Accumulation points in rings of integers

Question

I am just learning about discrete subgroups of $(\mathbb{R}^n,+)$ and I read that every $\mathbb{Z}$-basis of such a subgroup is even linearly independent over $\mathbb{R}$. Conversely, this implies that the ring of integers of an algebraic number field often has accumulation points. My question is: Is there a method/theory of explicitely finding such accumulation points together with the corresponding sequences? To be more concrete, consider $\mathbb{Z}[\sqrt{2}]$. We know that there are arbitrarily small (i.e. close to zero) real numbers of the form $n+m\sqrt{2}$ with $m,n\in\mathbb{Z}$. How can we find these $m, n\in\mathbb{Z}$ so that $|n+m\sqrt{2}|<\epsilon$ for given $\epsilon>0$?