Quantitative generalizations of Niederreiter's results on continued fractions (Q2857811)

Let \(f(d,N)\) be the number of numerators \(c\in\{1,2,\ldots, d-1\}\) such that all partial quotients in the continued fraction expansion \(c/d=[0;a_1,a_2,\ldots,a_n]\) are bounded by \(N\). In [Monatsh. Math. 101, 309--315 (1986; Zbl 0584.10004)] \textit{H. Niederreiter} proved that for any \(m\geq 2\) numbers \(f(2^m,4)\), \(f(3^m,4)\) and \(f(5^m,5)\) are positive. The paper under review contains a more general result: for any \(a>1\), \(m>1\) and \(N\geq N_0\) NEWLINE\[NEWLINEf(a^m,N)\geq 2^{-\log N(\log\log N)^2}m^{\log\log N}.NEWLINE\]NEWLINE In particular, \(f(a^m,N)\to\infty\) as \(m\to\infty\).