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 !