If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum\_{k=0}^{n-1}f(ak)\rightarrow \int\_0^1 f(x)dx.$$
Can we say something about the speed of convergence ?
Thank you for your help !



Speed of convergence for Weyl's Equidistribution theorem
As a complement to Gerry Myerson answer, you can bound the discrepancy $D\_N$ using the Erdos-Turan inequality
$$
D\_{N} \leq \frac{\log 2}{\pi (H + 1)} + \frac{1}{\pi N} \sum\_{h = 1}^{H} \frac{1}{h}
\bigg | \sum\_{n = 1}^{N} \exp(2\pi i h x\_n) \bigg |
$$
with $H$ arbitrary. The Erdos-Turan inequality is a reasonably easy consequence of Poisson-summation. To get nice constants as above one uses Beurling-Selberg majorants.


See: <http://www.math.northwestern.edu/~mlerma/papers/charla/appl_of_ef.pdf>
and the references there-in (in particular the survey of Valeer in the Bulletin). The result cited above is Theorem 3.2 in the link.


