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 Thin sets and stricttwoassociatedness
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Thin sets and stricttwoassociatedness Hare, Kathryn Elizabeth
Abstract
Let G be a compact, abelian group and let E be a subset of its discrete, abelian, dual group Ĝ. E is said to be a ∧(p) set if for some r < p there is a constant c(r) so that ‖ƒ‖ [sub p] ≤ c(r) ‖ƒ‖ [sub r] whenever the support of ƒ, the Fourier transform of ƒ, is a finite subset of E. The main result of this thesis, Theorem 3.5, is that if E is a ∧(p) set, p > 2, and E satisfies a necessary technical condition, then for each S⊂Gof positive measure there is a constant c(S, E) > 0 so that [See Thesis for Equation] whenever ƒ ∈ L² (G) and support ƒ ⊂ E. When such an inequality holds for all ƒ ∈ L² (G) with support f ⊂ E, then E and S are said to be strictly2associated. Actually we obtain the conclusion of strict2associatedness for a possibly larger class of sets than ∧(p) sets, p > 2, so that our theorem improves upon previously known work even when G is the circle group and E ⊂ Z. Most of Chapter 3 is dedicated to proving this result and showing that it is almost bestpossible. In the remainder of Chapter 3 we establish necessary and sufficient conditions for a conclusion stronger than, but similar to strict2associatedness. In Chapter 4 we prove that if E is any ∧(p) set, p > 0, (or any set with the same arithmetic structure as ∧(p) sets) and if E satisfies the same necessary technical condition as in Theorem 3.5, then E is strictly2associated with all open subsets oiG. The proofs of these theorems depend on the arithmetic structure of ∧(p) sets. This topic is discussed in detail in Chapter 2. It has long been known that ∧(p) sets in Z with p > 2, cannot contain arbitrarily long arithmetic progressions and have "uniformly large gaps". We prove that no ∧(p) set, p > 0, can contain arbitrarily large parallelepipeds, a generalization of arithmetic progressions. This is new for ∧(p) sets, p < 1, in groups other than the circle. We introduce a definition which extends the notion of "uniformly large gaps" to the general setting. Combinatorial arguments are used to prove that sets which do not contain arbitrarily large parallelepipeds have this property. Finally, parallelepipeds are used to show that ∧(p) sets are built up from finite sets in a controlled way. This last fact and the notion of "uniform
Item Metadata
Title 
Thin sets and stricttwoassociatedness

Creator  
Publisher 
University of British Columbia

Date Issued 
1986

Description 
Let G be a compact, abelian group and let E be a subset of its discrete, abelian, dual group Ĝ.
E is said to be a ∧(p) set if for some r < p there is a constant c(r) so that
‖ƒ‖ [sub p] ≤ c(r) ‖ƒ‖ [sub r]
whenever the support of ƒ, the Fourier transform of ƒ, is a finite subset of E.
The main result of this thesis, Theorem 3.5, is that if E is a ∧(p) set, p > 2, and E satisfies a necessary technical condition, then for each S⊂Gof positive measure there is a constant c(S, E) > 0 so that
[See Thesis for Equation]
whenever ƒ ∈ L² (G) and support ƒ ⊂ E. When such an inequality holds for all
ƒ ∈ L² (G) with support f ⊂ E, then E and S are said to be strictly2associated.
Actually we obtain the conclusion of strict2associatedness for a possibly larger
class of sets than ∧(p) sets, p > 2, so that our theorem improves upon previously
known work even when G is the circle group and E ⊂ Z. Most of Chapter 3 is
dedicated to proving this result and showing that it is almost bestpossible. In
the remainder of Chapter 3 we establish necessary and sufficient conditions for a conclusion stronger than, but similar to strict2associatedness.
In Chapter 4 we prove that if E is any ∧(p) set, p > 0, (or any set with the same arithmetic structure as ∧(p) sets) and if E satisfies the same necessary technical condition as in Theorem 3.5, then E is strictly2associated with all open subsets oiG.
The proofs of these theorems depend on the arithmetic structure of ∧(p) sets. This topic is discussed in detail in Chapter 2.
It has long been known that ∧(p) sets in Z with p > 2, cannot contain arbitrarily long arithmetic progressions and have "uniformly large gaps". We prove that no ∧(p) set, p > 0, can contain arbitrarily large parallelepipeds, a generalization of arithmetic progressions. This is new for ∧(p) sets, p < 1, in groups other than the circle.
We introduce a definition which extends the notion of "uniformly large gaps" to the general setting. Combinatorial arguments are used to prove that sets which do not contain arbitrarily large parallelepipeds have this property.
Finally, parallelepipeds are used to show that ∧(p) sets are built up from finite sets in a controlled way. This last fact and the notion of "uniform

Genre  
Type  
Language 
eng

Date Available 
20100806

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080427

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.