Upper and lower fast Khintchine spectra in continued fractions (Q284077)

This paper is a contribution to the multifractal analysis of Birkhoff sums (averages) of dynamical systems. (In history, the first multifractal analysis of Birkhoff averages maybe due to Besicovitch who studied the frequencies of digits in binary expansions.) More precisely, for an irrational number \(x\in [0,1)\), let \(x=[a_1(x),a_2(x),\ldots]\) be its continued fraction expansion. Let \(\psi:\mathbb {N}\to \mathbb {N}\) be a function with \(\psi(n)/n\to \infty\) as \(n\to \infty\). 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. In Theorem 1.2, the main result of the paper, the authors calculate the upper and lower fast Khintchine spectra.