{"paper":{"title":"Classifying $\\mathsf{GL}(n,\\mathbb Z)$-orbits of points and rational subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici, Leonardo Manuel Cabrer","submitted_at":"2015-07-24T12:52:26Z","abstract_excerpt":"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 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06826","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}