Proposition 7 (Kronecker)

Let $a_i = (a_{ij})$ $(1 \leq i \leq m,\ 1 \leq j \leq n)$ be $m$ points of $\mathbf{R}^n$ and let $b = (b_j)$ $(1 \leq j \leq n)$ be a point of $\mathbf{R}^n$. In order that, for each $\varepsilon > 0$, there exist $m$ integers $g_i$ $(1 \leq i \leq m)$ and $n$ integers $p_j$ $(1 \leq j \leq n)$ such that

it is necessary and sufficient that, for each finite sequence $(r_j)$ $(1 \leq j \leq n)$ of $n$ integers such that the $m$ numbers $\sum_{j=1}^n a_{ij} r_j$ $(1 \leq i \leq m)$ are all integers, the number $\sum_{j=1}^n b_j r_j$ should also be an integer.