We consider the problem of covering an arbitrary point pattern---a set of
$\lambda T$ points in the interval $[0,T]$---with a subset of $[0,T]$ that is
drawn from a predefined codebook. The subset is required to contain either all
or a certain proportion of the points in the pattern, depending on the problem
setting. Also, all subsets in this codebook must have Lebesgue measure not
exceeding $dT$ where $d \le 1$ is a given constant. The problem of interest here
is to find the trade-off between $d$ and the size of the codebook. We find
this
trade-off asymptotically as $T$ goes to infinity. When the subset is required
to
cover all the points, the answer turns out to be the same as in the case where
the points were randomly generated by a Poisson process of intensity $\lambda$.