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.