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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 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