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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 .

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