Is $\mathbb{Z}[\sqrt{2}]$ a lattice?

Question

Is $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ a discrete subgroup of $\mathbb{R}$?

How to prove that?

Answer 1

No. It fails to be discrete. In fact $\mathbb Z[\sqrt 2]$ is dense in $\mathbb R$.

It suffices to observe that the sequence $(\sqrt 2-1)^n\in \mathbb Z[\sqrt 2] $ converges to $0$.