{"paper":{"title":"Adelic versions of the Weierstrass approximation theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Giulio Peruginelli, Jean-Luc Chabert","submitted_at":"2015-11-11T11:44:20Z","abstract_excerpt":"Let $\\underline{E}=\\prod_{p\\in\\mathbb{P}}E_p$ be a compact subset of $\\widehat{\\mathbb{Z}}=\\prod_{p\\in\\mathbb{P}}\\mathbb{Z}_p$ and denote by $\\mathcal C(\\underline{E},\\widehat{\\mathbb{Z}})$ the ring of continuous functions from $\\underline{E}$ into $\\widehat{\\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\\rm Int}_{\\mathbb{Q}}(\\underline{E},\\widehat{\\mathbb{Z}}):=\\{f(x)\\in\\mathbb{Q}[x]\\mid \\forall p\\in\\mathbb{P},\\;\\;f(E_p)\\subseteq \\mathbb{Z}_p\\}$ is dense in the direct product $\\prod_{p\\in\\mathbb{P}}\\mathcal C(E_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03465","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"}