Effective dimension

In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an algorithmically random point on a line, which has Hausdorff dimension 0 (since it is a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1).

Rigorous definitions

This article will define effective dimension for subsets of Cantor space 2ω; closely related definitions exist for subsets of Euclidean space Rn.