Existence of large $\varepsilon$-Kronecker and $FZI_0(U)$ sets in discrete abelian groups
Volume 127 / 2012
Abstract
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 ${\bf E}$.
