{"paper":{"title":"Variations on Dirichlet's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Simmons, Lior Fishman","submitted_at":"2015-03-07T19:30:20Z","abstract_excerpt":"We give a necessary and sufficient condition for the following property of an integer $d\\in\\mathbb N$ and a pair $(a,A)\\in\\mathbb R^2$: There exist $\\kappa > 0$ and $Q_0\\in\\mathbb N$ such that for all $\\mathbf x\\in \\mathbb R^d$ and $Q\\geq Q_0$, there exists $\\mathbf p/q\\in\\mathbb Q^d$ such that $1\\leq q\\leq Q$ and $\\|\\mathbf x - \\mathbf p/q\\| \\leq \\kappa q^{-a} Q^{-A}$. This generalizes Dirichlet's theorem, which states that this property holds (with $\\kappa = Q_0 = 1$) when $a = 1$ and $A = 1/d$. We also analyze the set of exceptions in those cases where the statement does not hold, showing t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02203","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"}