Conceitos Básicos
Constructive and Hausdorff dimension faithfulness for Cantor coverings are equivalent.
Resumo
The content introduces the concept of constructive Φ-dimension, focusing on Cantor coverings. It explores the equivalence between faithfulness at the constructive and Hausdorff levels. The Point-to-Set Principle is applied to Cantor coverings, showing that if a family of covers is faithful with respect to constructive dimension, it is also faithful with respect to Hausdorff dimension. Kolmogorov complexity characterizations are provided for Cantor series dimensions. The comprehensive analysis delves into the relationship between different dimensions in mathematical spaces.
Estatísticas
cdimΦ(x) = lim inf K(X ↿ mk) / mk (Theorem 8)
|KA(X ↿ n) - KB(Y ↿ n)| = o(n) (Lemma 7)
dimΦ(F) = min A sup X cdimA Φ(X) (Theorem 7)