Upper and lower fast Khintchine spectra in continued fractions

For an irrational number \(x\in 0,1)\), let \(x=a_1(x), a_2(x),\ldots ]\) be its continued fraction expansion. Let \(\psi : \mathbb {N} \rightarrow \mathbb {N}\) be a function with \(\psi (n)/n\rightarrow \infty \) as \(n\rightarrow \infty \). The (upper, lower) fast Khintchine spectrum for \(\psi \) is defined as the Hausdorff dimension of the set of numbers \(x\in (0,1)\) for which the (upper, lower) limit of \(\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)\) is equal to 1. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.