On upper and lower fast Khintchine spectra of continued fractions

Abstract

Abstract Let ψ : ℕ → ℝ + {\psi:\mathbb{N}\to\mathbb{R}^{+}} be a function satisfying ϕ ⁢ ( n ) n → ∞ {\frac{\phi(n)}{n}\to\infty} as n → ∞ {n\to\infty} . We investigate from a multifractal analysis point of view the growth rate of the sums ∑ k = 1 n log ⁡ a k ⁢ ( x ) {\sum^{n}_{k=1}\log a_{k}(x)} relative to ψ ⁢ ( n ) {\psi(n)} , where [ a 1 ⁢ ( x ) , a 2 ⁢ ( x ) , … ] {[a_{1}(x),a_{2}(x),\dots]} denotes the continued fraction expansion of an irrational x ∈ ( 0 , 1 ) {x\in(0,1)} . For α ∈ [ 0 , ∞ ] {\alpha\in[0,\infty]} , the upper (resp. lower) fast Khintchine spectrum is considered as a function of α which is defined by the Hausdorff dimension of the set of all points x such that the upper (resp. lower) limit of 1 ψ ⁢ ( n ) ⁢ ∑ k = 1 n log ⁡ a k ⁢ ( x ) {\frac{1}{\psi(n)}\sum^{n}_{k=1}\log a_{k}(x)} is equal to α. These two spectra have been studied by Liao and Rams (2016) under some restrictions on the growth rate of ψ. In this paper, we completely determine the precise formulas of these two spectra without any conditions on ψ.