{"paper":{"title":"Metric results for dyadic approximation on the middle-third Cantor set","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Bing Li, Bo Wang, Xin-Rong Dai, Yu-Feng Wu","submitted_at":"2026-06-24T02:09:07Z","abstract_excerpt":"Let $C$ be the middle-third Cantor set and $\\mu$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $\\mu(W_2(\\tau))=0$ if $\\tau>1$ and $\\mu(W_2(\\tau))=1$ if $0<\\tau\\leq 1$, where $W_2(\\tau)=\\left\\{x\\in[0,1]: \\|2^nx\\|<n^{-\\tau}\\ {\\rm for\\ infinitely\\ many }\\ n\\in\\mathbb{N} \\right\\}$. We prove that the conjecture holds for $\\tau>\\frac{1}{\\gamma}-\\frac{1-\\gamma}{3-\\gamma}\\,(\\approx 1.429)$ and $0<\\tau<\\frac{\\gamma}{12}\\,(\\approx 0.052)$, where $\\gamma=\\frac{\\log2}{\\log3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($\\tau>\\frac{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25305","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/2606.25305/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"}