{"paper":{"title":"A dichotomy law for the Diophantine properties in $\\beta$-dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Bao-Wei Wang, Michael Coons, Mumtaz Hussain","submitted_at":"2016-04-04T06:05:45Z","abstract_excerpt":"Let $\\beta>1$ be a real number and define the $\\beta$-transformation on $[0,1]$ by $T_\\beta:x\\mapsto \\beta x\\bmod 1$. Further, define $$W_y(T_{\\beta},\\Psi):=\\{x\\in [0, 1]:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\}$$ and $$W(T_{\\beta},\\Psi):=\\{(x, y)\\in [0, 1]^2:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\},$$ where $\\Psi:\\mathbb{N}\\to\\mathbb{R}_{>0}$ is a positive function such that $\\Psi(n)\\to 0$ as $n\\to \\infty$. In this paper, we show that each of the above sets obeys a Jarn\\'ik-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depend"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00747","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"}