{"paper":{"title":"Sharp Bounds for Rational Points Near Space Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Andreas Seeger, Mingfeng Chen, Niclas Technau, Rajula Srivastava","submitted_at":"2026-08-10T01:49:37Z","abstract_excerpt":"Let $Q\\geq 1$ be large, and $\\delta \\in(0,1)$ be small. Denote by $\\mathcal C \\subset \\mathbb R^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\\mathbf{a}/q$ of height $q\\in[Q,2Q]$ are $\\delta/q$-near $\\mathcal C$? This manuscript provides an essentially optimal answer. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a hitherto hidden `major arc' type obstruction. We also establish matching upper bounds (up to endpoints). Our argument co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09009","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/2608.09009/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"}