I'm interested in discrete subspaces and (additive) subgroups of $\mathbb{R}^n$.
I was interested in what can be said about the distance between elements in a discrete subspace/subgroup, in the sense of some lower bound for the distance.
The two canonical examples I have in mind are $\{1/m : m\in \mathbb{N}\}$ and $\mathbb{Z}^m$ (which is the only example for a subgroup up to isomorphism), for the first no such bound exists, and for the latter it is simply $1$.
Is it typical for subspaces vs subgroups?
I guess that what is needed is a classification of the isomorphisms of $R^n$ with itself, or maybe start with $n=1$ for simplicity as I did first, to understand the subgroup case (maybe find very contracting maps). It did not get me far though.