The run-length function of the β-expansion of the unit

Cao, Chun-Yun; Chen, Yuanhong

doi:10.1016/j.jnt.2017.01.015

Cited by 6 publications

(3 citation statements)

References 11 publications

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“…As the same discussion at the first part of Section 3, it holds that dim H E P 0,0 is of full Lebesgue measure by using the result of Cao and Chen [4] that the set β ∈ (1, 2) : lim n→∞ r n (β) log β n = 1…”

Section: Classical Results Of β-Expansion In the Parameter Spacementioning

confidence: 92%

“…In this section, we recall some important results of β-expansion in the parameter space {β ∈ R : β > 1}. The readers can refer to [4,8,12,15,18] for more information.…”

Section: Classical Results Of β-Expansion In the Parameter Spacementioning

confidence: 99%

“…is of full Hausdorff dimension. Remark that the results of [8] and [4] can be easily generalized to the whole parameter space {β ∈ R : β > 1}. For simplicity, in this paper, we will also consider the parameter space (1,2).…”

Section: Introductionmentioning

confidence: 99%

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Diophantine approximation and run-length function on β-expansions

Zheng

2019

Journal of Number Theory

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For any β > 1, denoted by rn(x, β) the maximal length of consecutive zeros amongst the first n digits of the β-expansion of x ∈ [0, 1]. The limit superior (respectively limit inferior) of rn(x,β) n is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level setFurthermore, we show that the extremely divergent set E0,1 which is of zero Hausdorff dimension is, however, residual. The same problems in the parameter space are also examined.

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Section: Classical Results Of β-Expansion In the Parameter Spacementioning

confidence: 92%

“…In this section, we recall some important results of β-expansion in the parameter space {β ∈ R : β > 1}. The readers can refer to [4,8,12,15,18] for more information.…”

Section: Classical Results Of β-Expansion In the Parameter Spacementioning

confidence: 99%