Effective Hausdorff Dimension

We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans.