{"paper":{"title":"On a lower bound of Hausdorff dimension of weighted singular vectors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Jaemin Park, Taehyeong Kim","submitted_at":"2022-07-16T13:35:30Z","abstract_excerpt":"Let $w=(w_1,\\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\\sum_{i}w_i =1$ and $w_1\\geq \\cdots \\geq w_d$. A $d$-dimensional vector $x=(x_1,\\dots,x_d)\\in\\mathbb{R}^d$ is said to be $w$-singular if for every $\\epsilon>0$ there exists $T_0>1$ such that for all $T>T_0$ the system of inequalities \\[ \\max_{1\\leq i\\leq d}|qx_i - p_i|^{\\frac{1}{w_i}} < \\frac{\\epsilon}{T} \\quad\\text{and}\\quad 0<q<T \\] have an integer solution $(\\mathbf{p},q)=(p_1,\\dots,p_d,q)\\in \\mathbb{Z}^d \\times \\mathbb{Z}$. We prove that the Hausdorff dimension of the set of $w$-singular vectors in $\\mathbb{R}^d$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.07944","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.07944/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}