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Intermediate convergents and a metric theorem of Khinchin

By Alan K. Haynes

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 $s_n$ of $f$ evaluated at the partial quotients $a_1,..., a_n$ in the continued fraction expansion for $x$. Does the sequence $\{s_n/n\}$ have a limit as $n\rar\infty$? In 1935 A. Y. Khinchin proved that the answer is yes for almost every $x$, provided that the function $f$ does not grow too quickly. In this paper 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

Topics: Mathematics - Number Theory, Mathematics - Probability, 11B57, 11K50, 60G46
Publisher: 'Wiley'
Year: 2009
DOI identifier: 10.1112/blms/bdp011
OAI identifier: oai:arXiv.org:0907.0161

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