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

Let X≅SL2(ℝ)∕SL2(ℤ) be the space of unimodular lattices in ℝ2, and for any r≥0 denote by Kr⊂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 K0= ⋂rKr 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 ℝ2 centered at the origin to derive an asymptotic formula for the volume of sets Kr as r→0. Combined with a zero-one law for the set of the ψ-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 {Kr}.