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

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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 $x\in\mathbb{R}^n$ completely classifies the GL$(n,\mathbb{Z})$-orbit of $x$. This yields a short proof of J.S. Dani's theorem: the 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 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 GL$(n,\mathbb{Z})$-orbit of $F$.
    Mathematics Subject Classification: Primary: 37C85; Secondary: 11B57, 22F05, 37A45.


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