{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7LZQARL34EOXD6FNAZA2MZMPK2","short_pith_number":"pith:7LZQARL3","schema_version":"1.0","canonical_sha256":"faf300457be11d71f8ad0641a6658f56af30af9b682798d61be78cc4c247b616","source":{"kind":"arxiv","id":"2606.25305","version":1},"attestation_state":"computed","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 $C$. 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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 $C$. 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