On the speed of convergence of the nearest integer continued fraction

Author:
H. Jager

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

MSC:
Primary 10K10

MathSciNet review:
669647

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Abstract: Let and denote the convergents of, respectively, the regular and the nearest integer continued fraction expansion of the irrational number *x*. There exists a function such that . Adams proved that for almost all *x* one has , . Here we present a shorter proof of this result, based on a simple expression for and the ergodicity of the shift operator, connected with the nearest integer continued fraction.

**[1]**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**, 10.1090/S0025-5718-1979-0537978-9**[2]**Patrick Billingsley,*Ergodic theory and information*, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0192027****[3]**Oskar Perron,*Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche*, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954 (German). 3te Aufl. MR**0064172****[4]**G. J. Rieger,*Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen*, J. Reine Angew. Math.**310**(1979), 171–181 (German). MR**546670**, 10.1515/crll.1979.310.171**[5]**Andrew M. Rockett,*The metrical theory of continued fractions to the nearer integer*, Acta Arith.**38**(1980/81), no. 2, 97–103. MR**604225****[6]**H. C. Williams,*Some results concerning the nearest integer continued fraction expansion of √𝐷*, J. Reine Angew. Math.**315**(1980), 1–15. MR**564520**, 10.1515/crll.1980.315.1

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Additional Information

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

Keywords:
Nearest integer continued fraction,
individual ergodic theorem

Article copyright:
© Copyright 1982
American Mathematical Society