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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?

Friedrich Philipp
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    This is equivalent to whether (ka,kb) mod 1 is dense in $\mathbb{R}\backslash\mathbb{Z}$ and the answer is whenever $1,a,b$ are independent over $\mathbb{Q}$. Let me find a reference – Yanko Jun 17 '17 at 22:12
  • @user426577 Hi, thanks. You mean $\mathbb R^2/\mathbb Z^2$, right? And what do you mean by "independent over $\mathbb Q$? – Friedrich Philipp Jun 17 '17 at 22:14
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    yes ofc $\mathbb{R}^2\setminus \mathbb{Z}^2$. And I mean that $a$ is irrational and $b$ is irrational and there is no $q\in\mathbb{Q}$ so that $a=bq$ (I think that in this case they're not just dense they're equidistributed) – Yanko Jun 17 '17 at 22:15
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    https://mathoverflow.net/questions/18174/independence-over-q-and-kroneckers-result – Yanko Jun 17 '17 at 22:17
  • Thank you very much. So, my condition is also necessary for non-density. great! – Friedrich Philipp Jun 17 '17 at 22:26

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