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Suppose $f$ is a continuous function on $R^1$ with period 1. Prove that $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nf(n\alpha)=\int_0^1f(t)dt$$ for every irrational number $\alpha$. Hint: Do it first for $$f(t)=e^{2πikt}$$

How are these two thing connected? Intuitively the left hand side is acting like a Riemann sum but it isn't a Riemann sum. $n\alpha$ is fairly ambiguous and hard to understand. In the long run I understand that $(n\alpha)$ (the decimal value of $n\alpha$) populates the entire interval $(0,1$ fairly uniformly. I don't understand how that may help. I don't know any assistance

user160110
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  • Are you able to show that it's true for $f(t) = e^{2\pi i k t}$ as the hint suggests? –  Dec 01 '17 at 03:03
  • This might be helpful https://math.stackexchange.com/questions/272545/multiples-of-an-irrational-number-forming-a-dense-subset . – ChargeShivers Dec 01 '17 at 03:04
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    also https://math.stackexchange.com/questions/2033021/proof-of-lim-limitsm-to-infty-frac1m-sum-limitsn-1m-fan-int-limits?rq=1 – Trevor Gunn Dec 01 '17 at 03:06
  • and to a lesser extent https://math.stackexchange.com/questions/517953/how-prove-this-nice-limit-lim-n-to-infty-frac1n-sum-k-1nf-ka?rq=1 – Trevor Gunn Dec 01 '17 at 03:07
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    @TrevorGunn Your second link is a better candidate than the first. The answer to the first has a problem, as noted in the associated comment. –  Dec 01 '17 at 03:08
  • @Bungo is right, if you're looking at the first link, note that you should approximate your continuous function uniformly by trigonometric polynomials instead of trying to take a Fourier series. – Trevor Gunn Dec 01 '17 at 03:13

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