指数-$s$动力Borel-Cantelli引理与等待时间问题
An exponent-$s$ dynamical Borel-Cantelli lemma and the waiting time problem
AI总结：
研究将球的递减序列的动力Borel-Cantelli性质与等待时间问题的对应关系扩展到指数-$s$设置。通过证明$s$MSTP与等待时间指数的关系等，得到定量轨道近似陈述，讨论了相关性质在圆旋转上的尖锐性。
AI中文摘要：
加拉托洛和金证明了球的递减序列的动力Borel-Cantelli性质与等待时间问题紧密相关。在所有此类序列都是Borel-Cantelli的系统中，首次进入小球$B$所需的时间尺度为$\mu(B)^{-1}$，反之，等待时间估计为半径以可控方式减小的球序列产生Borel-Cantelli结果。我们将这种对应关系扩展到曾引入的指数-$s$设置。对于$s\geq1$，$s$-指数单调收缩目标性质（$s$MSTP）仅要求对于满足更强发散条件$\sum_n\mu(B_n)^s=\infty$的中心球的递减序列有Borel-Cantelli结论。我们证明$s$MSTP迫使以$-\log\mu(B(y,r))$为尺度测量的较低等待时间指数几乎处处位于区间$[1,s]$。该性质的定量（$s$-强）形式将上指数限制为$s$，反之，指数-$s$等待时间估计意味着对于半径服从与临界发散指数$s$匹配的校准衰减条件的中心球的递减序列有Borel-Cantelli性质。我们还得到了相应的定量轨道近似陈述$\liminf_n n^{\beta}\,d(T^nx,y)=0$对于$\beta < 1/(s\,\underline{d}_\mu(y))$，表明指数为$1$的通用下界将理论固定在$s\geq1$，并在圆旋转上讨论了尖锐性，其中根据库尔兹韦尔、金-徐和曾的结果，情况由旋转数的丢番图类型决定。
英文摘要：
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 $μ(B)^{-1}$, 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 $\sum_nμ(B_n)^s=\infty$. We prove that $s$MSTP forces the lower waiting time exponent, measured on the scale of $-\logμ(B(y,r))$, 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 $\liminf_n n^β\,d(T^nx,y)=0$ for $β<1/(s\,\underline{d}_μ(y))$, 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.
