Existence of large $\varepsilon$-Kronecker and $FZI_0(U)$ sets in discrete abelian groups

Colin C. Graham¹ ; Kathryn E. Hare²

¹ Department of Mathematics University of British Columbia Vancouver, BC, Canada Mailing address: P.O. Box 2031 Haines Junction, YT, Canada Y0B 1L0
² Department of Pure Mathematics University of Waterloo Waterloo, ON, Canada N2L 3G1

Colloquium Mathematicum, Tome 127 (2012) no. 1, pp. 1-15

Résumé

Let $G$ be a compact abelian group with dual group $\Gamma$ and let $\varepsilon>0$. A set ${\bf E}\subset\Gamma$ is a “weak $\varepsilon$-Kronecker set” if for every $\varphi:{\bf E}\to\mathbb T$ there exists $x$ in the dual of $\Gamma$ such that $|\varphi(\gamma)- \gamma(x)| \le \varepsilon$ for all $\gamma\in {\bf E}$. When $\varepsilon\sqrt2$, every bounded function on ${\bf E}$ is known to be the restriction of a Fourier–Stieltjes transform of a discrete measure. (Such sets are called $I_0$.)We show that for every infinite set ${\bf E}$ there exists a weak 1-Kronecker subset ${\bf F}$, of the same cardinality as ${\bf E}$, provided there are not “too many” elements of order 2 in the subgroup generated by ${\bf E}$. When there are “too many” elements of order 2, we show that there exists a subset ${\bf F}$, of the same cardinality as ${\bf E}$, on which every $\{-1,1\}$-valued function can be interpolated exactly. Such sets are also $I_0$. In both cases, the set ${\bf F}$ also has the property that the only continuous character at which ${\bf F}\cdot{\bf F}^{-1}$ can cluster in the Bohr topology is ${\bf1}$. This improves upon previous results concerning the existence of $I_0$ subsets of a given ${\bf E}$.

Détail

DOI : 10.4064/cm127-1-1

Keywords: compact abelian group dual group gamma varepsilon set subset gamma weak varepsilon kronecker set every varphi mathbb there exists dual gamma varphi gamma gamma varepsilon gamma varepsilon sqrt every bounded function known restriction fourier stieltjes transform discrete measure sets called every infinite set there exists weak kronecker subset cardinality provided there too many elements order subgroup generated there too many elements order there exists subset cardinality which every valued function interpolated exactly sets cases set has property only continuous character which cdot cluster bohr topology improves previous results concerning existence subsets given