Replacing a cube by a slab

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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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