Hausdorff dimension of the maximal run-length in dyadic expansion

Ruibiao Zou

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 881-888
  • ISSN: 0011-4642

Abstract

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For any x ∈ [ 0 , 1 ) , let x = [ ϵ 1 , ϵ 2 , ⋯ , ] be its dyadic expansion. Call r n ( x ) : = max { j ≥ 1 : ϵ i + 1 = ⋯ = ϵ i + j = 1 , 0 ≤ i ≤ n - j } the n -th maximal run-length function of x . P. Erdös and A. Rényi showed that lim n → ∞ 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.

How to cite

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Zou, Ruibiao. "Hausdorff dimension of the maximal run-length in dyadic expansion." Czechoslovak Mathematical Journal 61.4 (2011): 881-888. <http://eudml.org/doc/196705>.

@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 -

References

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