Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 881-888, 2011

Hausdorff dimension of the maximal
run-length in dyadic expansion

Ruibiao Zou

Ruibiao Zou, Hunan Agriculture University, Changsha, Hunan, China P. R., 410128, e-mail:

Abstract: For any $x\in[0,1)$, let $x=[\epsilon_1,\epsilon_2,\cdots,]$ be its dyadic expansion. Call $r_n(x):=\max\{j\geq1 \epsilon_{i+1}=\cdots=\epsilon_{i+j}=1$, $0\leq i\leq n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdos and A. Renyi 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.

Keywords: run-length function, Hausdorff dimension, dyadic expansion

Classification (MSC 2010): 11K55, 28A78, 28A80

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