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.