Kolgomorov Complexity and Hausdorff Dimension

Abstract In this paper various relationships between the Kolmogorov complexity of infinite strings and measures of information content are given. The general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the Hausdorff dimension or the entropy of that set. It turns out that Hausdorff dimension yields lower bounds to the Kolmogorov complexity, whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds. More detailed investigations result in a generalization of P. Martin-Lof′s theorems on the Kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings.