Consider a rotation of the form $x\mapsto e^{i\theta}x$, for $x$ on the unit circle. By iterating this rotation, one can approximate any other rotation $x\mapsto e^{i\phi}x$ arbitrarily well, as long as $\theta/2\pi$ is irrational. It is interesting to ask how many times this rotation must be iterated to attain a given accuracy. This is nicely explained in, e.g., Appendix A of this paper. Namely, let $\theta/2\pi$ be irrational with irrationality measure $\mu$. Then for any $\epsilon > 0$, for any $\phi$ and any $\delta > 0$, there is some $m$ with $|m\theta - \phi|\leq \delta$ and with $$ m = O\left(\frac{1}{\delta^{\mu + \epsilon}}\right). $$ The absolute value signs denote distance on the circle.

I would like to know the generalization of this result to the $n$-torus. Namely, consider a rotation $(e^{i\phi_1}, \ldots, e^{i\phi_n})$ where $\phi_1/2\pi, \ldots, \phi_n/2\pi$ are irrational with irrationality measures $\mu_1, \ldots, \mu_n$. Suppose further that $\phi_1/2\pi, \ldots, \phi_n/2\pi$ are linearly independent over $\mathbb{Q}$, so that powers of this rotation are dense in the set of all rotations of the $n$-torus. For a given $\delta$ and a given rotation $(e^{i\theta_1}, \ldots, e^{i\theta_n})$, we would like to find some $m$ such that $$ |m\phi_1 - \theta_1|, \ldots, |m\phi_n - \theta_n|\leq \delta. $$ My guess is that we have a similar bound: for any $\epsilon > 0$, $$ m = O\left(\frac{1}{\delta^{(1 + \mu_1)\cdots (1 + \mu_n) - 1 + \epsilon}}\right). $$ Is this guess correct?

Consider a rotation of the form $x\mapsto e^{i\theta}x$, for $x$ on the unit circle. By iterating this rotation, one can approximate any other rotation $x\mapsto e^{i\phi}x$ arbitrarily well, as long as $\theta/2\pi$ is irrational. It is interesting to ask how many times this rotation must be iterated to attain a given accuracy. This is nicely explained in, e.g., Appendix A of this paper. Namely, let $\theta/2\pi$ be irrational with irrationality measure $\mu$. Then for any $\epsilon > 0$, for any $\phi$ and any $\delta > 0$, there is some $m$ with $|m\theta - \phi|\leq \delta$ and with $$ m = O\left(\frac{1}{\delta^{\mu + \epsilon}}\right). $$ The absolute value signs denote distance on the circle.

I would like to know the generalization of this result to the $n$-torus. Namely, consider a rotation $(e^{i\phi_1}, \ldots, e^{i\phi_n})$ where $\phi_1/2\pi, \ldots, \phi_n/2\pi$ are irrational with irrationality measures $\mu_1, \ldots, \mu_n$. Suppose further that $\phi_1/2\pi, \ldots, \phi_n/2\pi$ are linearly independent over $\mathbb{Q}$, so that powers of this rotation are dense in the set of all rotations of the $n$-torus. For a given $\delta$ and a given rotation $(e^{i\theta_1}, \ldots, e^{i\theta_n})$, we would like to find some $m$ such that $$ |m\phi_1 - \theta_1|, \ldots, |m\phi_n - \theta_n|\leq \delta. $$ My guess is that we have a similar bound: for any $\epsilon > 0$, $$ m = O\left(\frac{1}{\delta^{(1 + \mu_1)\cdots (1 + \mu_n) - 1 + \epsilon}}\right). $$ Is this guess correct?

EDIT: this guess is almost certainly too naive. This is because I haven't said anything about the "extent to which" $\phi_1/2\pi, \ldots, \phi_n/2\pi$ are linearly independent over $\mathbb{Q}$ (or $\mathbb{Z}$). Namely, if I choose $\phi_1/2\pi, \ldots, \phi_n/2\pi$ that are arbitrarily "close" to being linearly dependent over $\mathbb{Q}$, then there should exist points on the $n$-torus that take arbitrarily long times to approximate to a given accuracy. This is probably what Anthony Quas was getting at in the comments.

Instead, let me ask a much simpler "averaged" version of this question. Suppose I choose all of the rotation angles $\phi_1, \ldots, \phi_n$ and target angles $\theta_1, \ldots, \theta_n$ uniformly at random from $[0, 2\pi)$. Then my intuition would say that "on average," I need to apply the rotation $O(1/\delta^n)$ (or more precisely, $\Theta(1/\delta^n)$) times to approximate the target to accuracy $\delta$. This is simply because I would expect the images of the rotation to be uniformly distributed, and I need to apply the rotation as many times as there are points in a grid of points on the $n$-torus whose spacing along each circle is of order $\delta$. Of course, this intuition breaks down if some of the angles $\phi_1, \ldots, \phi_n$ happen to be very small, but I am interested in the asymptotics as $\delta\to 0$.

Is this expectation correct? And as someone who knows nothing about ergodic theory, how should I formalize this statement?