Intermediate convergents and a metric theorem of Khinchin

AK Haynes

Research output: Contribution to journalArticle (Academic Journal)peer-review

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.
Translated title of the contributionIntermediate convergents and a metric theorem of Khinchin
Original languageEnglish
Pages (from-to)396 - 410
Number of pages15
JournalBulletin of the London Mathematical Society
Volume41, number 3
DOIs
Publication statusPublished - Jun 2009

Bibliographical note

Publisher: Oxford University Press

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