Denote by \({p_n\over q_n} (n=1,2, \dots)\) the convergents to a real number \(x\) for the optimal continued fraction expansion. Define the functions \[ \vartheta_n(x) = q^2_n \cdot\left | x-{p_n \over q_n} \right| \qquad (n=1,2, \dots). \] Assume that \(k_n\) stands for \(\Phi(n)\) or \(\Phi(r_n)\) \((n=1,2, \dots)\), where \(\Phi\) is a polynomial which maps \(\mathbb{N}\) into \(\mathbb{N}\) and \(r_k\) denotes the \(k\)-th rational prime. The author studies the metric behaviour of the sequence \((\vartheta_{k_n} (x))^\infty_{n=1}\) with respect to Lebesgue measure. The author’s results generalize some earlier results of W. Bosma and C. Kraaikamp [J. Number Theory 34, 251-270 (1990; Zbl 0697.10043)].