{"paper":{"title":"Counting algebraic points in expansions of o-minimal structures by a dense set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Pantelis E. Eleftheriou","submitted_at":"2017-08-13T17:17:03Z","abstract_excerpt":"The Pila-Wilkie theorem states that if a set $X\\subseteq \\mathbb R^n$ is definable in an o-minimal structure $\\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\\widetilde{\\mathcal R}=\\langle \\mathcal R, P\\rangle$ of $\\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\\mathcal R$, or it is independent, as follows. If $X$ is definable in $\\widetilde{\\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03936","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":""},"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"}