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?