Corollary 2

Let $\theta_1, \theta_2, \ldots, \theta_n$ be $n$ real numbers. In order that, given any $n$ real numbers $x_1, x_2, \ldots, x_n$ and a real number $\varepsilon > 0$, there should exist an integer $q$ and $n$ integers $p_j$ such that

it is necessary and sufficient that there exist no relation of the form $\sum_{j=1}^n r_j \theta_j = h$, where the $r_j$ are $n$ integers not all zero, and $h$ is an integer [which implies, in particular, that the $\theta_j$ and the ratios $\theta_j / \theta_k$ ($j \neq k$) must be irrational].

We may interpret this result as follows: for each integer $q \in \mathbf{Z}$, let $x_q$ denote the point with coordinates $q \theta_j - [q \theta_j]$ ($1 \leq j \leq n$); then Corollary 2 gives a necessary and sufficient condition for the set of points $x_q$ to be dense in the cube which is the product of $n$ intervals $[0, 1]$ in the factors of $\mathbf{R}^n$.