Let be the sequence of β-digits of a real number , with the golden number as basis. For any , any and any real number a, we consider the level set consisting of numbers x such that . We prove that the Hausdorff dimension of this set is independent of a and τ, and that it is equal to where .
Soit la suite de β-digits du nombre réel , avec le nombre d'or comme base. Pour tout , et , nous considérons l'ensemble de niveau qui est constitué des x tels que . Nous prouvons que la dimension de Hausdorff de cet ensemble est independante de a et τ, et qu'elle est égale à où .
Accepted:
Published online:
Aihua Fan 1, 2; Hao Zhu 1
@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}, }
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] Hausdorff dimension of level set of some Rademacher series, Pacific J. Math., Volume 12 (1962), pp. 35-46
[2] On fast Birkhoff averaging, Math. Proc. Cambridge Philos. Soc., Volume 135 (2003) no. 3, pp. 443-467
[3] Recurrence, dimension and entropy, J. London Math. Soc., Volume 64 (2001), pp. 229-244
[4] Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc., Volume 102 (1962), pp. 409-432
[5] Le système orthogonal de M. Rademacher, Studia Math., Volume 2 (1930), pp. 231-247
[6] Lacunary Taylor series and Fourier series, Bull. Amer. Math. Soc., Volume 70 (1964), pp. 199-213
[7] On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar., Volume 11 (1960), pp. 401-416
[8] Representation for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Volume 8 (1957), pp. 477-493
[9] Dimension of level sets of some Rademacher series, C. R. Acad. Sci. Paris, Ser. I, Volume 327 (1998), pp. 29-33
[10] 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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