Kolmogorov complexity and Hausdorff dimension
Information and Computation
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Gales suffice for constructive dimension
Information Processing Letters
The dimensions of individual strings and sequences
Information and Computation
The dimension of a point: computability meets fractal geometry
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
A correspondence principle for exact constructive dimension
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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We show that the classical Hausdorff and constructive dimensions of any union of 驴10-definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is 驴20-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.