I asked a very similar question already here. There I asked whether the set $\{(m+ka,n+kb) : m,n,k\in\mathbb Z\}$ is dense in $\mathbb R^2$ when $a,b\in\mathbb R\setminus\mathbb Q$. The answer was: Not in general. For example the set is not dense if $a=b$ or if there exists $t\in\mathbb R$ such that $t(a,b)\in\mathbb Z^2$. I could generalize this to the following:
The set is not dense when there exist $l_1,l_2\in\mathbb Z$ such that $l_1 a + l_2 b\in\mathbb Z$.
My question is: Does the converse hold?