{"paper":{"title":"On the homogeneous ergodic bilinear averages with M\\\"{o}bius and liouville weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.NT","math.PR"],"primary_cat":"math.CA","authors_text":"el Houcein el Abdalaoui","submitted_at":"2017-06-15T20:56:34Z","abstract_excerpt":"It is shown that the homogeneous ergodic bilinear averages with M\\\"{o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\\mathcal{A},\\mu)$, and $a,b \\in \\mathbb{Z}$, then for any $f,g \\in L^2(X)$, for almost all $x \\in X$, $${\\frac{1}{N}}\\sum_{n=1}^{N} \\nu(n) f(T^{an}x) g(T^{bn}x) \\longrightarrow 0, \\text{ as } N \\longrightarrow +\\infty, $$ where $\\nu$ is the Liouville function or the M\\\"{o}bius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan's estimation. Also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07280","kind":"arxiv","version":4},"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/1706.07280/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"}