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Dimension of some non-normal continued fraction sets

Published online by Cambridge University Press:  01 July 2008

LINGMIN LIAO ,
JIHUA MA  and
BAOWEI WANG
LINGMIN LIAO
Affiliation:
Department of Mathematics, Wuhan University, 430072 Wuhan, China. e-mail: jhma@whu.edu.cn
JIHUA MA
Affiliation:
Department of Mathematics, Wuhan University, 430072 Wuhan, China. e-mail: jhma@whu.edu.cn
BAOWEI WANG
Affiliation:
Department of Mathematics, Wuhan University, 430072 Wuhan, China. e-mail: jhma@whu.edu.cn
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Abstract

We consider certain sets of non-normal continued fractions for which the asymptotic frequencies of digit strings oscillate in one or other ways. The Hausdorff dimensions of these sets are shown to be the same value 1/2 as long as they are non-empty. An interesting example among them is the set of “extremely non-normal continued fractions” which was previously conjectured to be of Hausdorff dimension 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 215 - 225
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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