Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of $\Phi$-dimension using the notion of constructive $\Phi$-$s$-supergales. 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. We also provide a Kolmogorov complexity characterization of constructive $\Phi$-dimension. A class of covering sets $\Phi$ is said to be "faithful" to Hausdorff dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every set. Similarly, $\Phi$ is said to be "faithful" to constructive dimension if the constructive $\Phi$-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.

翻译：Hausdorff $\Phi$-维数是利用集合的一类受限覆盖发展起来的Hausdorff维数概念。我们利用构造性 $\Phi$-$s$-supergale 的概念，引入了 $\Phi$-维数的一个构造性类比。我们证明了 $\Phi$-维数的点对集原理，并由此得到了Hausdorff维数、连分数维数以及Cantor覆盖维数的点对集原理作为特例。我们还提供了构造性 $\Phi$-维数的一个Kolmogorov复杂性刻画。如果对于每个集合，其 $\Phi$-维数与Hausdorff维数均一致，则称覆盖集类 $\Phi$ 对Hausdorff维数是“忠实”的。类似地，如果对于每个集合，其构造性 $\Phi$-维数与构造性维数均一致，则称 $\Phi$ 对构造性维数是“忠实”的。利用Cantor覆盖的点对集原理以及一种用于构造满足特定Kolmogorov复杂性条件的序列的新技术，我们证明了Cantor覆盖在Hausdorff层面和构造性层面的“忠实性”概念是等价的。我们借鉴Albeverio、Ivanenko、Lebid和Torbin的结果，基于Cantor级数展开的项，推导出由该展开生成的覆盖具有构造性维数忠实性的充分必要条件。