2026-07-23T04:54:45Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20632 2024-08-23T05:50:30Z ddc:004 open_access

Point-To-Set Principle and Constructive Dimension Faithfulness Nandakumar, Satyadev Pulari, Subin S, Akhil Kolmogorov complexity Constructive dimension Faithfulness Point to set principle Continued fraction dimension Cantor series expansion 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Satyadev Nandakumar and Subin Pulari and Akhil S 2024 Is Part Of LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2024.76 urn:nbn:de:0030-drops-206321 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.76 eng https://creativecommons.org/licenses/by/4.0/legalcode