Let
$X\cong {SL}_{2}\left(\mathbb{R}\right)\u2215{SL}_{2}\left(\mathbb{Z}\right)$ 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 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
${\mathbb{R}}^{2}$
centered at the origin to derive an asymptotic formula for the volume of sets
${K}_{r}$ as
$r\to 0$.
Combined with a zeroone 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
$\left\{{K}_{r}\right\}$.
