On the Extremal Theory of Continued Fractions

Bazarova, A. and Berkes, István and Horváth, Lajos (2016) On the Extremal Theory of Continued Fractions. JOURNAL OF THEORETICAL PROBABILITY, 29 (1). pp. 248-266. ISSN 0894-9840

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Abstract

Letting (Formula presented.) denote the continued fraction expansion of an irrational number (Formula presented.), Khinchin proved that (Formula presented.) in measure, but not for almost every (Formula presented.). Diamond and Vaaler showed that, removing the largest term from (Formula presented.), the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of (Formula presented.) on the sum. In this paper we determine, for (Formula presented.) and (Formula presented.), the precise asymptotics of the sum of the (Formula presented.) largest terms of (Formula presented.) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.

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