Abstract. Let $E$ be a subset of a discrete abelian group $\Gamma $ with dual group $G$. We say $E$ is $I_0(U)$ if every bounded function on $E$ is the restriction of the Fourier--Stieltjes transform of a discrete measure on $U$. We show that every $I_0(G)$ set is a finite union of $I_0(U)$ sets (the number is not independent of the open set $U$, but the dependancy is made clear); if $G$ is connected then $E$ is $I_0(U)$ for all open $U$. Related results are given.

AMS Subject Classification (1991): 42A55, 43A46; 43A05, 43A25

Keyword(s): associated sets, Bohr group, Fatou--Zygmund property, Hadamard sets, I_0, sets, Sidon sets

Received May 14, 2008, and in revised form July 18, 2008. (Registered under 6068/2009.)