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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Hausdorff Φ-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of Φ-dimension using the notion of constructive Φ-s-supergales. We prove a Point-to-Set Principle for Φ-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive Φ-dimension.
A class of covering sets Φ is said to be "faithful" to Hausdorff dimension if the Φ-dimension and Hausdorff dimension coincide for every set. Similarly, Φ is said to be "faithful" to constructive dimension if the constructive Φ-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of "faithfulness" of Cantor coverings at the Hausdorff and constructive levels are equivalent.
We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin [Albeverio et al., 2020] to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion.

Satyadev Nandakumar, Subin Pulari, and Akhil S. Point-To-Set Principle and Constructive Dimension Faithfulness. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2024.76, author = {Nandakumar, Satyadev and Pulari, Subin and S, Akhil}, title = {{Point-To-Set Principle and Constructive Dimension Faithfulness}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {76:1--76:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.76}, URN = {urn:nbn:de:0030-drops-206321}, doi = {10.4230/LIPIcs.MFCS.2024.76}, annote = {Keywords: Kolmogorov complexity, Constructive dimension, Faithfulness, Point to set principle, Continued fraction dimension, Cantor series expansion} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

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.

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)

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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2023.70, author = {Nandakumar, Satyadev and S, Akhil and Vishnoi, Prateek}, title = {{Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {70:1--70:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.70}, URN = {urn:nbn:de:0030-drops-186041}, doi = {10.4230/LIPIcs.MFCS.2023.70}, annote = {Keywords: Algorithmic information theory, Kolmogorov complexity, Continued fractions, Effective Hausdorff dimension} }

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