Point-to-set Principle and Constructive Dimension Faithfulness
Satyadev Nandakumar, Subin Pulari, Akhil S
TL;DR
The paper develops a constructive analogue of Φ-dimension using Φ-s-supergales and proves a Point-to-Set Principle for Φ-dimension, enabling new connections between constructive and classical dimensions across Hausdorff, continued-fraction, and Cantor-covering contexts. It provides a Kolmogorov complexity characterization and a constructive framework that equates faithfulness of Cantor coverings at constructive and Hausdorff levels, via a novel Kolmogorov construction and a log-limit criterion. The Cantor series setting yields necessary and sufficient conditions for constructive faithfulness, linking representation-based invariance to information-theoretic density, and extending base representation results to a broad Cantor-series regime. Together, these results forge a bridge between geometric dimension theory and algorithmic information theory, with implications for faithfulness questions and potential applications to randomness representations.
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 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.