APPROXIMATION ORDERS OF A REAL NUMBER IN A FAMILY OF BETA-DYNAMICAL SYSTEMS

Xiaoqiong Wang () and Rao Li
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Xiaoqiong Wang: School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, P. R. China
Rao Li: School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, P. R. China

FRACTALS (fractals), 2023, vol. 31, issue 05, 1-10

Abstract: In this paper, we study the approximation orders of a real number x ∈ (0, 1) by the partial sums of its β-expansions as β varies in the parameter space {β ∈ ℠: β > 1}. More precisely, letting Sn(x,β) be the partial sum of the first n items of the β-expansion of x, we prove that for any real number x ∈ (0, 1), the approximation order of x by Sn(x,β) is β−n for Lebesgue almost all β > 1. Moreover, we obtain the size of the set of β > 1 for which x can be approximated with a more general order β−φ(n), where φ: ℕ → ℠+ is a positive function. We also determine the Hausdorff dimension of the set Cφ(α) = β > 1 :lim supn→∞ln(x,β) φ(n) = α, 0 ≤ α ≤∞, where ln(x,β) is the number of the longest consecutive zeros just after the nth digit in the β-expansion of x.

Keywords: Approximation Order; Beta-Expansion; Parameter Space; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23500482

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