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As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the M\\\"{o}bius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds."},"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":"1908.07554","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-20T18:12:01Z","cross_cats_sorted":[],"title_canon_sha256":"79242534f2eddd9e55f587436c3582213a500881271d708f21b45faedde43b9b","abstract_canon_sha256":"7805c58948751145f32e6ad163faf8ea3d6880b8772dd06a0580a7a6e51dbedb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:08.936296Z","signature_b64":"Gvon//8ScY99P5G2/vpmUnwscTHLHHTQqHh3XFYO7MmgkEM7BML+8Xm+wxPmVJ/wSFpzbsq7ZBlN9IHTLfFSDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbb97eb012de0f896284e4ba6a746e252e5ab6373bd0605416be5933cc8b16fd","last_reissued_at":"2026-07-04T23:59:08.935887Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:08.935887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reducing the Sarnak Conjecture to Toeplitz systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye, Zhengxing Lian","submitted_at":"2019-08-20T18:12:01Z","abstract_excerpt":"In this paper, we show that for any sequence ${\\bf a}=(a_n)_{n\\in \\Z}\\in \\{1,\\ldots,k\\}^\\mathbb{Z}$ and any $\\epsilon>0$, there exists a Toeplitz sequence ${\\bf b}=(b_n)_{n\\in \\Z}\\in \\{1,\\ldots,k\\}^\\mathbb{Z}$ such that the entropy $h({\\bf b})\\leq 2 h({\\bf a})$ and $\\lim_{N\\to\\infty}\\frac{1}{2N+1}\\sum_{n=-N}^N|a_n-b_n|<\\epsilon$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the M\\\"{o}bius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07554","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.07554/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.07554","created_at":"2026-07-04T23:59:08.935940+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.07554v2","created_at":"2026-07-04T23:59:08.935940+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.07554","created_at":"2026-07-04T23:59:08.935940+00:00"},{"alias_kind":"pith_short_12","alias_value":"XO4X5MAS3YHY","created_at":"2026-07-04T23:59:08.935940+00:00"},{"alias_kind":"pith_short_16","alias_value":"XO4X5MAS3YHYSYUE","created_at":"2026-07-04T23:59:08.935940+00:00"},{"alias_kind":"pith_short_8","alias_value":"XO4X5MAS","created_at":"2026-07-04T23:59:08.935940+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/XO4X5MAS3YHYSYUE4S5GU5DOEU","json":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU.json","graph_json":"https://pith.science/api/pith-number/XO4X5MAS3YHYSYUE4S5GU5DOEU/graph.json","events_json":"https://pith.science/api/pith-number/XO4X5MAS3YHYSYUE4S5GU5DOEU/events.json","paper":"https://pith.science/paper/XO4X5MAS"},"agent_actions":{"view_html":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU","download_json":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU.json","view_paper":"https://pith.science/paper/XO4X5MAS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.07554&json=true","fetch_graph":"https://pith.science/api/pith-number/XO4X5MAS3YHYSYUE4S5GU5DOEU/graph.json","fetch_events":"https://pith.science/api/pith-number/XO4X5MAS3YHYSYUE4S5GU5DOEU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU/action/storage_attestation","attest_author":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU/action/author_attestation","sign_citation":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU/action/citation_signature","submit_replication":"https://pith.science/pith/XO4X5MAS3YHYSYUE4S5GU5DOEU/action/replication_record"}},"created_at":"2026-07-04T23:59:08.935940+00:00","updated_at":"2026-07-04T23:59:08.935940+00:00"}