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!
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).