On the speed of convergence of the nearest integer continued fraction

Abstract:

Let ${p_n}/{q_n}$ and ${A_n}/{B_n}$ denote the convergents of, respectively, the regular and the nearest integer continued fraction expansion of the irrational number x. There exists a function $k(n)$ such that ${A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}$. Adams proved that for almost all x one has $\lim k(n)/n = \log 2/\log G$, $G = \frac {1}{2}(1 + \sqrt 5 )$. Here we present a shorter proof of this result, based on a simple expression for $k(n)$ and the ergodicity of the shift operator, connected with the nearest integer continued fraction.