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 .2 for d = 4 and the birthday paradox argument of .3 for d = 6 to conclude that the orbit of should be finite.
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