On the metric theory of the nearest integer continued fraction expansion

Abstract

Supposek _n denotes either ϕ(n) or ϕ(p _n) (n=1,2,...) where the polynomial ϕ maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let$\left( {\frac{{r_n }}{{q_n }}} \right)_{n = 1}^\infty$ denote the sequence of convergents to a real numberx and letc _n(x)) ^∞_n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constantsθ _n(x)) ^∞_n=1 by

$$\theta _n (x) = q_n^2 \left| {x - \frac{{r_n }}{{q_n }}} \right|. (n = 1,2,...)$$

In this paper we study the behaviour of the sequences$(\theta _{k_n } (x))_{n = 1}^\infty$ and$(c_{k_n } (x))_{n = 1}^\infty$ for almost allx with respect to the Lebesgue measure. In the special case wherek _n=n (n=1,2,...) these results are known and due to H. Jager, G. J. Rieger and others.