A couple months ago I shared a blog post on Dirichlet’s approximation theorem which we may use to prove that integer angles are dense in the unit circle. Now, I tried to go further and show that is dense in the unit circle. This particular result is interesting because it leads one to deduce that is dense in where is the nth prime.
This week I learned about the Weyl equidistribution theorem from which this particular result follows as an immediate corollary. However, I wondered whether there might be a simpler proof and after searching I came across a short paper on Kronecker’s density theorem which had exactly the result I wanted and a bit more. The purpose of this blog post is to present the proof contained in this paper.
Kronecker’s density theorem:
If , then is dense in the unit circle.
1. is dense in so it’s enough to prove that:
Equivalently we have:
2. Without loss of generality, we may assume that since given that ,
It’s sufficient to prove the case since if we have found such that
for any we can find such that:
To obtain it’s sufficient to take the integer part of the fraction in the inequality.
Now, it follows that we have:
3. For fixed the proof strategy is as follows:
a) starting at we move anti-clockwise around the unit circle in steps of arc length until we cross the positive x-axis.
b) so we reach where and
c) if we’re done. Otherwise, we repeat the procedure with .
d) The remainder of this proof will show that we can find an upper-bound for the number of iterations of required.
5. Before proceeding, it’s useful to note that there exists where at the first iteration we have:
So at the nth iteration we have:
6. If , then we’re done as we have:
Otherwise, let’s define the smallest such that:
If the process continued after the Mth iteration we would have:
Note 1: The arguments in this blog post are taken almost entirely from the original paper with some slight modifications.
Note 2: The fact that the proof is constructive is very nice as Kronecker was a very strong advocate of constructive proofs.