{"title":"A note on exceptional sets in Erdös–Rényi limit theorem","authors":"Chuntai Liu","doi":"10.1007/s43034-023-00294-w","DOIUrl":null,"url":null,"abstract":"<div><p>For <span>\\(x\\in [0,1]\\)</span>, the run-length function <span>\\(r_n(x)\\)</span> is defined as the length of longest run of 1’s among the first <i>n</i> dyadic digits in the dyadic expansion of <i>x</i>. We study the Hausdorff dimension of the exceptional set in Erdös–Rényi limit theorem. Let <span>\\(\\varphi :{{\\mathbb {N}}}\\rightarrow (0,\\infty )\\)</span> be a monotonically increasing function with <span>\\(\\lim _{n\\rightarrow \\infty }\\varphi (n)=\\infty \\)</span> and <span>\\(0\\le \\alpha \\le \\beta \\le \\infty \\)</span>, define </p><div><div><span>$$\\begin{aligned} E_{\\alpha ,\\beta }^\\varphi =\\Big \\{x\\in [0,1]:\\, \\liminf _{n\\rightarrow \\infty } \\dfrac{r_n(x)}{\\varphi (n)}=\\alpha , \\limsup _{n\\rightarrow \\infty } \\frac{r_n(x)}{\\varphi (n)}=\\beta \\Big \\}. \\end{aligned}$$</span></div></div><p>We prove that <span>\\(E_{\\alpha ,\\beta }^\\varphi \\)</span> has Hausdorff dimension one if <span>\\(\\lim _{n,p\\rightarrow \\infty }\\frac{\\varphi (n+p)-\\varphi (n)}{p}=0\\)</span> and that <span>\\(E_{0,\\infty }^\\varphi \\)</span> is residual in [0,1] when <span>\\(\\liminf _{n\\rightarrow {\\infty }}\\frac{\\varphi (n)}{n}=0\\)</span>.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00294-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For \(x\in [0,1]\), the run-length function \(r_n(x)\) is defined as the length of longest run of 1’s among the first n dyadic digits in the dyadic expansion of x. We study the Hausdorff dimension of the exceptional set in Erdös–Rényi limit theorem. Let \(\varphi :{{\mathbb {N}}}\rightarrow (0,\infty )\) be a monotonically increasing function with \(\lim _{n\rightarrow \infty }\varphi (n)=\infty \) and \(0\le \alpha \le \beta \le \infty \), define
We prove that \(E_{\alpha ,\beta }^\varphi \) has Hausdorff dimension one if \(\lim _{n,p\rightarrow \infty }\frac{\varphi (n+p)-\varphi (n)}{p}=0\) and that \(E_{0,\infty }^\varphi \) is residual in [0,1] when \(\liminf _{n\rightarrow {\infty }}\frac{\varphi (n)}{n}=0\).
期刊介绍:
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
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