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arXiv · 2203.14971

M\"{o}bius random law and infinite rank-one maps

e. H. el Abdalaoui·Cesar E. Silva
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Abstract

We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the M\"{o}bius function is a good weight for the ergodic theorem, to maps acting on $\sigma$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the M\"{o}bius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on M\"{o}bius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.

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e. H. el Abdalaoui, Cesar E. Silva. 2022-03-28. M\"{o}bius random law and infinite rank-one maps. https://arxiv.org/abs/2203.14971

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