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Let $\alpha$ be a number such that $\alpha/\pi$ is not rational. Prove that

(1) $$\lim_{N\to\infty}\sum_{n=1}^{N} e^{ik(x+n\alpha)}=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{ikt}\,dt,$$ (2) for any continuous function $f:R\to C$ with period $2\pi$, $$\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^{N} f(x+n\alpha)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)\,dt.$$

How can we prove this?

There is not $x$ on the right hand side...

XLDD
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  • This seems very much related to the equidistribution theorem: http://en.wikipedia.org/wiki/Equidistribution_theorem your choice of $x$ should not make a difference. – Ben Grossmann Jul 21 '13 at 01:51
  • In (1) I'm not sure how $,x,,,\alpha,$ on the left hand "disappear" on the right side... – DonAntonio Jul 21 '13 at 09:31

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