{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:3TPVKFJGDGARHSN2DHSVDJ4KMX","short_pith_number":"pith:3TPVKFJG","canonical_record":{"source":{"id":"1811.06718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-16T09:25:38Z","cross_cats_sorted":[],"title_canon_sha256":"2374248615b6da39bbe07096cbe29cf8e2b5c5ac31abdd8ec4b03d7a1feda822","abstract_canon_sha256":"ab57d76aad011411daddd55bbad40f072a64be2c7d9b87e0beb1b5da176e865d"},"schema_version":"1.0"},"canonical_sha256":"dcdf551526198113c9ba19e551a78a65d8537dc97c4d4b4dab9d83ac81743d9f","source":{"kind":"arxiv","id":"1811.06718","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.06718","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"arxiv_version","alias_value":"1811.06718v2","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.06718","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"pith_short_12","alias_value":"3TPVKFJGDGAR","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"3TPVKFJGDGARHSN2","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"3TPVKFJG","created_at":"2026-05-18T12:32:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:3TPVKFJGDGARHSN2DHSVDJ4KMX","target":"record","payload":{"canonical_record":{"source":{"id":"1811.06718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-16T09:25:38Z","cross_cats_sorted":[],"title_canon_sha256":"2374248615b6da39bbe07096cbe29cf8e2b5c5ac31abdd8ec4b03d7a1feda822","abstract_canon_sha256":"ab57d76aad011411daddd55bbad40f072a64be2c7d9b87e0beb1b5da176e865d"},"schema_version":"1.0"},"canonical_sha256":"dcdf551526198113c9ba19e551a78a65d8537dc97c4d4b4dab9d83ac81743d9f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:49.856095Z","signature_b64":"s+RdEnCgK3X7wIO6cutubIDfcA6uXgpaRcltGhwIi25ucaa6fG+LTyb2vo7iZfECfUXRqZRW1LLiHpsaDqO3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dcdf551526198113c9ba19e551a78a65d8537dc97c4d4b4dab9d83ac81743d9f","last_reissued_at":"2026-05-17T23:42:49.855474Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:49.855474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.06718","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CrTQk2kGVq5YxbTiDTx6k8eI4YZ/na6ZgO3uS0bt7wyMuNAJbKgit/VjEe0MvZzVjfVYTq1VXaAVvinFmXbUCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T10:57:47.545406Z"},"content_sha256":"4a4fb559d949074a145d4babd08653c97732290f43f57854175e3d1ea50ff7af","schema_version":"1.0","event_id":"sha256:4a4fb559d949074a145d4babd08653c97732290f43f57854175e3d1ea50ff7af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:3TPVKFJGDGARHSN2DHSVDJ4KMX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On self-affine tiles whose boundary is a sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"J\\\"org Thuswaldner, Shu-Qin Zhang","submitted_at":"2018-11-16T09:25:38Z","abstract_excerpt":"Let $M$ be a $3\\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\\mathcal{D}\\subset\\mathbb{Z}^3$ be a set with $|\\mathcal{D}|=|\\det M|$, called digit set. The set equation $MT = T+\\mathcal{D}$ uniquely defines a nonempty compact set $T\\subset \\mathbb{R}^3$. If $T$ has positive Lebesgue measure it is called a $3$-dimensional self-affine tile. In the present paper we study topological properties of $3$-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form $\\mathcal{D}=\\{0,v,2v,\\ldots, (|\\det M|-1)v\\}$ for some $v\\in\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06718","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R+mBXFNFEjk5xl6CRsEDjYG71EUqfBpBECswuQG8bBXvzmFjDVzKQvCMQDC2vFwhJc6t209Pz6NuFrFd3q2TCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T10:57:47.545754Z"},"content_sha256":"ab6cadbb97e54afe7506e0df779591c7123657c922ff97541b3cc28d9a3947a8","schema_version":"1.0","event_id":"sha256:ab6cadbb97e54afe7506e0df779591c7123657c922ff97541b3cc28d9a3947a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/bundle.json","state_url":"https://pith.science/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