{"paper":{"title":"Topological and Diophantine properties of lattice subset projections","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.NT","authors_text":"Wayne M Lawton","submitted_at":"2026-05-31T06:10:54Z","abstract_excerpt":"Fix $1 \\leq n < m, k = m-n.$ The Grassmannian $Gr(n,m)$ is a compact $kn$-dimensional manifold with a unique rotation invariant probability measure $\\sigma_n.$ For $W \\in Gr(n,m)$, $P_W : \\mathbb R^m \\mapsto W$ is orthogonal projection. A lattice subset $L \\subset \\mathbb Z^m \\subset \\mathbb R^m$ is called $k$-dense if it intersects $C(O) := \\bigcup_{V \\in O} V\\backslash \\{0\\}$ for every nonempty open $O \\subset Gr(k,m)$. We use Baire's category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim} := \\{W \\in Gr(n,m) : 0 \\mbox{ is a limit point of } P_W(L) \\}$ is a $G_\\delta$ set. We use K"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01040","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.01040/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}