  Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove
a precise estimate for the number of positive integers $q\leq Q$ that satisfy
$\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get
asymptotics as $Q$ gets large, provided $\F Q$ grows quickly enough in terms of
the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if
$(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then we only
need that $\F Q$ tends to infinity. Our result yields a new upper bound on sums
of reciprocals of products of fractional parts, and sheds some light on a
recent question of L\^{e} and Vaaler.
