$\varepsilon $-Kronecker and $I_{0}$ sets in abelian groups, III: interpolation by measures on small sets

Colin C. Graham¹ ; Kathryn E. Hare²

¹ University of British Columbia
² Department of Pure Mathematics University of Waterloo Waterloo, ON, N2L 3G1 Canada

Studia Mathematica, Tome 171 (2005) no. 1, pp. 15-32

Résumé

Let $U$ be an open subset of a locally compact abelian group $G$ and let $E$ be a subset of its dual group ${\mit\Gamma} $. We say $E$ is $I_0(U)$ if every bounded sequence indexed by $E$ can be interpolated by the Fourier transform of a discrete measure supported on $U$. We show that if $E\cdot {\mit\Delta} $ is $I_0$ for all finite subsets ${\mit\Delta} $ of a torsion-free ${\mit\Gamma} $, then for each open $U\subset G$ there exists a finite set $F\subset E$ such that $E\setminus F$ is $I_0(U)$. When $G$ is connected, $F$ can be taken to be empty. We obtain a much stronger form of that for Hadamard sets and $\varepsilon $-Kronecker sets, and a slightly weaker general form when ${\mit\Gamma} $ has torsion. This extends previously known results for Sidon, $\varepsilon $-Kronecker, and Hadamard sets.

Détail

DOI : 10.4064/sm171-1-2

Keywords: subset locally compact abelian group subset its dual group mit gamma say every bounded sequence indexed interpolated fourier transform discrete measure supported cdot mit delta finite subsets mit delta torsion free mit gamma each subset there exists finite set subset setminus connected taken empty obtain much stronger form hadamard sets varepsilon kronecker sets slightly weaker general form mit gamma has torsion extends previously known results sidon varepsilon kronecker hadamard sets