Möbius随机律与无限秩一映射
Möbius random law and infinite rank-one maps
摘要：
我们证明了Sarnak猜想对任何无限测度符号秩一映射成立。我们进一步推广Bourgain-Sarnak的结果，它说，默比乌斯函数是一个很好的权重遍历定理，映射作用在$σ$ -有限测度空间。我们还讨论和扩展Bourgain的定理，建立有一类映射的莫比乌斯不相交性的任何连续有界函数。我们的证明使我们能够得到一个扩展的Bourgain定理的莫比乌斯不相交有界秩一的地图和一个简单的和独立的证明这一事实。
英文摘要：
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 Möbius function is a good weight for the ergodic theorem, to maps acting on $σ$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the Möbius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on Möbius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.
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