# Article

 Title: Hausdorff dimension of the maximal run-length in dyadic expansion  (English) Author: Zou, Ruibiao Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 61 Issue: 4 Year: 2011 Pages: 881-888 Summary lang: English . Category: math . Summary: For any $x\in [0,1)$, let $x=[\epsilon _1,\epsilon _2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=\max \{j\geq 1\colon \epsilon _{i+1}=\cdots =\epsilon _{i+j}=1$, $0\leq i\leq n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\lim _{n\to \infty }{r_n(x)}/{\log _2 n}=1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than $\log _2 n$, is quantified by their Hausdorff dimension. Keyword: run-length function Keyword: Hausdorff dimension Keyword: dyadic expansion MSC: 11K55 MSC: 28A78 MSC: 28A80 . Date available: 2011-12-16T15:33:47Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/141793 . Reference: [1] Arratia, R., Gordon, L., Waterman, M. S.: The Erdös-Rényi law in distribution, for coin tossing and sequence matching.Ann. Stat. 18 (1990), 539-570. Zbl 0712.92016, MR 1056326 Reference: [2] Benjamini, I., Häggström, O., Peres, Y., Steif, J. E.: Which properties of a random sequence are dynamically sensitive? Ann.Probab. 31 (2003), 1-34. MR 1959784 Reference: [3] Billingsley, P.: Ergodic Theory and Information,.Wiley Series in Probability and Mathematical Statistics. New York: John Wiley and Sons (1965). Zbl 0141.16702, MR 0192027 Reference: [4] Khoshnevisan, D., Levin, D. A., Méndez-Hernández, P. J.: On dynamical Gaussian random walks.Ann. Probab. 33 (2005), 1452-1478. Zbl 1090.60066, MR 2150195 Reference: [5] Khoshnevisan, D., Levin, D. A., Méndez-Hernández, P. J.: Exceptional times and invariance for dynamical random walks.Probab. Theory Relat. Fields. 134 (2006), 383-416. Zbl 1130.60079, MR 2226886 Reference: [6] Khoshnevisan, D., Levin, D. A.: On dynamical bit sequences.arXiv:0706.1520v2. Reference: [7] Ma, J.-H., Wen, S.-Y., Wen, Z.-Y.: Egoroff's theorem and maximal run length.Monatsh. Math. 151 (2007), 287-292. Zbl 1170.28001, MR 2329089 Reference: [8] Révész, P.: Random Walk in Random and Non-Random Enviroments.Singapore. World Scientific (1990). MR 1082348 .

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