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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.

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up vote 5 down vote accepted

The rate of convergence is related to the discrepancy. Koksma's inequality says that if $f$ is of bounded variation $V(f)$, and $x_1,\dots,x_n$ are points in $[0,1)$ with discrepancy $D_n$, then $$\left|{1\over n}\sum_1^nf(x_i)-\int_0^1f(t)\,dt\right|\le V(f)D_n$$

The discrepancy, in turn, is a measure of departure from unifomity of distribution. It is defined by $$D_n=\sup_{0\lt a\le1}\left|{1\over n}\#\lbrace i\le n:x_i\lt a\rbrace-a\right|$$

The discrepancy of the sequence of fractional parts of $\alpha,2\alpha,3\alpha,\dots,n\alpha$ is closely related to the partial quotients in the continued fraction expansion of $\alpha$. For algebraic irrationals it is known that $nD_n=O(n^{\epsilon})$ for every $\epsilon\gt0$. But we don't know enough about continued fractions to distinguish transcendentals from algebraics (in general), so, in general, I don't think you'll be able to distinguuish the two classes by convergence rates.

All of this generalizes to higher dimensions and has been studied in some detail. I've only presented the tip of the iceberg. For more, see Kuipers and Niederreiter, Uniform Distribution of Sequences.

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