Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups

• Autorzy: Colin C. Graham , Kathryn E. Hare
• Języki publikacji: EN

Abstrakty

EN

Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a "weak ε-Kronecker set" if for every φ:E → 𝕋 there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.)
We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not "too many" elements of order 2 in the subgroup generated by E. When there are "too many" elements of order 2, we show that there exists a subset F, of the same cardinality as E, on which every {-1,1}-valued function can be interpolated exactly. Such sets are also I₀. In both cases, the set F also has the property that the only continuous character at which $F·F^{-1}$ can cluster in the Bohr topology is 1. This improves upon previous results concerning the existence of I₀ subsets of a given E.

Kategorie tematyczne

• 42A63: Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
• 42A55: Lacunary series of trigonometric and other functions; Riesz products
• 43A05: Measures on groups and semigroups, etc.
• 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
• 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 127
• Numer: 1
• Strony: 1-15

Daty

wydano

2012

Twórcy

autor

Colin C. Graham

• Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
• P.O. Box 2031 Haines Junction, YT, Canada Y0B 1L0

autor

Kathryn E. Hare

• Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Identyfikatory

DOI

10.4064/cm127-1-1