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Point-To-Set Principle and Constructive Dimension Faithfulness

Authors Satyadev Nandakumar , Subin Pulari , Akhil S

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Satyadev Nandakumar
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Subin Pulari
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Akhil S
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India

Acknowledgements

We gratefully acknowledge the anonymous reviewers for their invaluable insights and constructive feedback, which has significantly strengthened this manuscript.

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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) https://doi.org/10.4230/LIPIcs.MFCS.2024.76

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Mathematics of computing → Information theory
Keywords
  • Kolmogorov complexity
  • Constructive dimension
  • Faithfulness
  • Point to set principle
  • Continued fraction dimension
  • Cantor series expansion

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