Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is, $$ S=\{(m_1,m_2)\in\mathbb{Z}\times\mathbb{Z}:|m_1a_1+m_2a_2|<\epsilon\}. $$ Now, suppose we somehow sort the elements of this set, according to $|m_1|+|m_2|$. My questions are:

1) What is a typical distance between any two consecutive elements of $S$? That is, say we find $(m_1,m_2)$ pair; for the 'next' pair $(m_1',m_2')$; how would entries compare?

2) Is there a systematic way to generate all members of $S$?

3) In general, for such a problem, what kind of a toolset would be useful?

Of course the problem depends heavily on $a_1,a_2$, but say they are of the following values $a_1=\sqrt{2}$, $a_2=\sqrt{3}$; to have an understanding.