Hausdorff $Φ$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $Φ$-dimension, which we call constructive $Φ$-dimension. 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. We 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. 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. Hence we show the necessary and sufficient conditions for Hausdorff dimension faithfulness of Cantor coverings, thereby giving an information theoretic proof of the result by Albeverio, Ivanenko, Lebid, and Torbin.

翻译：豪斯多夫Φ-维数是一种利用集合覆盖的限制类别所定义的豪斯多夫维数概念。我们引入了一种有效的豪斯多夫Φ-维数版本，称之为构造性Φ-维数。我们证明了Φ-维数的点集原理，并从中得到豪斯多夫维数、连分数维数和康托尔覆盖维数的点集原理作为特例。我们还给出了构造性Φ-维数的柯尔莫哥洛夫复杂性刻画。如果对于每个集合，Φ-维数与豪斯多夫维数一致，则称覆盖集类Φ对豪斯多夫维数是“保真的”。类似地，如果对于每个集合，构造性Φ-维数与构造性维数一致，则称Φ对构造性维数是“保真的”。基于康托尔级数展开的项，我们推导了该展开所生成覆盖的构造性维数保真性的充分必要条件。利用康托尔覆盖的点集原理以及构造满足特定柯尔莫哥洛夫复杂性条件的序列的新技术，我们证明了康托尔覆盖在豪斯多夫层次和构造性层次上的“保真性”概念是等价的。因此，我们给出了康托尔覆盖的豪斯多夫维数保真性的充分必要条件，从而为Albeverio、Ivanenko、Lebid和Torbin的结果提供了信息论证明。