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- Title
EXCEPTIONAL SETS RELATED TO THE RUN-LENGTH FUNCTION OF BETA-EXPANSIONS.
- Authors
FANG, LULU; SONG, KUNKUN; WU, MIN
- Abstract
Let β > 1 and x ∈ [ 0 , 1 ] be real numbers. The run-length function of β -expansions denoted by r n (x , β) is defined as the maximal length of consecutive zeros in the first n digits of the β -expansion of x. It is known that for Lebesgue almost all x ∈ [ 0 , 1 ] , r n (x , β) increases to infinity with the logarithmic speed log β n as n goes to infinity. In this paper, we calculate the Hausdorff dimension of the subtle set for which r n (x , β) grows to infinity with other speeds. More precisely, we prove that for any 0 ≤ a ≤ b ≤ ∞ , the set x ∈ [ 0 , 1) : lim inf n → ∞ r n (x , β) ϕ (n) = a , lim sup n → ∞ r n (x , β) ϕ (n) = b has full Hausdorff dimension, where ϕ : ℝ + → ℝ + is a strictly increasing function satisfying that ϕ (n) / n is non-increasing, ϕ (n) → ∞ and ϕ (n) / n → 0 as n → ∞. This result significantly extends the existing results in this topic, such as the results in [J.-H. Ma, S.-Y. Wen and Z.-Y. Wen, Egoroff's theorem and maximal run length, Monatsh. Math.151(4) (2007) 287–292; R.-B. Zou, Hausdorff dimension of the maximal run-length in dyadic expansion, Czechoslovak Math. J.61(4) (2011) 881–888; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem, J. Math. Anal. Appl.436(1) (2016) 355–365; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem revisited, Monatsh. Math.182(4) (2017) 865–875; Y. Sun and J. Xu, A remark on exceptional sets in Erdős–Rényi limit theorem, Monatsh. Math.184(2) (2017) 291–296; X. Tong, Y.-L. Yu and Y.-F. Zhao, On the maximal length of consecutive zero digits of β -expansions, Int. J. Number Theory12(3) (2016) 625–633; J. Liu, and M.-Y. Lü, Hausdorff dimension of some sets arising by the run-length function of β -expansions, J. Math. Anal. Appl.455(1) (2017) 832–841; L.-X. Zheng, M. Wu and B. Li, The exceptional sets on the run-length function of β -expansions, Fractals25(6) (2017) 1750060; X. Gao, H. Hu and Z.-H. Li, A result on the maximal length of consecutive 0 digits in β -expansions, Turkish J. Math.42(2) (2018) 656–665, doi: 10.3906/mat-1704-119].
- Subjects
REAL numbers; MATHEMATICAL functions; FRACTAL dimensions; DYADIC communication; MATHEMATICS theorems
- Publication
Fractals, 2018, Vol 26, Issue 4, pN.PAG
- ISSN
0218-348X
- Publication type
Article
- DOI
10.1142/S0218348X18500494