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ACM Transactions on Computation Theory
Article . 2025 . Peer-reviewed
Data sources: Crossref
https://dx.doi.org/10.48550/ar...
Article . 2023
License: CC BY
Data sources: Datacite
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Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals

Authors: Satyadev Nandakumar; Akhil S; Prateek Vishnoi;

Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals

Abstract

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] 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. 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 effective dimension is invariant with respect to base conversion [ 8 ]. 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 [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) 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 [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

Country
Germany
Keywords

FOS: Computer and information sciences, 03D32, 68P30, 11A55, Effective Hausdorff dimension, Continued fractions, Computer Science - Information Theory, Information Theory (cs.IT), Kolmogorov complexity, Algorithmic information theory, 510, 004, ddc: ddc:004

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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