Probability Theory
Level sets of β-expansions
Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 709-712.

Let ${εn(x)}n⩾1$ be the sequence of β-digits of a real number $x∈(0,1)$, with the golden number $β=(5+1)/2$ as basis. For any $0⩽p⩽1/2$, any $0<τ⩽1$ and any real number a, we consider the level set consisting of numbers x such that $∑n=1∞(εn(x)−p)/nτ=a$. We prove that the Hausdorff dimension of this set is independent of a and τ, and that it is equal to $logf(p)/logβ$ where $f(p)=(1−p)1−p/((1−2p)1−2ppp)$.

Soit ${εn(x)}n⩾1$ la suite de β-digits du nombre réel $x∈(0,1)$, avec le nombre d'or $β=(5+1)/2$ comme base. Pour tout $0⩽p⩽1/2$, $0<τ⩽1$ et $a∈R$, nous considérons l'ensemble de niveau qui est constitué des x tels que $∑n=1∞(εn(x)−p)/nτ=a$. Nous prouvons que la dimension de Hausdorff de cet ensemble est independante de a et τ, et qu'elle est égale à $logf(p)/logβ$$f(p)=(1−p)1−p/((1−2p)1−2ppp)$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.026
Aihua Fan 1, 2; Hao Zhu 1

1 Department of Mathematics, Wuhan University, 430072, Wuhan, China
2 LAMFA, UMR 6140 CNRS, université de Picardie, 33, rue Saint Leu, 80039 Amiens, France
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Aihua Fan; Hao Zhu. Level sets of β-expansions. Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 709-712. doi : 10.1016/j.crma.2004.09.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.026/

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