Kolmogorov Complexity and Hausdorff Dimension

Kolmogorov Complexity and Hausdorff Dimension

A Kolmogorov complexity characterization of constructive Hausdorff dimension

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– Löf randomness. All Martin–Löf random sequences have constructive dimension 1, while in the case of ...

Constructive dimension equals Kolmogorov complexity

We derive the coincidence of Lutz’s constructive dimension and Kolmogorov complexity for sets of infinite strings from Levin’s early result on the existence of an optimal left computable cylindrical semi-measure M via simple calculations.

Historic set carries full hausdorff dimension

‎We prove that the historic set for ratio‎ ‎of Birkhoff average is either empty or full of Hausdorff dimension in a class of one dimensional‎ ‎non-uniformly hyperbolic dynamical systems.

Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws

We apply results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α, > 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y| = Ω(|x|), with K(y) > (1− )|y|. This result holds for both unbounded and space-bounded Kolmogorov complexity. We use the extraction procedure for ...