{"paper":{"title":"Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.DS"],"primary_cat":"math.NT","authors_text":"Damaris Schindler, Niclas Technau, Rajula Srivastava","submitted_at":"2023-10-05T19:51:22Z","abstract_excerpt":"Let $\\mathcal{M}\\subset \\mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\\mathcal{M}$ is nowhere completely flat. Let $N_{\\mathcal{M}}(\\delta,Q)$ denote the number of rational vectors $\\mathbf{a}/q$ within a distance of $\\delta/q$ from $\\mathcal{M}$ so that $q \\in [Q,2Q)$. We develop a novel method to analyse $N_{\\mathcal{M}}(\\delta,Q)$. The salient feature of our technique is the combination of powerful quantitative non-divergence estimates, in a form due to Bernik, Kleinbock, and Margulis, with Fourier analytic tools. The second ingredient enables us to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.03867","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/2310.03867/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"}