{"paper":{"title":"Density of Rational Points Near Flat/Rough Hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Niclas Technau, Rajula Srivastava","submitted_at":"2023-05-01T19:21:45Z","abstract_excerpt":"For $n\\geq 3$, let $\\mathscr{M} \\subseteq\\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\\in \\mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number $\\mathrm{N}_{\\mathscr{M}}(\\delta,Q)$ of rationals $\\mathbf{a}/q$, with denominator $q\\in [Q,2Q)$ and $\\mathbf{a} \\in \\mathbb{Z}^{n-1}$, lying at a distance at most $\\delta/q$ from $\\mathscr{M}$. This manuscript provides essentially sharp estimates for $\\mathrm{N}_{\\mathscr{M}}(\\delta,Q)$ throughout the range $\\delta \\in (Q^{\\varepsilon-1},1/2)$ for $d>1+\\tfrac{1}{2n-3}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.01047","kind":"arxiv","version":3},"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/2305.01047/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"}