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If we go beyond the box principle used in section 6.1 and section 6.2 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 section 6.2, 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

For **d = 4**, this argument requires , which is just short of what can be
proved, while for **d = 6**, the requirement is , which is just short of the
we obtain in section 6.4 by a ``random walk'' argument. For ,
however, even the most favorable assumptions would not seem to justify the expectation
that the orbit should be finite.

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