Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding of $\mathbb{Z}$ in the $n$ torus, i.e. $\mathbb{R}^n / \mathbb{Z}^n$, actually even more: it becomes equidistributed in the following sense $\frac{1}{N}\sum\limits_{k \leq N} f(k\alpha) \rightarrow \int\limits_{\mathbb{R^N} / \mathbb{Z}^n} f( x) \mathrm{d} x,$ where $\mathrm{d} x$ denotes the Haarmeasure. A similar result holds for the $\infty$ torus as well.
What can be said about the rate of convergence here? For $n=1$, can we distinguish wether $\alpha_1$ is transcendental or algebraic from the error term?
Motivation: I try to undestand effective version of universality theorems of L functions, see e.g. http://en.wikipedia.org/wiki/Zeta_function_universality.