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