Arbitrarily slow decay in the Möbius disjointness conjecture

Abstract

Sarnak's Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N}{\sum }_{n=1}^{N}f(T^{n}x)\mu (n)=o(1)$ 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. In fact, our methods yield a more general result, where in lieu of $\mu (n)$ one can put any bounded sequence such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero.