I am trying to understand the structure of Discrete subgroup of $\mathbb{R}^n$ given in Samuels' Algebraic number theory.
Let $H$ be a discrete subgroup of $\mathbb{R}^n$ (so $H\cap C$ is finite for compact $C\subset \mathbb{R}^n$).
Let $e_1,\cdots, e_r$ be a maximal independent set of vectors in $H$. Consider compact set
$$P=\{ x=\Sigma \alpha_ie_i : 1\le i\le r, 0\le \alpha_i <1\}.$$
Then $P\cap H$ is finite. For any $x\in H$, by maximality of $e_i$'s chose, we get some real $\lambda_i$'s such that
$$x=\lambda_1 e_1 + \cdots + \lambda_r e_r.$$
Consider following sequence of vectors in $H$:
$$x_j=jx-\Sigma_{i=1}^r [j\lambda_i]e_i$$
where $[a]$ denote greatest integer $\leq a$.
Q.1 What geometrically this sequence is concerned about?
Next, $x_j=\sum_{i=1}^r (j\lambda_i-[j\lambda_i])e_i$ and is in $P\cap H$ which is finite so there exists integers $j,k$ such that $x_j=x_k$, so $(j-k)\lambda_i=[j\lambda_i]-[k\lambda_i]$ hence $\lambda_i$'s are rational.
Thus $H$ is a finitely generated $\mathbb{Z}$-module generated by rational combination of $e_i$'s; let $d$ be a common denominator of these coefficients.
Q.2 How could we ensure about this common denominator for elements of $H$?
Q1. The $e_j$ span a lattice. Given $x$, we investigate the position of its multiples $jx$ within their "lattice cell". To this end, we transport it back to the "standard lattice cell" $P$ by subtracting integer multiples of the $e_j$.
Q2. A priori, the common denominator of the coefficients as found in the proof depends on $x$. However, upon closer inspection, the denominator of $\lambda_i$ is $\le|P\cap H|$, and this bound does not depend on $x$. In other words, we find that $\lambda_i\in\frac1{ |P\cap H|!}\Bbb Z$.