{"paper":{"title":"Joint normality of representations of numbers: an ergodic approach","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Vitaly Bergelson, Younghwan Son","submitted_at":"2022-08-18T02:26:48Z","abstract_excerpt":"We introduce an ergodic approach to the study of {\\em joint normality} of representations of numbers. For example, we show that for any integer $b \\geq 2$ almost every number $x \\in [0,1)$ is jointly normal with respect to the $b$-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with {\\em pointwise joint ergodicity}:\n  Let $T_b:[0,1] \\rightarrow [0,1]$ be the times $b$ map defined by $T_b x = bx \\, \\bmod \\, 1 $ and let $T_G:[0,1] \\rightarrow [0,1]$ be the Gauss map defined by $T_G(x) = \\{\\frac{1}{x}\\}$ for $x \\ne 0$ and $T_G (0) =0.$ (Her"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.08596","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/2208.08596/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"}