Arkiv för Matematik

  • Ark. Mat.
  • Volume 48, Number 2 (2010), 335-360.

The arithmetic-geometric scaling spectrum for continued fractions

Johannes Jaerisch and Marc Kesseböhmer

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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.

Article information

Source
Ark. Mat. Volume 48, Number 2 (2010), 335-360.

Dates
Received: 25 February 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907110

Digital Object Identifier
doi:10.1007/s11512-009-0102-8

Mathematical Reviews number (MathSciNet)
MR2672614

Zentralblatt MATH identifier
1261.11055

Rights
2009 © Institut Mittag-Leffler

Citation

Jaerisch, Johannes; Kesseböhmer, Marc. The arithmetic-geometric scaling spectrum for continued fractions. Ark. Mat. 48 (2010), no. 2, 335--360. doi:10.1007/s11512-009-0102-8. https://projecteuclid.org/euclid.afm/1485907110.


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