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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Gauss-Kusmin theorem
for optimal continued fractions


Authors: Karma Dajani and Cor Kraaikamp
Journal: Trans. Amer. Math. Soc. 351 (1999), 2055-2079
MSC (1991): Primary 28D05, 11K50
Published electronically: January 27, 1999
MathSciNet review: 1473436
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Abstract: A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal $S$-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal $S$-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal $S$-expansion.


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Additional Information

Karma Dajani
Affiliation: Faculteit Wiskunde en Informatica, Budapestlaan 6, P.O. Box 80.010, 3508TA Utrecht, The Netherlands
Email: dajani@math.ruu.nl

Cor Kraaikamp
Affiliation: Technische Universiteit Delft and Thomas Stieltjes Institute for Mathematics, Fac. ITS (SSOR), Mekelweg 4, 2628 CD Delft, The Netherlands
Email: c.kraaikamp@its.tudelft.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02177-7
PII: S 0002-9947(99)02177-7
Received by editor(s): December 12, 1996
Published electronically: January 27, 1999
Article copyright: © Copyright 1999 American Mathematical Society