T10:57:47Z","links":{"resolver":"https://pith.science/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX","bundle":"https://pith.science/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/bundle.json","state":"https://pith.science/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3TPVKFJGDGARHSN2DHSVDJ4KMX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3TPVKFJGDGARHSN2DHSVDJ4KMX","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":"ab57d76aad011411daddd55bbad40f072a64be2c7d9b87e0beb1b5da176e865d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-16T09:25:38Z","title_canon_sha256":"2374248615b6da39bbe07096cbe29cf8e2b5c5ac31abdd8ec4b03d7a1feda822"},"schema_version":"1.0","source":{"id":"1811.06718","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.06718","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"arxiv_version","alias_value":"1811.06718v2","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.06718","created_at":"2026-05-17T23:42:49Z"},{"alias_kind":"pith_short_12","alias_value":"3TPVKFJGDGAR","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"3TPVKFJGDGARHSN2","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"3TPVKFJG","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:ab6cadbb97e54afe7506e0df779591c7123657c922ff97541b3cc28d9a3947a8","target":"graph","created_at":"2026-05-17T23:42:49Z","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":"Let $M$ be a $3\\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\\mathcal{D}\\subset\\mathbb{Z}^3$ be a set with $|\\mathcal{D}|=|\\det M|$, called digit set. The set equation $MT = T+\\mathcal{D}$ uniquely defines a nonempty compact set $T\\subset \\mathbb{R}^3$. If $T$ has positive Lebesgue measure it is called a $3$-dimensional self-affine tile. In the present paper we study topological properties of $3$-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form $\\mathcal{D}=\\{0,v,2v,\\ldots, (|\\det M|-1)v\\}$ for some $v\\in\\ma","authors_text":"J\\\"org Thuswaldner, Shu-Qin Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-16T09:25:38Z","title":"On self-affine tiles whose boundary is a sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06718","kind":"arxiv","version":2},"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":"sha256:4a4fb559d949074a145d4babd08653c97732290f43f57854175e3d1ea50ff7af","target":"record","created_at":"2026-05-17T23:42:49Z","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":"ab57d76aad011411daddd55bbad40f072a64be2c7d9b87e0beb1b5da176e865d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-11-16T09:25:38Z","title_canon_sha256":"2374248615b6da39bbe07096cbe29cf8e2b5c5ac31abdd8ec4b03d7a1feda822"},"schema_version":"1.0","source":{"id":"1811.06718","kind":"arxiv","version":2}},"canonical_sha256":"dcdf551526198113c9ba19e551a78a65d8537dc97c4d4b4dab9d83ac81743d9f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dcdf551526198113c9ba19e551a78a65d8537dc97c4d4b4dab9d83ac81743d9f","first_computed_at":"2026-05-17T23:42:49.855474Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:49.855474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s+RdEnCgK3X7wIO6cutubIDfcA6uXgpaRcltGhwIi25ucaa6fG+LTyb2vo7iZfECfUXRqZRW1LLiHpsaDqO3AQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:49.856095Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.06718","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a4fb559d949074a145d4babd08653c97732290f43f57854175e3d1ea50ff7af","sha256:ab6cadbb97e54afe7506e0df779591c7123657c922ff97541b3cc28d9a3947a8"],"state_sha256":"f83d4e455dda33c8a9716bc4221adf51b3b582245ba5366535608f9a7730bd3b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P76cvhbYpuzyl4NEijuYIKmqkEyeyq+3h515vbGjUUqV0qte7//NYkc6pnTKZNsCZyI/YqpmX3EYHjs0+e+FBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T10:57:47.547578Z","bundle_sha256":"e4078c3438b492bd2bd9870d9af31b0db0bf6b79ea3165fa557f8ea5d74dad22"}}