{"paper":{"title":"Diophantine analysis of the expansions of a fixed point under continuum many bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Baowei Wang, Fan Lv, Jun Wu","submitted_at":"2021-02-28T16:14:46Z","abstract_excerpt":"In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_{\\beta}$ be the beta-transformation with base $\\beta>1$, $\\{x_{n}\\}_{n\\geq 1}$ be a sequence of real numbers in $[0,1]$ and $\\varphi\\colon \\mathbb{N}\\rightarrow (0,1]$ be a positive function. With a detailed analysis on the distribution of {\\em full cylinders} in the base space $\\{\\beta>1\\}$, it is shown that for any given $x\\in(0,1]$, for almost all or almost no bases $\\beta>1$, the orbit of $x$ under $T_{\\beta}$ can $\\varphi$-well approximate "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.00546","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2103.00546/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"}