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The limiting behaviour of certain sequences of continued fractions

Published online by Cambridge University Press:  17 April 2009

David Angell
David Angell
Affiliation:
School of Mathematics, University of N.S.W., Kensington, N.S.W. 2033, Australia
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Abstract

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We investigate the set of limit points of the continued fractions

where x1, x2, … is a given sequence of positive integers. We show that this set is closed, and that it may include any given countable subset of [0, 1] if the integers xk are chosen appropriately. Our main result, which has applications in transcendence theory, is that the sequence of continued fractions has no rational limit point when the sequence {xk} of partial quotients is bounded.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 1 , August 1988 , pp. 67 - 76
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Angell, D.D., Mahler's Method in Transcencence Theory (Ph.D. thesis, University of New South Wales, 1987).Google Scholar
[2]Loxton, J.H. and van der Poorten, A.J., ‘Arithmetic properties of certain functions in several variables III’, Bull. Austral. Math. Soc. 16 (1977), 15–47.CrossRefGoogle Scholar