Given a minimal dynamical system T : X → X and elements x, y ∈ X, one can always approximate y by elements of the orbit {Tn(x)}n≥1 . This raises the natural question to estimate the size of points approximable at various speed rates by an orbit {Tn(x)}n≥1 . Fan-Shmeling-Troubetzkoy study such problems in the case where X = T1 and T is the doubling of the angle and, re_ning this result, Liao-Seuret extended the result for expanding markov maps on [0, 1].

Many author established some variation, of such results, among which one can refer to Wang, Wu, Xu on the middle third Cantor set equipped with the Cantor measure and Hu-Li, who studied the case of exponentially mixing Alfhors regular systems.