Recurrence speed of multiples of an irrational number. (Q696376)

Let \(\theta\) be an irrational number in \((0,1)\) and \(x\) a real number in \((0,1)\). Define the transformation of \(x\) by \(T_\theta(x)= \theta+ x\); for its iteration we have \(T^n_\theta(x)= n\theta+ x\). For a real \(t\) set \(\| t\|= \min(t- n)\), where the minimum has to be taken over all integer \(n\). Denote \(Q_n\) the interval \(((i- 1)/2^n\), \(i/2^n)\) containing \(x\). Set \[ J_n(x)= \min(j\geq 1:\| x- T^j_\theta x\|< 1/2^n), \] and \[ K_n(x)= \min(j\geq 1: T^j_\theta(x)\in Q_n). \] The authors prove that \[ \log J_n(x)/n\to 1\quad\text{and}\quad \log K_n(x)\to 1,\tag{\(*\)} \] almost every in \(\theta\) as \(n\to\infty\). Reviewer's remark: The two asymptotic relations in \((*)\) are equivalent. The authors are almost immediate consequences of a classical result of Khitchine, stating that if \(A_n/B_n\) are the convergents of the continued fraction expansion of \(\theta\), then we have almost everywhere \[ \log B_n\sim(\pi^2/12)n\tag{\(**\)} \] (Khintchine proved \((**)\) only with an undetermined constant instead of \(\pi^2/12\)) and the ``Ostrowski Algorithm'' described in [\textit{A. M. Rockett} and \textit{P. Szüsz}, Continued fractions Singapore: World Scientific, (1992; Zbl 0925.11038)].