MinkowskiDimension

Minkowski dimension, also known as the box-counting dimension, is a way to assign a nonnegative real number to a bounded subset A of Euclidean space that describes how the detail of A scales as one zooms in. Let ε > 0. Define N(A, ε) as the minimal number of sets of diameter at most ε required to cover A (equivalently, the number of ε-sized boxes in a grid needed to cover A). The Minkowski dimension of A, denoted dim_M(A), is defined as lim_{ε→0} log N(A, ε) / log(1/ε) when the limit exists. If the limit does not exist, the upper and lower Minkowski dimensions are defined by limsup and liminf respectively.

An equivalent formulation uses ε-neighborhoods. For A ⊂ R^d, let A_ε = {x : dist(x, A) ≤ ε}. If Vol(A_ε) ~ C ε^{d − D} as ε → 0, then D is the Minkowski dimension (under suitable conditions).

Relationship to other dimensions: For bounded A ⊂ R^d, the Hausdorff dimension satisfies dim_H(A) ≤ dim_M(A). In general they can differ; the Minkowski dimension can be easier to compute for many sets and provides an upper scaling exponent for coverings.

Examples: The Cantor middle-thirds set has dim_M = dim_H = log 2 / log 3 ≈ 0.6309. The Sierpinski gasket has dim_M = dim_H = log 3 / log 2 ≈ 1.585. For many regular geometric objects, dim_M equals the topological dimension. Some irregular sets may have dim_M strictly larger than dim_H.

Applications: Minkowski dimension is used in fractal geometry as a practical descriptor of rough or fragmented sets, in image analysis, and in the study of dynamical systems and attractors where simple scaling laws govern the growth of coverings or neighborhoods.