Document Open Access Logo

Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals

Authors Satyadev Nandakumar , Akhil S , Prateek Vishnoi

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2023.70.pdf
  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Author Details

Satyadev Nandakumar
  • Department of Computer Science, Indian Institute of Technology Kanpur, India
Akhil S
  • Department of Computer Science, Indian Institute of Technology Kanpur, India
Prateek Vishnoi
  • Department of Computer Science, Indian Institute of Technology Kanpur, India

Acknowledgements

The authors would like to thank Subin Pulari for his comments and helpful discussions.

Cite As Get BibTex

Satyadev Nandakumar, Akhil S, and Prateek Vishnoi. Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.70

Abstract

We establish that constructive continued fraction dimension originally defined using s-gales [Nandakumar and Vishnoi, 2022] is robust, but surprisingly, that the effective continued fraction dimension and effective (base-b) Hausdorff dimension of the same real can be unequal in general.
We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable s-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest.
We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion [Hitchcock and Mayordomo, 2013]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [Nandakumar and Vishnoi, 2022]. In contrast, for any 0 < ε < 0.5, we prove the existence of a real whose effective Hausdorff dimension is less than ε, but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the "non-faithfulness" of certain families of covers, investigated by Peres and Torbin [Peres and Torbin] and by Albeverio, Ivanenko, Lebid and Torbin [Albeverio et al., 2020]. 
We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Mathematics of computing → Information theory
Keywords
  • Algorithmic information theory
  • Kolmogorov complexity
  • Continued fractions
  • Effective Hausdorff dimension

Metrics

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

References

  1. S. Albeverio, Ganna Ivanenko, Mykola Lebid, and Grygoriy Torbin. On the Hausdorff dimension faithfulness and the Cantor series expansion. Methods of Functional Analysis and Topology, 26(4):298-310, 2020. Google Scholar
  2. Sergio Albeverio, Yuri Kondratiev, Roman Nikiforov, and Grygoriy Torbin. On new fractal phenomena connected with infinite linear IFS. Math. Nachr., 290(8-9):1163-1176, 2017. URL: https://doi.org/10.1002/mana.201500471.
  3. 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
  4. Karma Dajani and Cor Kraaikamp. Ergodic Theory of Numbers. The Mathematical Association of America, 2002. Google Scholar
  5. Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010. URL: https://doi.org/10.1007/978-0-387-68441-3.
  6. M. Einsiedler and T. Ward. Ergodic Theory: with a view towards Number Theory. Graduate Texts in Mathematics. Springer London, 2010. URL: https://books.google.co.in/books?id=PiDET2fS7H4C.
  7. Kenneth Falconer. Fractal geometry. John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2003. Mathematical foundations and applications. URL: https://doi.org/10.1002/0470013850.
  8. John M. Hitchcock and Elvira Mayordomo. Base invariance of feasible dimension. Inform. Process. Lett., 113(14-16):546-551, 2013. URL: https://doi.org/10.1016/j.ipl.2013.04.004.
  9. Marius Iosifescu and Cor Kraaikamp. Metrical theory of continued fractions, volume 547 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2002. URL: https://doi.org/10.1007/978-94-015-9940-5.
  10. L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413-1416, 1973. Google Scholar
  11. Jack H. Lutz. Gales and the constructive dimension of individual sequences. In Automata, languages and programming (Geneva, 2000), volume 1853 of Lecture Notes in Comput. Sci., pages 902-913. Springer, Berlin, 2000. URL: https://doi.org/10.1007/3-540-45022-X_76.
  12. Jack H. Lutz. Dimension in complexity classes. SIAM J. Comput., 32(5):1236-1259, 2003. URL: https://doi.org/10.1137/S0097539701417723.
  13. Jack H. Lutz. The dimensions of individual strings and sequences. Inform. and Comput., 187(1):49-79, 2003. URL: https://doi.org/10.1016/S0890-5401(03)00187-1.
  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(6):602-619, 1966. Google Scholar
  16. Elvira Mayordomo. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inform. Process. Lett., 84(1):1-3, 2002. URL: https://doi.org/10.1016/S0020-0190(02)00343-5.
  17. Elvira Mayordomo. Effective dimension in some general metric spaces. In Benedikt Löwe and Glynn Winskel, editors, Proceedings 8th International Workshop on Developments in Computational Models, DCM 2012, Cambridge, United Kingdom, 17 June 2012, volume 143 of EPTCS, pages 67-75, 2012. URL: https://doi.org/10.4204/EPTCS.143.6.
  18. 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.
  19. 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
  20. Satyadev Nandakumar and Prateek Vishnoi. On continued fraction randomness and normality. Inform. and Comput., 285(part B):Paper No. 104876, 17, 2022. URL: https://doi.org/10.1016/j.ic.2022.104876.
  21. André Nies. Computability and randomness, volume 51. OUP Oxford, 2009. Google Scholar
  22. Yuval Peres and Gyorgiy Torbin. Continued fractions and dimensional gaps. In preparation. Google Scholar
  23. Adrian-Maria Scheerer. On the continued fraction expansion of absolutely normal numbers, 2017. URL: https://arxiv.org/abs/1701.07979.
  24. C. P. Schnorr. Zufälligkeit und Wahrscheinlichkeit. Springer-Verlag, Berlin, 1971. Google Scholar
  25. Ludwig Staiger. The Kolmogorov complexity of real numbers. Theor. Comput. Sci., 284(2):455-466, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00102-5.
  26. Joseph Vandehey. Absolutely abnormal and continued fraction normal numbers. Bulletin of the Australian Mathematical Society, 94(2):217-223, 2016. URL: https://doi.org/10.1017/S0004972716000101.
  27. Prateek Vishnoi. Algorithmic Information Theory & Continued Fractions. PhD thesis, Indian Institute of Technology Kanpur, 2023. Google Scholar
  28. Prateek Vishnoi. Normality, randomness and Kolmogorov complexity of continued fractions. In Theory and Applications of Models of Computation: 17th Annual Conference, TAMC 2022, Tianjin, China, September 16-18, 2022, Proceedings, pages 382-392. Springer, 2023. Google Scholar
  29. A. K. Zvonkin and L. A. Levin. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Surv., 25(6):83-124, 1970. URL: https://doi.org/10.1070/rm1970v025n06ABEH001269.
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