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The condition on in Proposition 6.1 can plausibly be replaced by . The points actually lie not just in a
cube but in a slab with 2 sides of bounded length, corresponding to the
conjugates and , and **d-2** sides of length corresponding
to the conjugates . The volume of is , which is
if . If we assume that lies in ``general position'' with
respect to , then we would expect it to contain
points of . It is possible that the slab is tilted in such a way as to
contain more than its fair share of points of , but we regard this as unlikely.
This cannot happen with the cube used in the proof of Proposition 6.1 and is the
reason it was used there. If the estimate could be
established rigorously then, for **d = 4**, the argument here could be applied if , which is almost what we get from the nonrigorous argument of section
6.4.

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**Up:** A probabilistic model
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