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Let r_k(\\ell) denote the cardinality of the largest subset of the set {0,1,2,\\ldots, \\ell -1\\} that contains no arithmetric progression of length k. The limit \\[ \\lim_{n\\rightarrow \\infty} \\frac{g_k^{(s)}(n)}{n} = (s-1) \\sum_{m=1}^{\\infty}\n  \\left(\\frac{1}{s} \\right)^{\\min \\left(r_k^{-1}(m)\\right)} \\] exists and converges to an irrational number."},"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":"1307.8135","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-07-30T20:14:33Z","cross_cats_sorted":[],"title_canon_sha256":"2440a05640356e8114d5df3b0738106b8ad7e5b05edc6f01950ed7963799e8ef","abstract_canon_sha256":"192e2f061899fe074d064b7d8ed85ad119e789c7e9b996677949fd6961377924"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:53.575990Z","signature_b64":"lBzCJ3lC20b6Ku6Jjj+fLK8UO//gvjIU0OwCjp365D6St/YXawMl8eqGNqLyT2Oy3qchQlH7t3efxBVJFvS/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0188bf13f1934ec963ee20a10d78bd3d5dc0cd7b0111f571494aa11a95052f6","last_reissued_at":"2026-05-18T01:15:53.575436Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:53.575436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Irrational numbers associated to sequences without geometric progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin O'Bryant, Melvyn B. Nathanson","submitted_at":"2013-07-30T20:14:33Z","abstract_excerpt":"Let s and k be integers with s \\geq 2 and k \\geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\\ell) denote the cardinality of the largest subset of the set {0,1,2,\\ldots, \\ell -1\\} that contains no arithmetric progression of length k. The limit \\[ \\lim_{n\\rightarrow \\infty} \\frac{g_k^{(s)}(n)}{n} = (s-1) \\sum_{m=1}^{\\infty}\n  \\left(\\frac{1}{s} \\right)^{\\min \\left(r_k^{-1}(m)\\right)} \\] exists and converges to an irrational number."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.8135","kind":"arxiv","version":1},"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":"1307.8135","created_at":"2026-05-18T01:15:53.575534+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.8135v1","created_at":"2026-05-18T01:15:53.575534+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.8135","created_at":"2026-05-18T01:15:53.575534+00:00"},{"alias_kind":"pith_short_12","alias_value":"UAMIX4J7DE2O","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"UAMIX4J7DE2OZFR6","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"UAMIX4J7","created_at":"2026-05-18T12:28:02.375192+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/UAMIX4J7DE2OZFR64IFBBV4L2P","json":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P.json","graph_json":"https://pith.science/api/pith-number/UAMIX4J7DE2OZFR64IFBBV4L2P/graph.json","events_json":"https://pith.science/api/pith-number/UAMIX4J7DE2OZFR64IFBBV4L2P/events.json","paper":"https://pith.science/paper/UAMIX4J7"},"agent_actions":{"view_html":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P","download_json":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P.json","view_paper":"https://pith.science/paper/UAMIX4J7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.8135&json=true","fetch_graph":"https://pith.science/api/pith-number/UAMIX4J7DE2OZFR64IFBBV4L2P/graph.json","fetch_events":"https://pith.science/api/pith-number/UAMIX4J7DE2OZFR64IFBBV4L2P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P/action/storage_attestation","attest_author":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P/action/author_attestation","sign_citation":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P/action/citation_signature","submit_replication":"https://pith.science/pith/UAMIX4J7DE2OZFR64IFBBV4L2P/action/replication_record"}},"created_at":"2026-05-18T01:15:53.575534+00:00","updated_at":"2026-05-18T01:15:53.575534+00:00"}