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Algebraic numbers and harmonic analysis in the pseries case Aubertin, Bruce Lyndon
Abstract
For the case of compact groups G = Π∞ j=l Z(p)j which are direct products of countably many copies of a cyclic group of prime order p, links are established between the theories of uniqueness and spectral synthesis on the one hand, and the theory of algebraic numbers on the other, similar to the wellknown results of Salem, Meyer et al on the circle. Let p ≥ 2 be a prime and let k{x⁻¹} denote the pseries field of formal Laurent series z = Σhj=₋∞ ajxj with coefficients in the field k = {0, 1,…, p1} and the integer h arbitrary. Let L(z) =  ∞ if aj = 0 for all j; otherwise let L(z) be the largest index h for which ah ≠ 0. We examine compact sets of the form [Algebraic equation omitted] where θ ε k{x⁻¹}, L(θ) > 0, and I is a finite subset of k[x]. If θ is a Pisot or Salem element of k{x⁻¹}, then E(θ,I) is always a set of strong synthesis. In the case that θ is a Pisot element, more can be proved, including a version of Bochner's property leading to a sharper statement of synthesis, provided certain assumptions are made on I (e.g., I ⊃ {0,1,x,...,xL(θ)1}). Let G be the compact subgroup of k{x⁻¹} given by G = {z: L(z) < 0}. Let θ ɛ k{x⁻¹}, L(θ) > 0, and suppose L(θ) > 1 if p = 3 and L(θ) > 2 if p = 2. Let I = {0,1,x,...,x²L(θ)1}. Then E = θ⁻¹Ε(θ,I) is a perfect subset of G of Haar measure 0, and E is a set of uniqueness for G precisely when θ is a Pisot or Salem element. Some byways are explored along the way. The exact analogue of Rajchman's theorem on the circle, concerning the formal multiplication of series, is obtained; this is new, even for p = 2. Other examples are given of perfect sets of uniqueness, of sets satisfying the Herz criterion for synthesis, and sets of multiplicity, including a class of Msets of measure 0 defined via Riesz products which are residual in G. In addition, a class of perfect M₀sets of measure 0 is introduced with the purpose of settling a question left open by W.R. Wade and K. Yoneda, Uniqueness and quasimeasures on the group of integers of a pseries field, Proc. A.M.S. 84 (1982), 202206. They showed that if S is a character series on G with the property that some subsequence {SpNj} of the pnth partial sums is everywhere pointwise bounded on G, then S must be the zero series if SpNj → 0 a.e.. We obtain a strong complement to this result by establishing that series S on G exist for which Sn → 0 everywhere outside a perfect set of measure 0, and for which sup SpN becomes unbounded arbitrarily slowly.
Item Metadata
Title 
Algebraic numbers and harmonic analysis in the pseries case

Creator  
Publisher 
University of British Columbia

Date Issued 
1986

Description 
For the case of compact groups G = Π∞ j=l Z(p)j which are direct products
of countably many copies of a cyclic group of prime order p, links are established between the theories of uniqueness and spectral synthesis on the one hand, and the theory of algebraic numbers on the other, similar to the wellknown results of Salem, Meyer et al on the circle.
Let p ≥ 2 be a prime and let k{x⁻¹} denote the pseries field of
formal Laurent series z = Σhj=₋∞ ajxj with coefficients in the field
k = {0, 1,…, p1} and the integer h arbitrary. Let L(z) =  ∞ if aj = 0 for all j; otherwise let L(z) be the largest index h for which ah ≠ 0. We examine compact sets of the form
[Algebraic equation omitted]
where θ ε k{x⁻¹}, L(θ) > 0, and I is a finite subset of k[x].
If θ is a Pisot or Salem element of k{x⁻¹}, then E(θ,I) is always a set of strong synthesis. In the case that θ is a Pisot element, more can be proved, including a version of Bochner's property leading to a sharper statement of synthesis, provided certain assumptions are made on I (e.g., I ⊃ {0,1,x,...,xL(θ)1}).
Let G be the compact subgroup of k{x⁻¹} given by G = {z: L(z) < 0}.
Let θ ɛ k{x⁻¹}, L(θ) > 0, and suppose L(θ) > 1 if p = 3 and L(θ) > 2 if p = 2. Let I = {0,1,x,...,x²L(θ)1}. Then E = θ⁻¹Ε(θ,I) is a perfect subset of G of Haar measure 0, and E is a set of uniqueness
for G precisely when θ is a Pisot or Salem element.
Some byways are explored along the way. The exact analogue of Rajchman's theorem on the circle, concerning the formal multiplication of series, is obtained; this is new, even for p = 2. Other examples are given of perfect sets of uniqueness, of sets satisfying the Herz criterion
for synthesis, and sets of multiplicity, including a class of Msets of measure 0 defined via Riesz products which are residual in G.
In addition, a class of perfect M₀sets of measure 0 is introduced
with the purpose of settling a question left open by W.R. Wade and
K. Yoneda, Uniqueness and quasimeasures on the group of integers of a
pseries field, Proc. A.M.S. 84 (1982), 202206. They showed that if
S is a character series on G with the property that some subsequence
{SpNj} of the pnth partial sums is everywhere pointwise bounded on G,
then S must be the zero series if SpNj → 0 a.e.. We obtain a strong
complement to this result by establishing that series S on G exist for
which Sn → 0 everywhere outside a perfect set of measure 0, and for
which sup SpN becomes unbounded arbitrarily slowly.

Genre  
Type  
Language 
eng

Date Available 
20101204

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 
10.14288/1.0080435

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
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.