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The distribution of the largest digit in continued fraction expansions

Published online by Cambridge University Press:  01 January 2009

JUN WU  and
JIAN XU
JUN WU
Affiliation:
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China. e-mail: [email protected]
JIAN XU
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, P.R. China. e-mail: [email protected]
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Abstract

Let [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved thatOkano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the setis of Hausdorff dimension 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 207 - 212
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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