On the speed of convergence of the nearest integer continued fraction

Author:
H. Jager

Journal:
Math. Comp. **39** (1982), 555-558

MSC:
Primary 10K10

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669647-9

MathSciNet review:
669647

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Abstract | References | Similar Articles | Additional Information

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.

- William W. Adams,
*On a relationship between the convergents of the nearest integer and regular continued fractions*, Math. Comp.**33**(1979), no. 148, 1321–1331. MR**537978**, DOI https://doi.org/10.1090/S0025-5718-1979-0537978-9 - Patrick Billingsley,
*Ergodic theory and information*, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0192027** - Oskar Perron,
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*Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen*, J. Reine Angew. Math.**310**(1979), 171–181 (German). MR**546670**, DOI https://doi.org/10.1515/crll.1979.310.171 - Andrew M. Rockett,
*The metrical theory of continued fractions to the nearer integer*, Acta Arith.**38**(1980/81), no. 2, 97–103. MR**604225**, DOI https://doi.org/10.4064/aa-38-2-97-103 - H. C. Williams,
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Additional Information

Keywords:
Nearest integer continued fraction,
individual ergodic theorem

Article copyright:
© Copyright 1982
American Mathematical Society