Citation
Buck, Robert Jay (1968) A Generalized Hausdorff Dimension for Functions and Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/60CE6945. https://resolver.caltech.edu/CaltechTHESIS:11232015082345589
Abstract
Let E be a compact subset of the ndimensional unit cube, 1_{n}, and let C be a collection of convex bodies, all of positive ndimensional 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:H_{β, C}(E) > 0),
where H_{β, C} is the outer measure
inf(Ʃm(C_{i})^{β}:UC_{i} Ↄ E, C_{i} ϵ C).
Only the one and twodimensional 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 leftopen 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:
Inf(Ʃ(diam. (C_{i}))^{β}: UC_{i} Ↄ E, C_{i} ϵ C),
for the purpose of studying the influence of the shape of the covering sets on the dimension d_{C}(E).
If E is a closed set in 1_{1}, let M(E) be the class of all nondecreasing 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)
where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that
d_{C}(E) = sup (d_{C}(μ):μ ϵ M(E)).
This notion of dimension is extended to a certain class Ӻ of subadditive 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 d_{C}(f) where f ϵ Ӻ. Specifically, the set of points in 1_{1},
(*) {d_{B}(F), d_{C}(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 doublystarred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1_{2}, doublystarred 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), nondecreasing in x and y, supported on 1_{2} 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
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_{C}_{1}(f) = d_{C}_{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).
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): 
 
Thesis Committee: 
 
Defense Date:  1 April 1968  
Funders: 
 
Record Number:  CaltechTHESIS:11232015082345589  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:11232015082345589  
DOI:  10.7907/60CE6945  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9284  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  25 Nov 2015 17:08  
Last Modified:  28 Mar 2024 22:05 
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