M{ö}bius orthogonality in density for zero entropy dynamical systems

Abstract

It is proved that whenever a zero entropy dynamical system $(X,T)$ has only countably many ergodic measures and $\mu$ stands for the arithmetic M{ö}bius function, then there exists a subset $A$ of integers depending only on the system, of logarithmic density one, such that for each $f$ continuous on $X$, $\frac{1}{N}{\sum }_{n\le N}f(T^{n}x)\mu (n)\to 0$ as $N\to \infty$, $N\in A$, uniformly in $x\in X$. In particular, the density version of M{ö}bius orthogonality holds.