Point-to-set Principle and Constructive Dimension Faithfulness

We introduce a constructive analogue of $\Phi$-dimension, a notion of Hausdorff dimension developed using a restricted class of coverings of a set. A class of coverings $\Phi$ is said to be "faithful" to Hausdorff dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every set. We prove a Point-to-Set Principle for $\Phi$-dimension, through which we get Point-to-Set Principles for Hausdorff Dimension, continued-fraction dimension and dimension of Cantor Coverings as special cases. 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. This condition yields two general classes of representations of reals, one whose constructive dimensions that are equivalent to the constructive Hausdorff dimensions, and another, whose effective dimensions are different from the effective Hausdorff dimensions, completely classifying Cantor series expansions of reals.