Some metrical observations on the approximation of an irrational number by its nearest mediants (Q1182526)

For any irrational number \(x\in[0,1]\) denote by \(x=[a_1,a_2,\ldots]\) its expansion as a regular continued fraction and by \((p_n/q_n)\), \(n=1,2,\ldots\) the sequence of its associated convergents. The numbers \(\frac{A_n}{B_n}:= \frac{p_n+q_{n-1}}{q_n+q_{n-1}}\) and \(\frac{C_n}{D_ n}:= \frac{p_n-p_{n-1}}{q_ n-q_{n-1}}\) form the extreme mediants. Then the exact limit distribution of the sequences \(B_n | xB_n - A_n|\) resp. \(D_n | xD_n - C_n|\) is calculated. Furthermore a special case of a theorem of \textit{P. Erdős} [Acta Arith. 5, 359--369 (1959; Zbl 0097.03502)] is obtained by a combination of an old theorem of Fatou and Koksma and ergodic theory (the same result was obtained by \textit{S. Itô} [Osaka J. Math. 26, 557--578 (1989; Zbl 0702.11046)]).