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On the Extremal Theory of Continued Fractions

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Abstract

Letting \(x=[a_1(x), a_2(x), \ldots ]\) denote the continued fraction expansion of an irrational number \(x\in (0, 1)\), Khinchin proved that \(S_n(x)=\sum \nolimits _{k=1}^n a_k(x) \sim \frac{1}{\log 2}n\log n\) in measure, but not for almost every \(x\). Diamond and Vaaler showed that, removing the largest term from \(S_n(x)\), the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of \(S_n (x)\) on the sum. In this paper we determine, for \(d_n\rightarrow \infty \) and \(d_n/n\rightarrow 0\), the precise asymptotics of the sum of the \(d_n\) largest terms of \(S_n(x)\) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.

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Acknowledgments

We would like to thank the referee for her/his remarks leading to a substantial improvement of the paper.

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Authors and Affiliations

  1. Institute of Statistics, Graz University of Technology, Kopernikusgasse 24, 8010, Graz, Austria

    Alina Bazarova & István Berkes

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112-0090, USA

    Lajos Horváth

Authors
  1. Alina Bazarova
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  2. István Berkes
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  3. Lajos Horváth
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Corresponding author

Correspondence to István Berkes.

Additional information

Alina Bazarova received research support from Austrian Science Fund (FWF) Grant W1230.

István Berkes received research support from Austrian Science Fund (FWF) Grant P24302-N18 and from OTKA Grant K 108615.

Lajos Horváth received research support from NSF Grant DMS-13-05858.

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Bazarova, A., Berkes, I. & Horváth, L. On the Extremal Theory of Continued Fractions. J Theor Probab 29, 248–266 (2016). https://doi.org/10.1007/s10959-014-0577-5

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  • DOI: https://doi.org/10.1007/s10959-014-0577-5

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