Corollary

Let $(a_i)_{1 \leq i \leq p}$ be a free system of $p$ points of $\mathbf{R}^n$ and let

be a linear combination of the $a_i$, with real coefficients. Then the subgroup $G$ of $\mathbf{R}^n$ generated by the $p + 1$ points $a_1, a_2, \ldots, a_p$ and $b$ is discrete if and only if the numbers $t_i$ are rational.

(*) We recall (Chapter IV, § 8, no. 2) that, for each real number $x, [x]$ is the integral part of $x$, i.e., the largest rational integer $\leq x$.

Proposition 1 shows that the condition is necessary. Also it is sufficient, for if it is satisfied we can write $t_i = m_i/d$, where $d$ and the $m_i$ are integers ($1 \leq i \leq p$); $b$ is therefore a linear combination, with integer coefficients, of the $p$ points $(1/d)a_i$; hence $G$ is a subgroup of the discrete group generated by these $p$ points, and so is itself discrete.

The result of Proposition 1 can be expressed as follows: if $q$ points $x_i$ ($1 \leq i \leq q$) of a discrete subgroup $G$ of $\mathbf{R}^n$ form a system which is dependent with respect to $\mathbf{R}$, then they form a system which is dependent with respect to $\mathbf{Q}$. It follows immediately that the rational rank of a discrete subgroup of $\mathbf{R}^n$ is equal to its rank.

The Corollary to Proposition 1, applied to the case where the $a_i$ are the $n$ vectors $e_i$ of the canonical basis of $\mathbf{R}^n$, gives us the following proposition: