ON THE LARGEST PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS

Lulu Fang and Jian Liu
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Lulu Fang: School of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China2School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China
Jian Liu: College of Information Engineering, Nanjing University of Finances and Economics, Nanjing 210023, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 04, 1-14

Abstract: Let [a1(x),a2(x),…,an(x),…] be the continued fraction expansion of an irrational x ∈ (0, 1). For any n ≥ 1, write Tn(x) =max1≤k≤n{ak(x)}. This paper is concerned with the Hausdorff dimension of the set E(ψ) := x ∈ (0, 1) :limn→∞Tn(x) ψ(n) = 1 , where ψ : ℕ → ℠+ is a function such that ψ(n) →∞ as n →∞. We calculate the Hausdorff dimension of E(ψ) for a very large class of functions with certain growth rates, which improves the existing results of Wu and Xu (2009), Liao and Rams (2016) and Chang and Chen (2018).

Keywords: Continued Fractions; Largest Partial Quotients; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21500997

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