On eventually always hitting points

We consider dynamical systems (X, T, μ) which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls (Bn)n=1∞, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a k< m with T^k(x) ∈ B_m. We also give an asymptotic estimate as m→ ∞ on the number of k< m with T^k(x) ∈ B_m. As an application, we prove for almost every point x an asymptotic estimate on the number of k≤ m such that a_k≥ m^t, where t∈ (0 , 1) and a_k are the continued fraction coefficients of x.