Classical fractal dimensions have recently been effectivized by characterizing them in terms of real-valued functions called gales, and imposing computability and complexity constraints on these gales. This paper surveys these developments and their applications in algorithmic information theory and computational complexity theory.

A notion of resource‐bounded Baire category is developed for the classPC[0,1]of all polynomial‐time computable real‐valued functions on the unit interval. The meager subsets ofPC[0,1]are characterized in terms of resource‐bounded Banach‐Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function inPC[0,1]is nowhere differentiable. This is a complexity‐theoretic extension of the analogous classical result that Banach proved for the classC[0, 1] in 1931. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

This paper initiates the study of sets in Euclidean spaces ℝn that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, n ], we are interested in the connectivity properties of the set DIMI, consisting of all points in ℝn whose dimensions lie in I, and of its dual DIMIstr, consisting of all points whose strong dimensions lie in I. If I is [0, 1) or.