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

Let $X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)$ be the space of unimodular lattices in $\mathbb R^2$, 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 subset{s} 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 $\mathbb R^2$ 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 $ψ$-Dirichlet numbers established by Kleinbock and Wadleigh, this gives a new dynamical Borel-Cantelli lemma for the geodesic flow on $X$ with respect to the family of shrinking targets $\{K_r\}$.