Sarnak's M\"{ o}bius 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 Ces\`aro mean of the corresponding sequence of absolute values does not tend to zero.
Categories: math.DS math.NT