Hausdorff dimension of the maximal run-length in dyadic expansion

@article{Zou2011,
abstract = {For any $x\in [0,1)$, let $x=[\epsilon _1,\epsilon _2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=\max \lbrace j\ge 1\colon \epsilon _\{i+1\}=\cdots =\epsilon _\{i+j\}=1$, $0\le i\le n-j\rbrace $ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\lim _\{n\rightarrow \infty \}\{r_n(x)\}/\{\log _2 n\}=1$ 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.},
author = {Zou, Ruibiao},
journal = {Czechoslovak Mathematical Journal},
keywords = {run-length function; Hausdorff dimension; dyadic expansion; run-length function; Hausdorff dimension; dyadic expansion},
language = {eng},
number = {4},
pages = {881-888},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hausdorff dimension of the maximal run-length in dyadic expansion},
url = {http://eudml.org/doc/196705},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Zou, Ruibiao
TI - Hausdorff dimension of the maximal run-length in dyadic expansion
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 881
EP - 888
AB - For any $x\in [0,1)$, let $x=[\epsilon _1,\epsilon _2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=\max \lbrace j\ge 1\colon \epsilon _{i+1}=\cdots =\epsilon _{i+j}=1$, $0\le i\le n-j\rbrace $ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\lim _{n\rightarrow \infty }{r_n(x)}/{\log _2 n}=1$ 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.
LA - eng
KW - run-length function; Hausdorff dimension; dyadic expansion; run-length function; Hausdorff dimension; dyadic expansion
UR - http://eudml.org/doc/196705
ER -