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Randomness and Effective Dimension of Continued Fractions

Authors Satyadev Nandakumar, Prateek Vishnoi

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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 AsGet 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

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