A metrical result on the approximation by continued fractions (Q2840017)

For a real number \(x\), the \(n\)th approximation coefficient is given by NEWLINE\[NEWLINE \theta_n = q_n^2 \left| x - {p_n \over q_n}\right|, NEWLINE\]NEWLINE where \(p_n/q_n\) denotes the \(n\)th convergent of \(x\). In the paper under review, the author calculates the distribution function for the difference \(| \theta_{n+1} - \theta_{n-1}|\) for almost all \(x\). As a consequence, the average of these differences is shown to converge to \((2\gamma + 1 - \log 2\pi)/2 \log 2\) for almost every \(x\), where \(\gamma\) is the Euler constant.