A Kolmogorov complexity characterization of constructive Hausdorff dimension

Abstract

Lutz [7] has recently developed a constructive version of Hausdorff dimension, using it to assign to every sequence A ∈ C a constructive dimension dim(A) ∈ [0,1]. Classical Hausdorff dimension [3] is an augmentation of Lebesgue measure, and in the same way constructive dimension augments Martin– Lof randomness. All Martin–Lof random sequences have constructive dimension 1, while in the case of non-random sequences a finer distinction is obtained. Martin–Lof randomness has a useful interpretation in terms of information content, since a sequence A is random if and only if there is a constant c such that