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The arithmetic-geometric scaling spectrum for continued fractions

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Arkiv för Matematik

Abstract

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given number which tends to infinity.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Bremen, Bibliothekstrasse 1, DE-28356, Bremen, Germany

    Johannes Jaerisch

  2. Fachbereich Mathematik, Universität Bremen, Bibliothekstrasse 1, DE-28356, Bremen, Germany

    Marc Kesseböhmer

Authors
  1. Johannes Jaerisch
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  2. Marc Kesseböhmer
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Correspondence to Marc Kesseböhmer.

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Jaerisch, J., Kesseböhmer, M. The arithmetic-geometric scaling spectrum for continued fractions. Ark Mat 48, 335–360 (2010). https://doi.org/10.1007/s11512-009-0102-8

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  • DOI: https://doi.org/10.1007/s11512-009-0102-8

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