{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZAKHGV4H4SYJU2IAZEOZBWQSSO","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":"4bf341715e306522016db3ef88955f437cf335b2332e9fe2d9b593e5b0369af1","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-05-16T05:13:02Z","title_canon_sha256":"ea161455bbf3a62523fe0b2c5694d5d4c5280a25503ef9c8bd1ec4c938f693b4"},"schema_version":"1.0","source":{"id":"1205.3552","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.3552","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"arxiv_version","alias_value":"1205.3552v1","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.3552","created_at":"2026-05-18T03:48:26Z"},{"alias_kind":"pith_short_12","alias_value":"ZAKHGV4H4SYJ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZAKHGV4H4SYJU2IA","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZAKHGV4H","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:5a6493a26432a0b6dafef0b05d1977c781471ba8ce62360e0a3afa4519b0a7c3","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":"In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\\det (A)|=3$ and a collinear digit set ${\\mathcal{D}}=\\{0,1,b\\}v$, where $b>1$ and $v\\in {\\mathbb{R}}^2$ such that $\\{v, Av\\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,{\\mathcal{D}})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\\det (A)|> 3$ are obtained as well.","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":"2012-05-16T05:13:02Z","title":"Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.3552","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:c265628300e4835a96c2ed2ff200b853ef44e6f4dc454658a049f884c7a4546d","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":"4bf341715e306522016db3ef88955f437cf335b2332e9fe2d9b593e5b0369af1","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2012-05-16T05:13:02Z","title_canon_sha256":"ea161455bbf3a62523fe0b2c5694d5d4c5280a25503ef9c8bd1ec4c938f693b4"},"schema_version":"1.0","source":{"id":"1205.3552","kind":"arxiv","version":1}},"canonical_sha256":"c814735787e4b09a6900c91d90da129397edf59ec1db261c0c5db543fc63bb86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c814735787e4b09a6900c91d90da129397edf59ec1db261c0c5db543fc63bb86","first_computed_at":"2026-05-18T03:48:26.887669Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:26.887669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8bUO4XJMX0JOs6UpldPsx28xArp0VXYdER2tTG5od3A/UfRuAHrEjZzWPzf/55aMbDt8fpVeqBS6PGPAK4T0AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:26.888326Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.3552","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c265628300e4835a96c2ed2ff200b853ef44e6f4dc454658a049f884c7a4546d","sha256:5a6493a26432a0b6dafef0b05d1977c781471ba8ce62360e0a3afa4519b0a7c3"],"state_sha256":"65467eafc481605e69c12996c812940fad58eda5fc929fd2ca1a70cbb084200b"}