eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
73:1
73:13
10.4230/LIPIcs.MFCS.2020.73
article
Randomness and Effective Dimension of Continued Fractions
Nandakumar, Satyadev
1
Vishnoi, Prateek
1
Computer Science and Engineering, Indian Institute of Technology Kanpur, India
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.73/LIPIcs.MFCS.2020.73.pdf
Continued fractions
Martin-Löf randomness
Computable randomness
effective Fractal dimension