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

Abstract

Abstract.In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If An/B Pk/<tk denote the convergents of the nearest integer and regular continued fractions of an irrational number a, then for all n there is a k(n) such that AnjBn = Pk(n)/lk(ny It ¡s shown thatfor almost all a. 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).

On certain statistical properties of continued fractions with even and with odd partial quotients

“…The cases α = 1 and α = 1 2 are best known, corresponding to the RCF and NICF (continued fraction to the nearest integer). The latter was introduced by Minnigerode [9] and was also studied in [1,13,18]. Our calculations show explicit connections with Ustinov's RCF distribution.…”

“…In both ECF and OCF situations, we take x 1 , x 2 , x 3 , x 4 ∈ (0, 1]. 1 In the OCF case, the ratio Q/Q ′ of successive denominators can in fact be any rational number in the interval (0, G), but since in the definition of n R we are interested only in Q R < Q ′ , we can restrict to x 3 1 in the definition of L O,± . The golden ratios G = (1 + √ 5)/2 and g = 1/G = (−1 + √ 5)/2 will be used often.…”