On upper and lower fast Khintchine spectra of continued fractions (Q2136085)

One can begin with authors' abstract: ``Let \(\psi : \mathbb{N}\to\mathbb{R}^+\) be a function satisfying \(\frac{\phi (n)}{n} \to \infty\) as \(n\to\infty\). We investigate from a multifractal analysis point of view the growth rate of the sums \(\sum^n_{k=1} \log a_k (x)\) relative to \(\psi (n)\), where \([a_1 (x),a_2 (x),\ldots]\) denotes the continued fraction expansion of an irrational \(x\in (0,1)\). For \(\alpha\in [0,\infty]\), the upper (resp. lower) fast Khintchine spectrum is considered as a function of \(\alpha\) which is defined by the Hausdorff dimension of the set of all points \(x\) such that the upper (resp. lower) limit of \(\frac{1}{\psi (n)}\sum^n_{k=1}\log a_k (x)\) is equal to \(\alpha\). These two spectra have been studied by \textit{L. Liao} and \textit{M. Rams} [Monatsh. Math. 180, No. 1, 65--81 (2016; Zbl 1345.28011)] under some restrictions on the growth rate of \(\psi\). In this paper, we completely determine the precise formulas of these two spectra without any conditions on \(\psi\).'' Notions as continued fraction expansions, the Gauss map, and the Khintchine exponent of a number, as well as the Khintchine constant and the Khintchine spectrum, etc., are described and auxiliary references are given. The fast Khintchine exponent and the fast Khintchine spectrum are considered. The notions of the upper and lower fast Khintchine spectra are explained, the motivation of their investigations and a brief descriptions of related researches are given. A new phenomenon in continued fractions is noted.