{"paper":{"title":"Note on the points with dense orbit under $\\times 2$ and $\\times 3$ maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Wenya Wang","submitted_at":"2014-10-01T07:31:08Z","abstract_excerpt":"It was conjectured by Furstenberg that for any $x\\in [0,1]\\backslash Q$, $$ \\dim_H \\bar{\\{2^nx ({\\text{mod}}\\ 1): n\\ge 1\\}}+ \\dim_H \\bar{\\{3^nx ({\\text{mod}}\\ 1): n\\ge 1\\}}\\ge 1. $$ When $x$ is a normal number, the above result holds trivially. In this note, we give explicit non-normal numbers for which the above dimensional formula holds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0129","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"}