The arithmetic-geometric scaling spectrum for continued fractions

Johannes Jaerisch Fachbereich Mathematik, Universität Bremen Marc Kesseböhmer Fachbereich Mathematik, Universität Bremen

TBD mathscidoc:1701.333167

Arkiv for Matematik, 48, (2), 335-360, 2009.2
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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@inproceedings{johannes2009the,
  title={The arithmetic-geometric scaling spectrum for continued fractions},
  author={Johannes Jaerisch, and Marc Kesseböhmer},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203627015495976},
  booktitle={Arkiv for Matematik},
  volume={48},
  number={2},
  pages={335-360},
  year={2009},
}
Johannes Jaerisch, and Marc Kesseböhmer. The arithmetic-geometric scaling spectrum for continued fractions. 2009. Vol. 48. In Arkiv for Matematik. pp.335-360. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203627015495976.
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