{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:VV4SLBAWB5JTSDTUCBCIF4MRQN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"39967bb3af7d5e14209a9dadb26aed865f1472d414c453509229eedf08f63154","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-08-18T16:08:18Z","title_canon_sha256":"a6f7976be06699121a5439075e0e6d058553cfa6bc4914e56a003f89835e69d3"},"schema_version":"1.0","source":{"id":"1208.3759","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.3759","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"arxiv_version","alias_value":"1208.3759v1","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.3759","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"pith_short_12","alias_value":"VV4SLBAWB5JT","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VV4SLBAWB5JTSDTU","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VV4SLBAW","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:2e56733707468fc6b7b1d0e68cc25369e530674736c5a6ad28ecb3da9794caee","target":"graph","created_at":"2026-05-18T03:48:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the connectedness of the planar self-affine sets $T(A,{\\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\\det(A)|=3$ and a non-collinear digit set ${\\mathcal D}=\\{0, v, kAv\\}$ where $k\\in {\\mathbb Z}\\setminus\\{0\\}$ and $v\\in {\\mathbb Z}^2$ such that $\\{v, Av\\}$ is linearly independent. By checking the characteristic polynomials of $A$ case by case, we obtain a criterion concerning only $k$ to determine the connectedness of $T(A,{\\mathcal{D}})$.","authors_text":"Jun Jason Luo, King-Shun Leung","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-08-18T16:08:18Z","title":"Connectedness of planar self-affine sets associated with non-collinear digit sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3759","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2c9d0d875c10a7de212db43598caa99ba3c457916d53cb9927e110d09d98f89e","target":"record","created_at":"2026-05-18T03:48:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"39967bb3af7d5e14209a9dadb26aed865f1472d414c453509229eedf08f63154","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-08-18T16:08:18Z","title_canon_sha256":"a6f7976be06699121a5439075e0e6d058553cfa6bc4914e56a003f89835e69d3"},"schema_version":"1.0","source":{"id":"1208.3759","kind":"arxiv","version":1}},"canonical_sha256":"ad792584160f53390e74104482f191837be9817db4fcc4185f8eed6d70aad267","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad792584160f53390e74104482f191837be9817db4fcc4185f8eed6d70aad267","first_computed_at":"2026-05-18T03:48:26.330267Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:26.330267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Cqc6ECvGr9vY4cCUMoeCen4BsPEL6foZoowW6pd7z09ah3+O4QUAA0AUfsWCKHHhN4p8CT3iP99K8A6VG8MCCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:26.330811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.3759","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c9d0d875c10a7de212db43598caa99ba3c457916d53cb9927e110d09d98f89e","sha256:2e56733707468fc6b7b1d0e68cc25369e530674736c5a6ad28ecb3da9794caee"],"state_sha256":"51adfe0a44bd097fd6f19b6e3c4771976c916b554cf9bc03c85c4d0a09ff0ffa"}