A dynamical Borel–Cantelli lemma via improvements to Dirichlet's theorem

Abstract

Let $X\cong {SL}_2\left(ℝ\right)∕{SL}_2\left(\mathbb{Z}\right)$ 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 $\left\{K_r\right\}$.