Upper and lower fast Khintchine spectra in continued fractions

For an irrational number x epsilon [0, 1), let x = [a(1)(x), a(2)(x),...] be its continued fraction expansion. Let psi : N -> N be a function with psi (n)/n -> infinity as n -> infinity. The (upper, lower) fast Khintchine spectrum for psi is defined as the Hausdorff dimension of the set of numbers x epsilon (0, 1) for which the (upper, lower) limit of 1/psi(n) Sigma(n)(j = 1) 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.