I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.


What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ under the action of $SL(2,\mathbb{Z}) $ look like ? 


1) Obviously if the vector space spanned by the coordinates of X has dimension $1$, $SL(2,\mathbb{Z}) \cdot X$ is a lattice. 


2) The $SL(2,\mathbb{Z}) $-action is known to be ergodic on $\mathbb{R}^2$, which implies that there is a set of full measure of which every element has a dense orbit. Is this set exactly the remaining cases ? Namely when the coordinates of $X$ are $\mathbb{Q}$-linearly independent ? This reduces to the question :


If $\theta \notin \mathbb{Q}$, is $SL(2,\mathbb{Z}) \cdot (1, \theta) $ dense in $\mathbb{R}^2$ ?


3) Is the question solved in the case of the $SL(n,\mathbb{Z}) $ action on $\mathbb{R}^n$ ?


Any good reference is welcome !



Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
There are very precise results on the repartition of orbits of $SL(2,\mathbb{Z})$ of irrational points in $\mathbb{R}^2$, such as [this one](http://webusers.imj-prg.fr/~antonin.guilloux/articles/diophuni.pdf). 


In particular these orbits are dense, and this can be seen easily like this.


If $\theta$ is irrational, there are coprime $(a,b)$ such that $a+b\theta$ is less than $\epsilon$ for any $\epsilon>0$. Then there are $(c,d)$ such that $ad-bc=1$, an the various $c+d\theta$ form an $\epsilon$-net in $\mathbb{R}$, as they are $c\_0+d\_0\theta + k(a+b\theta)$ for $k\in\mathbb{Z}$.


So the closure of the $SL(2,\mathbb{Z})$)-orbit of $(1,\theta)$ contains at least the vertical axis $0\times\mathbb{R}$.


But this closure is invariant under $SL(2,\mathbb{Z})$, hence contains all images of the vertical line, that is all rational lines, hence the closure is the whole plane, and the orbit is dense.


ADDED: As you asked also about the situation for higher $n$, I can direct you to [this](http://arxiv.org/abs/1507.06826) very recent arXiv preprint.


