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We first show that the subgroup of the abelian real group $\mathbb{R}$generated by the coordinates of a point in $x = (x_1,\dots,x_n)\in\mathbb{R}^n$completely classifies the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x$. This yieldsa short proof of J.S.Dani's theorem: the $\mathsf{GL}(n,\mathbb Z)$-orbit of$x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R} \setminus \mathbb Q$ forsome $i,j=1,\dots,n$. We then classify $\mathsf{GL}(n,\mathbb Z)$-orbits ofrational 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 $F$.

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