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

Abstract

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 yields a short proof of J.S.Dani's theorem: the $\mathsf{GL}(n,\mathbb{Z})$-orbit of $x\in {\mathbb{R}}^n$ is dense iff $x_i/x_j\in \mathbb{R}\setminus \mathbb{Q}$ 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 $F$.