#### Vol. 9, No. 2, 2020

 Recent Issues Volume 9, Issue 2 Volume 9, Issue 1 Volume 8, Issue 4 Volume 8, Issue 3 Volume 8, Issue 2 Volume 8, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 Previously Published To Appear founded and published with the scientific support and advice of the Moscow Institute of Physics and Technology Other MSP Journals
A dynamical Borel–Cantelli lemma via improvements to Dirichlet's theorem

### Dmitry Kleinbock and Shucheng Yu

Vol. 9 (2020), No. 2, 101–122
DOI: 10.2140/moscow.2020.9.101
##### Abstract

Let $X\cong {SL}_{2}\left(ℝ\right)∕{SL}_{2}\left(ℤ\right)$ be the space of unimodular lattices in ${ℝ}^{2}$, and for any $r\ge 0$ denote by ${K}_{r}\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least ${e}^{-r}$. These are compact nested subsets of $X$, with ${K}_{0}={\bigcap }_{r}{K}_{r}$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in ${ℝ}^{2}$ centered at the origin to derive an asymptotic formula for the volume of sets ${K}_{r}$ as $r\to 0$. Combined with a zero-one law for the set of the $\psi$-Dirichlet numbers established by Kleinbock and Wadleigh (Proc. Amer. Math. Soc. 146 (2018), 1833–1844), this gives a new dynamical Borel–Cantelli lemma for the geodesic flow on $X$ with respect to the family of shrinking targets $\left\{{K}_{r}\right\}$.

##### Keywords
Siegel transform, dynamical Borel–Cantelli lemma
##### Mathematical Subject Classification 2010
Primary: 11J04, 37A17
Secondary: 11H60, 37D40