{"paper":{"title":"Measure complexity and M\\\"obius disjointness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Wen Huang, Xiangdong Ye, Zhiren Wang","submitted_at":"2017-07-20T02:21:18Z","abstract_excerpt":"In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's M\\\"{o}bius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity.\n  Moreover, it is proved that the following classes of topological dynamical systems $(X,T)$ meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$ is a homotopically trivial $C^\\infty$ skew product system on $\\mathbb{T}^2$ over an irratio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06345","kind":"arxiv","version":1},"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"}