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.

翻译：我们提出了$\Phi$-维数的一种构造性类比，这是一种利用集合的有界覆盖类发展的豪斯多夫维数概念。若对于任意集合，$\Phi$-维数与豪斯多夫维数均相等，则称此类覆盖$\Phi$对豪斯多夫维数是“忠实的”。我们证明了$\Phi$-维数的点集原理，并通过该原理得到了豪斯多夫维数、连分数维数及康托尔覆盖维数的点集原理作为特例。利用康托尔覆盖的点集原理和构建满足特定柯尔莫哥洛夫复杂性条件序列的新技术，我们证明了豪斯多夫层面与构造层面上的康托尔覆盖忠实性概念等价。我们改编了Albeverio、Ivanenko、Lebid和Torbin的结果，推导出由康托尔级数展开生成的覆盖的构造维数忠实性的充要条件。该条件给出了实数的两类推广表示：一类实数的构造维数等价于构造豪斯多夫维数，另一类实数的有效维数不同于有效豪斯多夫维数，从而完整分类了实数的康托尔级数展开。