{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:EYVCLJWGVJI6S6B62XRXDWYEQD","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":"4acfb5be0b259d8c80ac5be9b2abb9bd90d2ed291f90e24f5412c3ee9ba3c558","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-23T13:30:29Z","title_canon_sha256":"488f08344aff779de98afbaee838756536aeebb78ffc181fb9585d16cf383d2f"},"schema_version":"1.0","source":{"id":"1408.5495","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.5495","created_at":"2026-05-18T02:44:25Z"},{"alias_kind":"arxiv_version","alias_value":"1408.5495v1","created_at":"2026-05-18T02:44:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5495","created_at":"2026-05-18T02:44:25Z"},{"alias_kind":"pith_short_12","alias_value":"EYVCLJWGVJI6","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"EYVCLJWGVJI6S6B6","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"EYVCLJWG","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:aafcafa91a9743efc3806c2740804250fea7e1da90ff7424c33534e2b88c2fb0","target":"graph","created_at":"2026-05-18T02:44:25Z","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 $\\omega$ be a factor of Doubling sequence $D_\\infty=x_1x_2\\cdots$, then it occurs in the sequence infinitely many times. Let $\\omega_p$ be the $p$-th occurrence of $\\omega$ and $G_p(\\omega)$ be the gap between $\\omega_p$ and $\\omega_{p+1}$. In this paper, we discuss the structure of the gap sequence $\\{G_p(\\omega)\\}_{p\\geq1}$. We prove that all factors can be divided into two types, one type has exactly two distinct gaps $G_1(\\omega)$ and $G_2(\\omega)$, the other type has exactly three distinct gaps $G_1(\\omega)$, $G_2(\\omega)$ and $G_4(\\omega)$. We determine the expressions of gaps comple","authors_text":"Hanxiong Zhang, Yuke Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-23T13:30:29Z","title":"Envelope Word and Gap Sequence in Doubling Sequence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5495","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":"sha256:7cdc1009b435826d638d876cb863a4773ded684337c3f0cfe9a2aa96c787d65e","target":"record","created_at":"2026-05-18T02:44:25Z","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":"4acfb5be0b259d8c80ac5be9b2abb9bd90d2ed291f90e24f5412c3ee9ba3c558","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-23T13:30:29Z","title_canon_sha256":"488f08344aff779de98afbaee838756536aeebb78ffc181fb9585d16cf383d2f"},"schema_version":"1.0","source":{"id":"1408.5495","kind":"arxiv","version":1}},"canonical_sha256":"262a25a6c6aa51e9783ed5e371db0480f888b58719ce881b2531cb1e4b66d52e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"262a25a6c6aa51e9783ed5e371db0480f888b58719ce881b2531cb1e4b66d52e","first_computed_at":"2026-05-18T02:44:25.039444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:25.039444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KsPYGSLN9r8dQz7sYL30vVHcIQbar2A7nJi4p5tOdywGpswjXeR1cuQNPJZotKPjgD0cB14S2EUBO9lkXlHsDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:25.039856Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.5495","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7cdc1009b435826d638d876cb863a4773ded684337c3f0cfe9a2aa96c787d65e","sha256:aafcafa91a9743efc3806c2740804250fea7e1da90ff7424c33534e2b88c2fb0"],"state_sha256":"c24fe51380e4edd59bf67b9426d9b4838343fab115e78fa3ccd256512c4b54c8"}