On the order of the largest partial quotient of the regular continued fraction expansion (Q1817763)

Let \(x\in (0,1)\) be an irrational number and \([a_1(x), a_2(x),\dots]\) be its regular continued fraction expansion. Put \(f(z)= \exp (-\frac{1}{z\log 2})\) for \(z> 0\), \[ f_n(x,z)= \tfrac{1}{n} \# \Bigl\{k\leq n: \max_{1\leq j\leq k} a_j(x)< kz \Bigr\}. \] Then \[ \lim\mu \Bigl(x: \max_{1\leq i\leq n} a_i(x)< nz\Bigr)= f(z) \] (\(\mu\) being the Gauss measure) [cf. \textit{J. Galambos}, Q. J. Math., Oxf., II. Ser. 23, 147-151 (1972; Zbl 0234.10041)]. In connection with this result the author shows that if \(z>0\) then for almost all \(x\in (0,1)\) we have \[ \liminf_{n\to\infty} f_n(x,z)= 0, \qquad \limsup_{n\to\infty} f_n(x,z)= 1. \] Moreover, for all \(z>0\) the sequence \((f_n (x,z))_{n=1}^\infty\) has no subsequence which converges almost everywhere.