On the distribution of the convergents of almost all real numbers (Q2542313)

Let \(n_1 < n_2 < \ldots\) be an infinite sequence of integers. The necessary and sufficient condition that for almost all \(\alpha\) infinitely many \(n_i\) should occur among the convergents of \(\alpha\) is that \(\sum^\infty_{i=1} \varphi(n_i)/n^2_i = \infty\), where \(\varphi\) is the Euler \(\varphi\)-function. The necessity is obvious, but the proof of the sufficiency is complicated. In fact it is proved that if \(\sum^\infty_{i=1} \varphi (n_i)/n^2_i= \infty\) then for every \(\epsilon >0\) and almost all \(\alpha\) \[ |\alpha -a/n_i| < \epsilon /n^2_i, \quad (a,n_i) = 1 \] has infinitely many solutions.