On the Extremal Theory of Continued Fractions |
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Authors: | Alina Bazarova István Berkes Lajos Horváth |
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Institution: | 1.Institute of Statistics,Graz University of Technology,Graz,Austria;2.Department of Mathematics,University of Utah,Salt Lake City,USA |
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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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