{"paper":{"title":"Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Baowei Wang, Bing Li, Jun Wu, Tomas Persson","submitted_at":"2013-01-16T06:16:55Z","abstract_excerpt":"We consider the distribution of the orbits of the number 1 under the $\\beta$-transformations $T_\\beta$ as $\\beta$ varies. Mainly, the size of the set of $\\beta>1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. That is, the dimension of the following set $$ E\\big({\\ell_n}_{n\\ge 1}, x_0\\big)=\\Big{\\beta>1: |T^n_{\\beta}1-x_0|<\\beta^{-\\ell_n}, {for infinitely many} n\\in \\N\\Big} $$ is determined, where $x_0$ is a given point in $[0,1]$ and ${\\ell_n}_{n\\ge 1}$ is a sequence of integers tending to infinity as $n\\to \\infty$. For the proof of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3595","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"}