Skip Navigation


Bulletin of the London Mathematical Society Advance Access originally published online on March 22, 2009
Bulletin of the London Mathematical Society 2009 41(3):396-410; doi:10.1112/blms/bdp011
This Article
Right arrow FREE Full Text (PDF) Freely available
Right arrow All Versions of this Article:
41/3/396    most recent
bdp011v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Haynes, A. K.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2009 London Mathematical Society

Intermediate convergents and a metric theorem of Khinchin

Alan K. Haynes

Department of Mathematics
University of York
Heslington
York
YO10 5DD
United Kingdom

Received 14 August 2008.

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 -> {infty}? 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.

2000 Mathematics Subject Classification 11B57, 11K50, 60G46.

Research supported by EPSRC grant EP/F027028/1.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.