Comptes Rendus
Probability Theory
Level sets of β-expansions
[Ensembles de niveau des β-développements.]
Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 709-712.

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).

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).

Reçu le :
Accepté le :
Publié le :
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
@article{CRMATH_2004__339_10_709_0,
     author = {Aihua Fan and Hao Zhu},
     title = {Level sets of \protect\emph{\ensuremath{\beta}}-expansions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {709--712},
     publisher = {Elsevier},
     volume = {339},
     number = {10},
     year = {2004},
     doi = {10.1016/j.crma.2004.09.026},
     language = {en},
}
TY  - JOUR
AU  - Aihua Fan
AU  - Hao Zhu
TI  - Level sets of β-expansions
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 709
EP  - 712
VL  - 339
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2004.09.026
LA  - en
ID  - CRMATH_2004__339_10_709_0
ER  - 
%0 Journal Article
%A Aihua Fan
%A Hao Zhu
%T Level sets of β-expansions
%J Comptes Rendus. Mathématique
%D 2004
%P 709-712
%V 339
%N 10
%I Elsevier
%R 10.1016/j.crma.2004.09.026
%G en
%F CRMATH_2004__339_10_709_0
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/

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique