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 {A_n}/{B_n} , p k / q k {p_k}/{q_k} denote the convergents of the nearest integer and regular continued fractions of an irrational number α \alpha , then for all n there is a k ( n ) k(n) such that A n / B n = p k ( n ) / q k ( n ) {A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}} . It is shown that \[ lim n → ∞ n k ( n ) = log ⁡ ( 1 + 5 2 ) log 2 \lim \limits _{n \to \infty } \;\frac {n}{{k\left ( n \right )}} = \frac {{\log \left ( {\frac {{1 + \sqrt 5 }}{2}} \right )}}{{\log \;2}} \] for almost all α \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).