On a relationship between the convergents of the nearest integer and regular continued fractions

Abstract:

In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If ${A_n}/{B_n}$, ${p_k}/{q_k}$ denote the convergents of the nearest integer and regular continued fractions of an irrational number $\alpha$, then for all n there is a $k(n)$ such that ${A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}$. It is shown that \[ \lim \limits _{n \to \infty } \;\frac {n}{{k\left ( n \right )}} = \frac {{\log \left ( {\frac {{1 + \sqrt 5 }}{2}} \right )}}{{\log \;2}}\] for almost all $\alpha$. This problem is reduced to a special case of a general result concerning the frequency of partial quotients in the regular continued fraction (Theorem 2).