Divergence points with fast growth orders of the partial quotients in continued fractions (Q624208)

Writing as usual \(a_k(x)\) for the digits in the continued fraction expansion of a real number \(x\), we may associate to any non-empty interval \(S\) and function \(\phi\) with \(\phi(n)/n\to\infty\) a set of divergence points \(E_{\phi}(S)\), defined as the set of \(x\in[0,1)\) for which the set of accumulation points of the sequence with \(n\)th term \((1/\phi(n))\sum_{k=1}^n\log a_k(x)\) coincides with \(S\). The main results show that for \(\phi(n)=n^{\alpha}\) and any non-empty \(S\), \(\alpha>1\), the Hausdorff dimension of \(E_{\alpha}(S)\) is \(\frac12\); if \(\phi(n)=\beta^n\) with \(\beta>1\) then the corresponding Hausdorff dimension is \(1/(1+\beta)\).