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 ${\sum }_n\mu (B_n{)}^s=\mathrm{\infty }$. 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 ${\mathrm{lim\hspace{0.17em}inf}}_n{n}^{\beta }d(T^{n}x,y)=0$ for $\beta <1\mathrm{/}(s{\underset{‾}{d}}_{\mu }(y))$, 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.