Randomness and Effective Dimension of Continued Fractions
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.
Continued fractions
Martin-Löf randomness
Computable randomness
effective Fractal dimension
Theory of computation~Constructive mathematics
Theory of computation~Computability
73:1-73:13
Regular Paper
The authors wish to thank Subin Pulari and Yann Bugeaud for helpful discussions, and anonymous reviewers for their suggestions.
Satyadev
Nandakumar
Satyadev Nandakumar
Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Prateek
Vishnoi
Prateek Vishnoi
Computer Science and Engineering, Indian Institute of Technology Kanpur, India
10.4230/LIPIcs.MFCS.2020.73
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Satyadev Nandakumar and Prateek Vishnoi
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