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Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals

Authors Satyadev Nandakumar , Akhil S , Prateek Vishnoi

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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 AsGet 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

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