Document Open Access Logo

Randomness and Effective Dimension of Continued Fractions

Authors Satyadev Nandakumar, Prateek Vishnoi

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2020.73.pdf
  • Filesize: 474 kB
  • 13 pages

Document Identifiers

Author Details

Satyadev Nandakumar
  • Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Prateek Vishnoi
  • Computer Science and Engineering, Indian Institute of Technology Kanpur, India

Acknowledgements

The authors wish to thank Subin Pulari and Yann Bugeaud for helpful discussions, and anonymous reviewers for their suggestions.

Cite As Get BibTex

Satyadev Nandakumar and Prateek Vishnoi. Randomness and Effective Dimension of Continued Fractions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 73:1-73:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.73

Abstract

Recently, Scheerer [Adrian-Maria Scheerer, 2017] and Vandehey [Vandehey, 2016] showed that normality for continued fraction expansions and base-b expansions are incomparable notions. This shows that at some level, randomness for continued fractions and binary expansion are different statistical concepts. In contrast, we show that the continued fraction expansion of a real is computably random if and only if its binary expansion is computably random.
To quantify the degree to which a continued fraction fails to be effectively random, we define the effective Hausdorff dimension of individual continued fractions, explicitly constructing continued fractions with dimension 0 and 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Computability
Keywords
  • Continued fractions
  • Martin-Löf randomness
  • Computable randomness
  • effective Fractal dimension

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. G. J. Chaitin. A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery, 22:329-340, 1975. Google Scholar
  2. K. Dajani and C. Kraaikamp. Ergodic Theory of Numbers. Mathematical Association of America, 2004. Google Scholar
  3. Rodney G. Downey and Denis H. Hirschfeldt. Algorithmic Randomness and Complexity. Springer, 2010. Google Scholar
  4. Thomas C. Einsiedler, Manfred \and Ward. Ergodic Theory: With a View Towards Number Theory. Springer, 2010. Google Scholar
  5. Henning Fernau and Ludwig Staiger. Iterated function systems and control languages. Inf. Comput., 168(2):125-143, 2001. URL: https://doi.org/10.1006/inco.2000.2912.
  6. P. Gács. On the symmetry of algorithmic information. Soviet Mathematics Doklady, 15:1477, 1974. Google Scholar
  7. F. Hausdorff. Dimension und äusseres Mass. Mathematische Annalen, 79:157-179, 1919. Google Scholar
  8. A. Ya. Khinchin. Continued Fractions. Dover Publications, 1997. Google Scholar
  9. A. N. Kolmogorov. Logical basis for information theory and probability theory. IEEE Transactions on Information Theory, IT-14:662-664, 1968. Google Scholar
  10. L. A. Levin. Laws of information conservation (nongrowth) and aspects of the foundation of probability theory. Problems of Information Transmission, 10:206-210, 1974. Google Scholar
  11. Ming Li and P Vitanyi. An Introduction to Kolmogorov Complexity and its Applications. Springer, 4 edition, 2019. Google Scholar
  12. J. H. Lutz. Gales and the constructive dimension of individual sequences. In Proceedings of the 27th International Colloquium on Automata, Languages, and Programming, pages 902-913, 2000. Revised as [J. H. Lutz, 2003]. Google Scholar
  13. J. H. Lutz. The dimensions of individual strings and sequences. Information and Computation, 187:49-79, 2003. URL: http://www.cs.iastate.edu/~lutz/=PAPERS/diss.ps.
  14. Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similar fractals. SIAM Journal on Computing, 38:1080-1112, 2008. Google Scholar
  15. P. Martin-Löf. The definition of random sequences. Information and Control, 9:602-619, 1966. Google Scholar
  16. Elvira Mayordomo. Effective hausdorff dimension in general metric spaces. Theory Comput. Syst., 62(7):1620-1636, 2018. URL: https://doi.org/10.1007/s00224-018-9848-3.
  17. S. Nandakumar. An effective ergodic theorem and some applications. In Proceedings of the 40th Annual Symposium on the Theory of Computing, pages 39-44, 2008. Google Scholar
  18. André Nies. Computability and Randomness. Oxford University Press, Oxford, U.K., 2009. Google Scholar
  19. Adrian-Maria Scheerer. On the continued fraction expansion of absolutely normal numbers. CoRR, 2017. Google Scholar
  20. C. P. Schnorr. A unified approach to the definition of random sequences. Mathematical Systems Theory, 5:246-258, 1971. Google Scholar
  21. Joseph Vandehey. Absolutely abnormal and continued fraction normal numbers. Bulletin of the Australian Mathematical Society, 94(2), 2016. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact