  We study the distribution modulo $1$ of the values taken on the integers of
$r$ linear forms in $d$ variables with random coefficients. We obtain quenched
and annealed central limit theorems for the number of simultaneous hits into
shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev
theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent
for the infinite number of hits.
