{"paper":{"title":"Exponents for three-dimensional simultaneous Diophantine approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nikolay Moshchevitin","submitted_at":"2010-09-06T08:23:25Z","abstract_excerpt":"Let $\\Theta = (\\theta_1,\\theta_2,\\theta_3)\\in \\mathbb{R}^3$. Suppose that $1,\\theta_1,\\theta_2,\\theta_3$ are linearly independent over $\\mathbb{Z}$. For Diophantine exponents $$ \\alpha(\\Theta) = \\sup \\{\\gamma >0:\\,\\,\\, \\limsup_{t\\to +\\infty} t^\\gamma \\psi_\\Theta (t) <+\\infty \\} ,$$ $$\\beta(\\Theta) = \\sup \\{\\gamma >0:\\,\\,\\, \\liminf_{t\\to +\\infty} t^\\gamma \\psi_\\Theta (t) <+\\infty\\} $$ we prove $$ \\beta (\\Theta) \\ge {1/2} ({\\alpha (\\Theta)}/{1-\\alpha(\\Theta)} +\\sqrt{{\\alpha(\\Theta)}/{1-\\alpha(\\Theta)})^2 +{4\\alpha(\\Theta)}/{1-\\alpha(\\Theta)}}) \\alpha (\\Theta) $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0987","kind":"arxiv","version":2},"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"}