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 .4.
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