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Algebraic numbers and harmonic analysis in the p-series case

University of British Columbia

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 well-known results of Salem, Meyer et al on the circle. Let p ≥ 2 be a prime and let k{x⁻¹} denote the p-series field of formal Laurent series z = Σhj=₋∞ ajxj with coefficients in the field k = {0, 1,…, p-1} 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 M-sets of measure 0 defined via Riesz products which are residual in G. In addition, a class of perfect M₀-sets of measure 0 is introduced with the purpose of settling a question left open by W.R. Wade and K. Yoneda, Uniqueness and quasi-measures on the group of integers of a p-series field, Proc. A.M.S. 84 (1982), 202-206. They showed that if S is a character series on G with the property that some subsequence {SpNj} of the pn-th 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.

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2010-12-04

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http://hdl.handle.net/2429/30282

Mathematics

University of British Columbia

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