On the speed of convergence of the nearest integer continued fraction
Author:
H. Jager
Journal:
Math. Comp. 39 (1982), 555558
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.
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 William W. Adams, "On a relationship between the convergents of the nearest integer and regular continued fractions," Math. Comp., v. 33, 1979, pp. 13211331. MR 537978 (82g:10078)
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 P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965. MR 0192027 (33:254)
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 O. Perron, Die Lehre von den Kettenbrüchen, Vol. I, 3rd ed., Teubner, Leipzig, 1954. MR 0064172 (16:239e)
 [4]
 G. J. Rieger, "Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen," J. Reine Angew. Math., v. 310, 1979, pp. 171181. MR 546670 (81c:10066)
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 A. M. Rockett, "The metrical theory of continued fractions to the nearer integer," Acta Arith., v. 38, 1980, pp. 97103. MR 604225 (82d:10074)
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 H. G. Williams, "Some results concerning the nearest integer continued fraction expansion of ," J. Reine Angew. Math., v. 315, 1980, pp. 115. MR 564520 (81f:10015)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206696479
PII:
S 00255718(1982)06696479
Keywords:
Nearest integer continued fraction,
individual ergodic theorem
Article copyright:
© Copyright 1982 American Mathematical Society
