Classifying $mathsf{GL}(n,mathbb Z)$-orbits of points and rational subspaces

We first show that the subgroup of the abelian real group $mathbb{R}$ generated by the coordinates of a point in $x = (x_1,dots,x_n)inmathbb{R}^n$ completely classifies the $mathsf{GL}(n,mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Danis theorem: the $mathsf{GL}(n,mathbb Z)$-orbit of $xinmathbb{R}^n$ is dense iff $x_i/x_jin mathbb{R} setminus mathbb Q$ for some $i,j=1,dots,n$. We then classify $mathsf{GL}(n,mathbb Z)$-orbits of rational affine subspaces $F$ of $mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the $mathsf{GL}(n,mathbb Z)$-orbit of $F$.