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Intermediate convergents and a metric theorem of Khinchin

Intermediate convergents and a metric theorem of Khinchin A landmark theorem in the metric theory of continued fractions begins this way: Select a non‐negative real function f defined on the positive integers and a real number x, and form the partial sums sn of f evaluated at the partial quotients a1, …, an in the continued fraction expansion for x. Does the sequence {sn/n} have a limit as n → ∞? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Intermediate convergents and a metric theorem of Khinchin

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References (9)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdp011
Publisher site
See Article on Publisher Site

Abstract

A landmark theorem in the metric theory of continued fractions begins this way: Select a non‐negative real function f defined on the positive integers and a real number x, and form the partial sums sn of f evaluated at the partial quotients a1, …, an in the continued fraction expansion for x. Does the sequence {sn/n} have a limit as n → ∞? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jun 1, 2009

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