Contents Next: A random walk Up: A probabilistic model Previous: Replacing a cube

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 .

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 .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.

Contents Next: A random walk Up: A probabilistic model Previous: Replacing a cube