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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