Title: Arbitrarily slow decay in the logarithmically averaged Sarnak conjecture Show Chinese title

Title: 对数平均的Sarnak猜想中的任意缓慢衰减

Abstract: In 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. Nonetheless, all of our examples satisfy the conjecture. Show Chinese abstract

Abstract: 2017年Tao提出了一个带有对数平均的Sarnak莫比乌斯不相关性猜想的变体：对于任何零熵动力系统$(X,T)$，$\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$对于每个$f\in \mathcal{C}(X)$和每个$x\in X$。我们构造了例子表明这个$o(1)$可以趋于零任意缓慢。尽管如此，我们所有的例子都满足该猜想。