If we go beyond the box principle used in .1 and .2 and imagine that
the points of the orbit are distributed ``randomly'' in the
slab , we can allow a larger value of . Suppose that with , so the volume of satisfies and hence, as in .2, contains points of the lattice . Now suppose that the points
are randomly chosen from these points.
Then, by the ``birthday paradox'' we will have for some
with probability tending to 1 as . That is, the probability that n
randomly selected points among are distinct is
as since .