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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"},"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":"1408.5495","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-23T13:30:29Z","cross_cats_sorted":[],"title_canon_sha256":"488f08344aff779de98afbaee838756536aeebb78ffc181fb9585d16cf383d2f","abstract_canon_sha256":"4acfb5be0b259d8c80ac5be9b2abb9bd90d2ed291f90e24f5412c3ee9ba3c558"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:25.039856Z","signature_b64":"KsPYGSLN9r8dQz7sYL30vVHcIQbar2A7nJi4p5tOdywGpswjXeR1cuQNPJZotKPjgD0cB14S2EUBO9lkXlHsDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"262a25a6c6aa51e9783ed5e371db0480f888b58719ce881b2531cb1e4b66d52e","last_reissued_at":"2026-05-18T02:44:25.039444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:25.039444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Envelope Word and Gap Sequence in Doubling Sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Hanxiong Zhang, Yuke Huang","submitted_at":"2014-08-23T13:30:29Z","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. 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