$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, II: sparseness of products of $\varepsilon$-Kronecker sets

Abstract

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.