On the Coverage of $k$-Dimensional Space by $k$-Dimensional Spheres

Let $n k$-dimensional spheres, each of content $a_n$, be distributed within a $k$-dimensional cube according to density $f$. We derive necessary and sufficient conditions on $a_n$ in order that the probability that the cube is completely covered at least $\ell$ times by the spheres, tend to one as $n\rightarrow\infty$. (Here $\ell$ is an arbitrary positive integer.) In the special case $f\equiv$ const., we obtain upper and lower bounds of the same order of magnitude for the probability of incomplete coverage.