Skip Navigation


Bulletin of the London Mathematical Society Advance Access originally published online on February 20, 2008
Bulletin of the London Mathematical Society 2008 40(1):18-22; doi:10.1112/blms/bdm103
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
40/1/18    most recent
bdm103v1
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 Wang, B.-W.
Right arrow Articles by Wu, J.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2008 London Mathematical Society

A problem of Hirst on continued fractions with sequences of partial quotients

Bao-Wei Wang and Jun Wu

Department of Mathematics
Huazhong University of Science and Technology
Wuhan
Hubei 430074
P. R. China
bwei_wang@yahoo.com.cn

Received 5 January 2007. Revision received 16 April 2007.

Let B denote an infinite sequence of positive integers b1 < b2 < ..., and let {tau} denote the exponent of convergence of the series {sum}n = 1{infty} 1/bn; that is, {tau} = inf {s ≥ 0 : {sum}n = 1{infty} 1/bns < {infty}}. Define E(B) = {x isin [0, 1]: an(x) isin B (n ≥ 1) and an(x) -> {infty} as n -> {infty}}. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227] proved the inequality dimH E(B) ≤ {tau}/2 and conjectured (see ibid., p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990) p. 278]) that equality holds. In this paper, we give a positive answer to this conjecture.

2000 Mathematics Subject Classification 11K55, 11K50, 28A80.

This work was supported by the NSFC grant 10571138.


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.