{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:2QPDMLQQU2YOEOX626GTLADOFM","short_pith_number":"pith:2QPDMLQQ","schema_version":"1.0","canonical_sha256":"d41e362e10a6b0e23afed78d35806e2b1e363d053c4c0aa08144382e62120e25","source":{"kind":"arxiv","id":"1312.2233","version":1},"attestation_state":"computed","paper":{"title":"Invariant games and non-homogeneous Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Eric Duch\\^ene, Julien Cassaigne, Michel Rigo","submitted_at":"2013-12-08T16:23:44Z","abstract_excerpt":"We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\\ge 0}$ and $(B_n)_{n\\ge 0}$ for which there exists an invariant game having exactly $\\{(A_n,B_n)\\mid n\\ge 0\\}\\cup \\{(B_n,A_n)\\mid n\\ge 0\\}$ as set of $\\mathcal{P}$-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can theref"},"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":"1312.2233","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-08T16:23:44Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"4cacdc142095431e7dc415356eecaeda83fb9369db9af040239934b333bdd2b5","abstract_canon_sha256":"6da6f37b0e5e17f9c5495f0f91c3832289b2067c21dbccfaa6c4af22664984cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:10.907177Z","signature_b64":"PGWCX6fSGfW7dDbPuWkzfD1XTPHp8gDHqhWf+8p2qaboqiedHScfEE2vYxpPwBrk/0IpsMOq5ejaZ4jBAkhHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d41e362e10a6b0e23afed78d35806e2b1e363d053c4c0aa08144382e62120e25","last_reissued_at":"2026-05-18T03:05:10.906644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:10.906644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant games and non-homogeneous Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Eric Duch\\^ene, Julien Cassaigne, Michel Rigo","submitted_at":"2013-12-08T16:23:44Z","abstract_excerpt":"We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\\ge 0}$ and $(B_n)_{n\\ge 0}$ for which there exists an invariant game having exactly $\\{(A_n,B_n)\\mid n\\ge 0\\}\\cup \\{(B_n,A_n)\\mid n\\ge 0\\}$ as set of $\\mathcal{P}$-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can theref"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2233","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":"1312.2233","created_at":"2026-05-18T03:05:10.906747+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.2233v1","created_at":"2026-05-18T03:05:10.906747+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2233","created_at":"2026-05-18T03:05:10.906747+00:00"},{"alias_kind":"pith_short_12","alias_value":"2QPDMLQQU2YO","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"2QPDMLQQU2YOEOX6","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"2QPDMLQQ","created_at":"2026-05-18T12:27:32.513160+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/2QPDMLQQU2YOEOX626GTLADOFM","json":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM.json","graph_json":"https://pith.science/api/pith-number/2QPDMLQQU2YOEOX626GTLADOFM/graph.json","events_json":"https://pith.science/api/pith-number/2QPDMLQQU2YOEOX626GTLADOFM/events.json","paper":"https://pith.science/paper/2QPDMLQQ"},"agent_actions":{"view_html":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM","download_json":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM.json","view_paper":"https://pith.science/paper/2QPDMLQQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.2233&json=true","fetch_graph":"https://pith.science/api/pith-number/2QPDMLQQU2YOEOX626GTLADOFM/graph.json","fetch_events":"https://pith.science/api/pith-number/2QPDMLQQU2YOEOX626GTLADOFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM/action/storage_attestation","attest_author":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM/action/author_attestation","sign_citation":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM/action/citation_signature","submit_replication":"https://pith.science/pith/2QPDMLQQU2YOEOX626GTLADOFM/action/replication_record"}},"created_at":"2026-05-18T03:05:10.906747+00:00","updated_at":"2026-05-18T03:05:10.906747+00:00"}