Correspondence principles for effective dimensions

We show that the classical Hausdorff and constructive dimensions of any union of Π0 1-definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is Σ0 2-definable, then the computable dimension also equals the Hausdorff dimension. This second result is implicit in the work of Staiger (1998). Staiger also proved related results using entropy rates of decidable languages. We show that Staiger’s computable entropy rate provides an equivalent definition of computable dimension. We also prove that a constructive version of Staiger’s entropy rate coincides with constructive dimension.