## On the speed of convergence of the nearest integer continued fraction

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- by H. Jager PDF
- Math. Comp.
**39**(1982), 555-558 Request permission

## 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.

## References

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## Additional Information

- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp.
**39**(1982), 555-558 - MSC: Primary 10K10
- DOI: https://doi.org/10.1090/S0025-5718-1982-0669647-9
- MathSciNet review: 669647