For a topological dynamical system $(X,T)$ (X is a compact Hausdorff space, and T is a continuous map from X to X), it is called strong mixing if For any nonempty open set U and V, $N(U,V):=\{n\in \mathbb{Z}_+:U\cap T^{-n}V\neq \emptyset\}$ (Here $\mathbb{Z}_+$ is the set of non-negative integers) is cofinite in $\mathbb{Z}_+$.
My Question is, if we have a cofinite set A in hand, can we find a strong mixing system (X,T) and two nonempty open set U, V in X such that N(U,V)=A?