$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, I: arithmetic properties 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 if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.