On metric diophantine approximation and subsequence ergodic theory (Q1383520)

Let \(x= [c_0;c_1,c_2,\dots]\) be the regular continued fraction expansion of \(x\), let \(\frac{r_n}{q_n}\) \((n=1,2,\dots)\) be convergents of \(x\). Define the functions \[ \vartheta_n(x)= q_n^2\Biggl| x-\frac{r_n}{q_n} \Biggr| \qquad (n=1,2,\dots). \] In the paper [\textit{W. Bosma, H. Jager}, and \textit{F. Wiedijk}, Indag. Math. 45, 281-299 (1983; Zbl 0519.10043)] distribution of the sequence (*) \((\vartheta_n(x))_{n=1}^\infty\) is studied with respect to Lebesgue measure. This investigation is deepened in this paper by using ergodic theory. Some further sequences connected with (*) are investigated. Such sequences are \((\vartheta_{k_n} (x))_{n=1}^\infty\), where \((k_n)_{n=1}^\infty\) is a sequence of positive integers, for instance \(k_n= \Phi(n)\) or \(k_n= \Phi(p_n)\) (\(p_n\) is the \(n\)-th prime), \(\Phi\) being a polynomial which maps \(\mathbb{N}\) into \(\mathbb{N}\).