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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.

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  • $\begingroup$ You had best assume your irrationals linearly independent over the rationals, lest your set $S$ just consist of multiples of one ordered pair. $\endgroup$ – Gerry Myerson Sep 11 '18 at 3:29
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    $\begingroup$ You could divide both numbers by $a_2$ without changing the problem substantially. Then you see the issue is related to irrational rotations of the circle and continued fractions. $\endgroup$ – Anthony Quas Sep 11 '18 at 4:24
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    $\begingroup$ Since epsilon is fixed, the poster also has to divide epsilon by a_2. Even so, the division is a useful reduction and highlights the need to consider rational independence/ irrational rotations of the circle. Indeed, rational approximations to a_1/a_2 are what are being considered here, and the tolerance epsilon/a_2 shows how far one needs to go. Farey fractions are related. Gerhard "Giving A Finer Problem Approximation" Paseman, 2018.09.11. $\endgroup$ – Gerhard Paseman Sep 11 '18 at 14:54

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