A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:Hβ, C</sub>(E) > 0),</sub>

where H_{β, C} is the outer measure</sub>

inf(Ʃm(C<sub>i)^β:UC<sub>i Ↄ E, C<sub>i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:</sub>

Inf(Ʃ(diam. (C<sub>i))^β: UC<sub>i Ↄ E, C<sub>i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).</sub>

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)</sub>

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (dC</sub>(μ):μ ϵ M(E)).</sub>

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of dC</sub>(f) where f ϵ Ӻ. Specifically, the set of points in 11,</sub>

(*) {d_B(F), d_C(f)): f ϵ Ӻ}</sub>

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C</p></sub>

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = dC</sub>2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_{i (I = 1, 2).}