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Let $\alpha$ be an irrational number. Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a continuous periodic function with period 1. Show that $\lim_{N \rightarrow \infty} \frac{1}{N} \Sigma_{n=1}^N f(n\alpha) = \int_0^1 f(x)\,dx$

The beginning (but probably not the end) of my confusion with this problem has to do with the irrational inputs. Why would that be necessary? Any help is appreciated!

Martin Sleziak
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user329311
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    Possibly related http://math.stackexchange.com/questions/384765/prove-that-lim-n-rightarrow-infty1-n-sum-n-1n-fnx-int-01ftdt?rq=1 – ForgotALot Apr 19 '16 at 06:14
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    If it's rational, then you only sample at a finite number of points (namely a number of points corresponding to the denominator of the rational number). For instance, if $\alpha = 1/2$, then $n\alpha \mod 1$ looks like $1/2, 0, 1/2, 0, ...$ By being irrational, this is really a statement about equidistribution. – davidlowryduda Apr 19 '16 at 06:18
  • @ForgotALot Yes, you can even say that it is a duplicate of this question. – Jean Marie Apr 19 '16 at 06:24
  • @mixedmath You give in fact the hint to a probabilistic proof of the question... – Jean Marie Apr 19 '16 at 06:26
  • Thank you for the help! Sorry about the redundancy, I did a search but wasn't able to find the question @ForgotALot linked to – user329311 Apr 19 '16 at 07:31
  • @ForgotALot The answer is not complete, it can only be applied to the functions whose Fourier series converges uniformly, this is not true for all continuous functions. – Xiang Yu Apr 19 '16 at 08:09

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You can use the equidistribution theorem to conclude that the sequence $\{n\alpha\},n=1,2,\dots$ is equidistributed on the unit interval $[0,1]$ (note that we need the condition that $\alpha$ is irrational), where $\{n\alpha\}:=n\alpha-\lfloor n\alpha\rfloor$ is the fraction part of $n\alpha$. Then apply the Riemann integral criterion for equidistribution (see here).

Xiang Yu
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