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On the metric theory of the optimal continued fraction expansion

Published online by Cambridge University Press:  17 April 2009

R. Nair
R. Nair
Affiliation:
Department of Pure MathematicsUniversity of LiverpoolPO Box 147Liverpool L69 3BXUnited Kingdom
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Abstract

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Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants by

In this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 1 , August 1997 , pp. 69 - 79
Copyright
Copyright © Australian Mathematical Society 1997

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