- #1
phreak
- 134
- 1
Homework Statement
Given [tex]k\in \mathbb{Z} \setminus \{ 0 \}[/tex], prove that [tex]\lim_{n\to \infty} \frac{1}{N} \sum_{n=1}^N e^{2\pi i k n \alpha}=0[/tex], for all [tex]\alpha \in \mathbb{R} \setminus \mathbb{Q}[/tex].
Homework Equations
The Attempt at a Solution
Well, I had an idea, but I'm not sure how well it works. Even if it did, I'm not sure how to make it rigorous. Basically, my idea was to say that you work your way around the unit circle, and that whenever we come arbitrarily close to hitting the same point again, the sum (between the two points) gets arbitrarily close to zero. Am I on the right track?
Edit: Alternately, could we use some convergence theorem using counting measure? Also, the limit is supposed to be 0, not 1. I changed it in the code, but it isn't working.