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 .
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