{"paper":{"title":"Multiplicative approximation by the Weil height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeffrey D. Vaaler, Robert Grizzard","submitted_at":"2017-10-23T17:29:57Z","abstract_excerpt":"Let $K/\\mathbb{Q}$ be an algebraic extension of fields, and let $\\alpha \\not= 0$ be contained in an algebraic closure of $K$. If $\\alpha$ can be approximated by roots of numbers in $K^{\\times}$ with respect to the Weil height, we prove that some nonzero integer power of $\\alpha$ must belong to $K^{\\times}$. More generally, let $K_1, K_2, \\dots , K_N$, be algebraic extensions of $\\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $\\alpha \\not= 0$ can be approximated by a product of roots of numbers from each $K_n$ w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08399","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":""},"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"}