Proposition 3.2
If |x| =1, , then .
Proof
We claim that if , then the condition of Lemma 3.3
holds for n large enough.
If and is irrational, then by Kronecker's theorem
is dense on the unit circle,
and then for every , the disk of radius is contained in the union on the right side of for n large enough.
If is rational, then the are the vertices of a regular k-gon, and since .
In that case the union on the right side of
contains a disk of radius r, where r, 1, and R are the sides of a triangle, and the angle between the sides of lengths r and 1 is .
Therefore, by the Law of Cosines,
and so
Since , we find that
for , since
is an increasing function of R.