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A Generalized Hausdorff Dimension for Functions and Sets


Buck, Robert Jay (1968) A Generalized Hausdorff Dimension for Functions and Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/60CE-6945.


Let E be a compact subset of the n-dimensional unit cube, 1n, 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

dC(E) = sup(β:Hβ, C(E) > 0),

where Hβ, C is the outer measure

inf(Ʃm(Ci)β:UCi E, Ci 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 1n, whose closure intersects 1n - 1’n. For n = 2, the outer measure Hβ, C is adopted in place of the usual:

Inf(Ʃ(diam. (Ci))β: UCi E, Ci ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension dC(E).

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),

dC(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

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

dC(E) = sup (dC(μ):μ ϵ M(E)).

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

(*) {dB(F), dC(f)): f ϵ Ӻ}

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

dC(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C


∆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 dC1(f) = dC2(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 Ci (I = 1, 2).

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bohnenblust, Henri Frederic
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 April 1968
Funding AgencyGrant Number
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:11232015-082345589
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9284
Deposited On:25 Nov 2015 17:08
Last Modified:28 Mar 2024 22:05

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