no. 10
Probability Theory
Level sets of β-expansions
[Ensembles de niveau des β-développements.]

Présenté par : Yves Meyer

Fan, Aihua1, 2 ; Zhu, Hao1
1 Department of Mathematics, Wuhan University, 430072, Wuhan, China
2 LAMFA, UMR 6140 CNRS, université de Picardie, 33, rue Saint Leu, 80039 Amiens, France
Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 709-712.

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 où f(p)=(1−p)1−p/((1−2p)1−2ppp).

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).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.09.026
Fan, Aihua 1, 2 ; Zhu, Hao 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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Fan, Aihua; Zhu, Hao. Level sets of β-expansions. Comptes Rendus. Mathématique, Tome 339 (2004) no. 10, pp. 709-712. doi : 10.1016/j.crma.2004.09.026. http://www.numdam.org/articles/10.1016/j.crma.2004.09.026/

[1] Beyer, W.A. Hausdorff dimension of level set of some Rademacher series, Pacific J. Math., Volume 12 (1962), pp. 35-46

[2] Fan, A.H.; Schmeling, J. On fast Birkhoff averaging, Math. Proc. Cambridge Philos. Soc., Volume 135 (2003) no. 3, pp. 443-467

[3] Fan, A.H.; Feng, D.J.; Wu, J. Recurrence, dimension and entropy, J. London Math. Soc., Volume 64 (2001), pp. 229-244

[4] Garsia, A.M. Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc., Volume 102 (1962), pp. 409-432

[5] Kaczmarz, S.; Steinhaus, H. Le système orthogonal de M. Rademacher, Studia Math., Volume 2 (1930), pp. 231-247

[6] Kahane, J.P. Lacunary Taylor series and Fourier series, Bull. Amer. Math. Soc., Volume 70 (1964), pp. 199-213

[7] Parry, W. On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar., Volume 11 (1960), pp. 401-416

[8] Rényi, A. Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Volume 8 (1957), pp. 477-493

[9] Wu, J. Dimension of level sets of some Rademacher series, C. R. Acad. Sci. Paris, Ser. I, Volume 327 (1998), pp. 29-33

[10] Xi, L.F. Hausdorff dimension of level sets of Rademacher series, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000), pp. 953-958

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