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![[Annotate]](/organics/icons/sannotate.gif)
![[Shownotes]](../gif/annotate/sshow-141.gif)
Let us explore the consequences of the plausible assumption
. By (6.3), we
expect that
, where the O constant is uniform in
. Thus the region
, being the product of two intervals of lengths 1 and
and
disks of radius
, has volume
. Since the points of
are distributed with density
, and assuming
is in general
position, there are
points of
in
. If
is sufficiently large, it is thus possible to
push through the argument of section 6.2 for d = 4 and the birthday paradox argument of
section 6.3 for d = 6 to conclude that the orbit of
should be finite.
More precisely,
, so
, where
If
is large, then the largest terms in
are the d-1 terms
involving the conjugate
; their product is about
. If the
conjugates on the unit circle are nicely distributed, then we would expect the product of
terms involving them not to be too small and so ``typically''
and
from (6.4). Values of
which
are untypically small will lead to large values of
.

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Next: The expected size
Up: A probabilistic model
Previous: A random walk