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Thin sets and strict-two-associatedness Hare, Kathryn Elizabeth


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 strictly-2-associated. Actually we obtain the conclusion of strict-2-associatedness 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 best-possible. In the remainder of Chapter 3 we establish necessary and sufficient conditions for a conclusion stronger than, but similar to strict-2-associatedness. 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 strictly-2-associated 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

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