Arbitrarily slow decay in the Mobius disjointness conjecture

Sarnak's Mobius disjointness conjecture asserts that for any zero entropy dynamical system (X, T), (1/N)Sigma(N)(n=1) f (T(n)x)mu(n) = o(1) for every f is an element of C(X) and every x is an element of 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 a(n) such that the Cesaro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence a(n) but (X, T) does not.