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

Let {εn(x)}n1 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 0p1/2, any 0<τ1 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)=(1p)1p/((12p)12ppp).

Soit {εn(x)}n1 la suite de β-digits du nombre réel x(0,1), avec le nombre d'or β=(5+1)/2 comme base. Pour tout 0p1/2, 0<τ1 et aR, 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)/logβf(p)=(1p)1p/((12p)12ppp).

Received:
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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