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Limit theorems for counting large continued fraction digits

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Abstract

We establish a central limit theorem for counting large continued fraction digits (an), that is, we count occurrences {an>bn}, where (bn) is a sequence of positive integers. Our result improves a similar result by Philipp, which additionally assumes that bn tends to infinity. Moreover, we give a refinement of the famous Borel–Bernstein theorem for continued fractions regarding the event that the nth continued fraction digit lies infinitely often between dn and dn(1 + 1/cn) for given sequences (cn) and (dn). Also, for these sets, we obtain a central limit theorem. As an interesting side result, we explicitly determine the first φ-mixing coefficient for the Gauss system.

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Authors and Affiliations

  1. Universität Bremen, Fachbereich 3 – Mathematik und Informatik, Bibliothekstr. 5, 28359, Bremen, Germany

    Marc Kesseböhmer

  2. Centro di Ricerca Ennio De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126, Pisa, PI, Italy

    Tanja I. Schindler

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  1. Marc Kesseböhmer
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  2. Tanja I. Schindler
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Correspondence to Marc Kesseböhmer.

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This research was supported by the German Research Foundation (DFG) grant “Renewal Theory and Statistics of Rare Events in Infinite Ergodic Theory” (Geschäftszeichen KE 1440/2-1). The research of TS was conducted at University of Bremen and at the Australian National University. While at University of Bremen TS was supported by the Studienstiftung des Deutschen Volkes.

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Kesseböhmer, M., Schindler, T.I. Limit theorems for counting large continued fraction digits. Lith Math J 60, 189–207 (2020). https://doi.org/10.1007/s10986-020-09479-5

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