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

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2009 © Institut Mittag-Leffler


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.

Export citation


  • Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, Springer, Berlin–Heidelberg, 1975.
  • Cusick, T. W., Hausdorff dimension of sets of continued fractions, Q. J. Math. 41 (1990), 277–286.
  • Falconer, K., Fractal Geometry, 2nd ed., Mathematical Foundations and Applications, Wiley, Hoboken, NJ, 2003.
  • Fan, A.-H., Liao, L.-M., Wang, B.-W. and Wu, J., On Khintchin exponents and Lyapunov exponents of continued fractions, Ergodic Theory Dynam. Systems 29 (2009), 73–109.
  • Good, I. J., The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941), 199–228.
  • Hensley, D., Continued fraction Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory 40 (1992), 336–358.
  • Hirst, K. E., A problem in the fractional dimension theory of continued fractions, Q. J. Math. 21 (1970), 29–35.
  • Hirst, K. E., Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973), 221–227.
  • Jarník, V., On the metric theory of Diophantine approximations [Przyczynek do metrycznej teorji przyblizeń diofantowych], Prace Mat.-Fiz. 36 (1929), 91–106 (Polish).
  • Kesseböhmer, M. and Stratmann, B. O., A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math. 605 (2007), 133–163.
  • Kesseböhmer, M. and Stratmann, B. O., Homology at infinity; fractal geometry of limiting symbols for modular subgroups, Topology 46 (2007), 469–491.
  • Kesseböhmer, M. and Zhu, S., Dimension sets for infinite IFSs: the Texan conjecture, J. Number Theory 116 (2006), 230–246.
  • Khinchin, A., Continued Fractions, 4th ed., Nauka, Moscow, 1978 (Russian). English transl. of 3rd. ed.: Univ. of Chicago Press, Chicago–London, 1964.
  • Kuzĭmin, R., Sur un problème de Gauss, C. R. Acad. Sci. URSS 1928 (1928), 375–380.
  • Mauldin, R. D. and Urbański, M., Graph Directed Markov Systems, Cambridge Tracts in Mathematics 148, Cambridge University Press, Cambridge, 2003.
  • Ramharter, G., Eine Bemerkung über gewisse Nullmengen von Kettenbrüchen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 28 (1986), 11–15.
  • Ramharter, G., On the fractional dimension theory of a class of expansions, Q. J. Math. 45 (1994), 91–102.
  • Rockafellar, R. T., Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton, 1970.
  • Walters, P., An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer, New York, 1982.
  • Wirsing, E., On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces, Acta Arith. 24 (1973/74), 507–528.