I have a problem proving this.
If $f$ is a continuous, piecewise smooth function defined on $\mathbb{R}$ and $2\pi$-periodic, with $\frac{\alpha }{\pi }$ irrational, we have to show that
$$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^nf(x+\alpha j)=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)~\mathrm{d}t$$
for all $x$. Does anyone have a lead to help get me started on the demonstration?
The result is true only when $\alpha$ is an irrational multiple of $2\pi$ (there are easy counterexamples otherwise).
When $\alpha$ is an irratoinal multiple, all you need to do is to use the result that $j\alpha \mod 2\pi$ is equidistributed. You can find a proof here (Fractional part of $n\alpha$ is equidistributed)