Looking for references on 'non-discrete lattices'

Looking for references on 'non-discrete lattices'

A lattice in $\mathbb{R}^n$ is a discrete subgroup that spans
$\mathbb{R}^n$. Recently I've been running into a similar sort of object
consisting of more than $n$ vectors in $\mathbb{R}^n$ and their
$\mathbb{Z}$-linear combinations.
I think this is best seen via an example.
Consider the boring lattice generated by $(1,0)$ and $(0,1)$. This is the
good ol' standard coordinate lattice we know and love.
Now look at the 'non-discrete lattice' generated by $(1,0), (0,1),
(1,\sqrt 2)$. This is not a discrete subgroup of $\mathbb{R}^2$ because
it's 'horizontally dense,' i.e. if $(x,y)$ is in the 'lattice,' then there
are infinitely many other points in $(x\pm \epsilon, y)$ in the 'lattice.'
This is very different than the non-disrete lattice generated by $(1,0),
(0,1), (\sqrt 2, \sqrt 2)$, which is everywhere dense.
These examples are a bit contrived, but in higher dimensions/more general
fields, I have a hard time determining the resulting characteristics of
the 'lattice.' For example, is it easy to tell how 'dense' the resulting
'lattice' is? Does this come up/are there known applications of this to,
say, number theory?
To be specific:
Is there a name for the concept of what I've been calling a 'non-discrete
lattice?'
Can you point me to reference material on 'non-discrete lattices?'