I am new to lattices and advanced concepts in number theory. I'm currently working on understanding sufficient criteria for determining if a given basis set $S = \{\xi_{1}, \ldots, \xi_{n}\}$ where each vector $\xi_{i} \in \mathbb{R}^{n}$, generates an integer lattice $L \subset \mathbb{R}^{n}$. Specifically, this lattice $L$ is formed by taking integer linear combinations of the vectors in the basis $S$: $L = \lbrace z_{i}\xi_{i}: z_{i} \in Z \rbrace$.
A sufficient criterion for $L$ to be an integer lattice should guarantee that $L$ is a discrete additive subgroup of $\mathbb{R}^n$. This is especially important in cases where the basis vectors $\xi_{i}\in \mathbb{R}^{n}$ may include irrational components.
Here is my attempt to identify a problematic scenario. Suppose there is a component $j\in[1,n]$ such that for two distinct basis vectors $\xi_{k}\in\mathbb{R}^{n}$ and $\xi_{l}\in\mathbb{R}^{n}$ where $k\neq l$, the component $\xi_{k,j}$ is a rational number, while $\xi_{l,j}$ is an irrational number. I am inclined to say that this condition implies that the set of $j-$th coordinates of all the lattice points, denoted by: $$L_{j} = \left\lbrace \sum_{i: i\neq k, i\neq l} z_i \xi_{i,j} + z_{k}\xi_{k,j} + z_{l}\xi_{l,j}~:~ z_{1}\in Z, z_{2}\in Z, \ldots, z_{n}\in Z \right\rbrace,$$ forms a dense set in $\mathbb{R}$ due to the implications of the Kronecker's Approximation Theorem (an intuitive proof of which can be found in this StackOverflow post.) Consequently, $L_{j}$ is not a discrete additive subgroup of $R_n$. Is this conclusion accurate?
What concerns me is that we've only demonstrated that the projection of the lattice $L$ along the $j-th$ component forms a dense set, which can also occur in valid lattices. So, are there additional conditions that $S$ must satisfy to be considered a valid lattice basis? Any helpful references would be much appreciated.
