{"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"}