Arbitrarily slow decay in the M"obius disjointness conjecture

Sarnak’s M"obius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, any point $x$ in $X$ and any continuous function $f$ on $X$, the correlation of $f(T^n x)$ with the M"obius function tends to 0. We construct examples showing that this correlation can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of μ(n) one can put any bounded sequence such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. This is a joint work with Amir Algom.