A Generalized Hausdorff Dimension for Functions and Sets
<p>Let E be a compact subset of the n-dimensional unit cube, 1<sub>n</sub>, 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...
| Summary: | <p>Let E be a compact subset of the n-dimensional unit cube, 1<sub>n</sub>, 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</p>
<p>d<sub>C</sub>(E) = sup(β:H<sub>β, C</sub>(E) > 0),</p>
<p>where H<sub>β, C</sub> is the outer measure </p>
<p>inf(Ʃm(C<sub>i</sub>)<sup>β</sup>:UC<sub>i</sub> <u>Ↄ</u> E, C<sub>i</sub> ϵ C).</p>
<p>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’<sub>n</sub>, of 1<sub>n</sub>, whose closure intersects 1<sub>n</sub> - 1’<sub>n</sub>. For n = 2, the outer measure H<sub>β, C</sub> is adopted in place of the usual:</p>
<p>Inf(Ʃ(diam. (C<sub>i</sub>))<sup>β</sup>: UC<sub>i</sub> <u>Ↄ</u> E, C<sub>i</sub> ϵ C), </p>
<p>for the purpose of studying the influence of the shape of the covering sets on the dimension d<sub>C</sub>(E).</p>
<p>If E is a closed set in 1<sub>1</sub>, 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),</p>
<p>d<sub>C</sub>(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)</p>
<p>where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that</p>
<p>d<sub>C</sub>(E) = sup (d<sub>C</sub>(μ):μ ϵ M(E)).</p>
<p>This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d<sub>C</sub>(E) as a function of the covering class C is reduced to the study of d<sub>C</sub>(f) where f ϵ Ӻ. Specifically, the set of points in 1<sub>1</sub>,</p>
<p>(*) {d<sub>B</sub>(F), d<sub>C</sub>(f)): f ϵ Ӻ}</p>
<p>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 1<sub>2</sub>, 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.</p>
<p>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 1<sub>2</sub> with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula</p>
<p>d<sub>C</sub>(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C</p>
<p>where</p>
<p>∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).</p>
<p>A characterization of the equivalence d<sub>C</sub><sub>1</sub>(f) = d<sub>C</sub><sub>2</sub>(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<sub>i</sub> (I = 1, 2). </p>
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