02:23:35.682151+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.5063v2","created_at":"2026-05-18T02:23:35.682151+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5063","created_at":"2026-05-18T02:23:35.682151+00:00"},{"alias_kind":"pith_short_12","alias_value":"UWHSM4TTMXVB","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"UWHSM4TTMXVBPXFX","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"UWHSM4TT","created_at":"2026-05-18T12:27:25.539911+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU","json":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU.json","graph_json":"https://pith.science/api/pith-number/UWHSM4TTMXVBPXFX34NLW7JDRU/graph.json","events_json":"https://pith.science/api/pith-number/UWHSM4TTMXVBPXFX34NLW7JDRU/events.json","paper":"https://pith.science/paper/UWHSM4TT"},"agent_actions":{"view_html":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU","download_json":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU.json","view_paper":"https://pith.science/paper/UWHSM4TT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.5063&json=true","fetch_graph":"https://pith.science/api/pith-number/UWHSM4TTMXVBPXFX34NLW7JDRU/graph.json","fetch_events":"https://pith.science/api/pith-number/UWHSM4TTMXVBPXFX34NLW7JDRU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU/action/storage_attestation","attest_author":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU/action/author_attestation","sign_citation":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU/action/citation_signature","submit_replication":"https://pith.science/pith/UWHSM4TTMXVBPXFX34NLW7JDRU/action/replication_record"}},"created_at":"2026-05-18T02:23:35.682151+00:00","updated_at":"2026-05-18T02:23:35.682151+00:00"}