{"paper":{"title":"Badly approximable numbers over imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Robert Hines","submitted_at":"2017-07-22T23:34:31Z","abstract_excerpt":"We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the \"badly approximable\" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\\geq C/|q|^2$ for all $p/q\\in K$), by boundedness of the partial quotients in the continued fraction expansion of $z$. Applying this algorithm to \"tagged\" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in $\\mathbb{C}$ whose points are badly approximable over $K$, with effective constants.\n  By other methods (the Dani correspondence), we prove the e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07231","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"}