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Abstract Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$ , $$({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$$ 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 $$a_{n}$$ such that the Cesàro 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.more » « less
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