An exponent-$s$ dynamical Borel-Cantelli lemma and the waiting time problem

Galatolo and Kim proved that the dynamical Borel-Cantelli property for decreasing sequences of balls is tightly connected with the waiting time problem. In systems where all such sequences are Borel-Cantelli, the time needed to enter a small ball $B$ for the first time scales as $\mu(B)^{-1}$, and conversely, waiting time estimates yield Borel-Cantelli results for sequences of balls whose radii decrease in a controlled way. We extend this correspondence to the exponent-$s$ setting introduced by Tseng. For $s\ge 1$, the $s$-exponent monotone shrinking target property ($s$MSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition $\sum_n\mu(B_n)^s=\infty$. We prove that $s$MSTP forces the lower waiting time exponent, measured on the scale of $-\log\mu(B(y,r))$, to lie in the interval $[1,s]$ almost everywhere. That a quantitative ($s$-strong) form of the property bounds the upper exponent by $s$ and that, conversely, an exponent-$s$ waiting time estimate implies the Borel-Cantelli property for decreasing sequences of centered balls whose radii obey the calibrated decay condition matching the critical divergence exponent $s$. We also obtain the corresponding quantitative orbit approximation statement $\liminf_n n^{\beta}\,d(T^nx,y)=0$ for $\beta<1/(s\,\underline{d}_\mu(y))$, show that the universal lower bound with exponent $1$ pins the theory to $s\ge 1$, and discuss sharpness on circle rotations, where by results of Kurzweil, Kim-Seo and Tseng the picture is governed by the Diophantine type of the rotation number.