An exponent-s dynamical Borel-Cantelli lemma and the waiting time problem
Abstract
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≥ 1, the s-exponent monotone shrinking target property (sMSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition ∑nμ(Bn)s=∞. We prove that sMSTP 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 liminfn nβ d(Tnx,y)=0 for β<1/(s underlinedμ(y)), show that the universal lower bound with exponent $1$ pins the theory to s≥ 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.
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BibTeX
@misc{bajovic2026exponents,
title = {{An exponent-$s$ dynamical Borel-Cantelli lemma and the waiting time problem}},
author = {Dušan Bajović and Boris Petković},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.11180},
eprint = {2607.11180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11180},
}
RIS
TY - UNPB
TI - An exponent-s dynamical Borel-Cantelli lemma and the waiting time problem
AU - Dušan Bajović
AU - Boris Petković
PY - 2026
DA - 2026/07/14
PB - arXiv
DO - 10.48550/arxiv.2607.11180
UR - https://arxiv.org/abs/2607.11180
AB - 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≥ 1, the s-exponent monotone shrinking target property (sMSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition ∑_nμ(B_n)^s=∞. We prove that sMSTP 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 underlined_μ(y)), show that the universal lower bound with exponent $1$ pins the theory to s≥ 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.
ER -