Is $GL_2(\mathbb Z)\cdot X$ a dense subset of $\mathbb R^2$?
It looks like the answer is positive. You can get a dense orbit from the subgroup $SL_2(\mathbb{Z})$ acting on any vector $v \in \mathbb{R}^2$ that isn't a scalar multiple of a vector in $\mathbb{Z} \times \mathbb{Z}$. See an elementary proof by S.G. Dani and Arnaldo Nogueira here. I'm not sure if there are proofs from scratch that are more brief.