Recall that was defined by

(dot product)

and our goal is to show that there is a such that
.
Our strategy is to use an appropriately chosen to construct a unit such . Recall that we used Blichfeld's lemma to find an such that , and

Let be representative generators for the finitely many nonzero principal ideals of of norm at most . Modify the to have the property that is minimal among generators of (this is possible because ideals are discrete). Note that the set depends only on . Since , we have , for some , so there is a unit such that .

Let

The inequality of the lemma now follows. That only depends on and our choice of follows from the formula for and how we chose the .

The amazing thing about Lemma 12.2.1 is that the bound on the right hand side does not depend on the . Suppose we could somehow cleverly choose the positive real numbers in such a way that

and

Then the facts that
and would together imply
that (since is closer to than is to 0),
which is exactly what we aimed to prove. We finish the proof by
showing that it is possible to choose such . Note that if we
change the , then could change, hence the such that
is a unit could change, but the don't change, just the
subscript . Also note that if , then we are trying to prove
that
is a lattice in
, which is
automatically true, so we may assume that .

where and for , and and for ,

The condition that is that the are not all the same, and in our new coordinates the lemma is equivalent to showing that , subject to the condition that . Order the so that . By hypothesis there exists a such that , and again re-ordering we may assume that . Set . Then and , so

Since , we have as .

William Stein 2004-05-06