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This has been extended to many other recurrence relations $\\{G_n\\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \\in [G_n, G_{n+1})$ converges to a Gaussian as $n\\to\\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $C_\\mu n+d_\\mu+o(1)$ for constants $C_\\mu > 0$ and $d_\\mu$ depending on $g$ and the recurr"},"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":"1606.08110","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-27T02:00:17Z","cross_cats_sorted":[],"title_canon_sha256":"7c01bd4b893c77602e2db951821143ab518ebdd13eddb17074e643ee2b009544","abstract_canon_sha256":"ab89919468247dde20926dd6f703c067a5b327c1fb1719e8dcaa91766275c530"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:19.714962Z","signature_b64":"XNbEszYUhtIZTBv4zuPLlJVyXJctHxJBjGPOQQilpm2w3IFLI6T2LjL+uAA5xqeov2qIZlFrUW9KZZyRU2AqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88ca280c0c59046b89eab7237faec1389f35e4bb0a2c923d9ec3d77f817468e2","last_reissued_at":"2026-05-18T01:10:19.714179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:19.714179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ray Li, Steven J. Miller","submitted_at":"2016-06-27T02:00:17Z","abstract_excerpt":"Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\\{1,2,3,5,\\dots\\}$. This has been extended to many other recurrence relations $\\{G_n\\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \\in [G_n, G_{n+1})$ converges to a Gaussian as $n\\to\\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $C_\\mu n+d_\\mu+o(1)$ for constants $C_\\mu > 0$ and $d_\\mu$ depending on $g$ and the recurr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08110","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.08110","created_at":"2026-05-18T01:10:19.714299+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.08110v2","created_at":"2026-05-18T01:10:19.714299+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.08110","created_at":"2026-05-18T01:10:19.714299+00:00"},{"alias_kind":"pith_short_12","alias_value":"RDFCQDAMLECG","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RDFCQDAMLECGXCPK","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RDFCQDAM","created_at":"2026-05-18T12:30:41.710351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC","json":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC.json","graph_json":"https://pith.science/api/pith-number/RDFCQDAMLECGXCPKW4RX7LWBHC/graph.json","events_json":"https://pith.science/api/pith-number/RDFCQDAMLECGXCPKW4RX7LWBHC/events.json","paper":"https://pith.science/paper/RDFCQDAM"},"agent_actions":{"view_html":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC","download_json":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC.json","view_paper":"https://pith.science/paper/RDFCQDAM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.08110&json=true","fetch_graph":"https://pith.science/api/pith-number/RDFCQDAMLECGXCPKW4RX7LWBHC/graph.json","fetch_events":"https://pith.science/api/pith-number/RDFCQDAMLECGXCPKW4RX7LWBHC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC/action/storage_attestation","attest_author":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC/action/author_attestation","sign_citation":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC/action/citation_signature","submit_replication":"https://pith.science/pith/RDFCQDAMLECGXCPKW4RX7LWBHC/action/replication_record"}},"created_at":"2026-05-18T01:10:19.714299+00:00","updated_at":"2026-05-18T01:10:19.714299+00:00"}