On upper and lower fast Knintchine spectra of continued fractions

2021/10/02 by Lulu Fang, Lei Shang, Fang, Lulu +3
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Number Theory (math.NT) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2110.00787

openalex publication_date 2021/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ψ:ℕ→ ℝ⁺ be a function satisfying ϕ(n)/n→ ∞ as n → ∞. We investigate from a multifractal analysis point of view the growth rate of the sums ∑ⁿ_k=1log a_k(x) relative to ψ(n), where [a₁(x),a₂(x), a₃(x)⋯] denotes the continued fraction expansion of x∈ (0,1). The upper (resp. lower) fast Khintchine spectrum is defined by the Hausdorff dimension of the set of all points x for which the upper (resp. lower) limit of (1)/(ψ(n))∑ⁿ_k=1log a_k(x) is 1. The precise formulas of these two spectra are completely determined, which strengthens a result of Liao and Rams (2016).

Citations

• On the sum of digits of real numbers represented in the dyadic system.
• Fractal Geometry: Mathematical Foundations and Applications.
• On Khintchine exponents and Lyapunov exponents of continued fractions
• Spectrum of weighted Birkhoff average

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• Upper and lower fast Khintchine spectra in continued fractions
• On the fast Khintchine spectrum in continued fractions
• On Khintchine exponents and Lyapunov exponents of continued fractions
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• Multifractal analysis of maximal product of consecutive partial quotients in continued fractions