  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 $\psi$-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 $\{K_r\}$.
