Some thin sets in discrete abelian groups

Abstract:

Let $\Gamma$ be a discrete abelian group, and $E \subset \Gamma$. For $F \subset E$, we say that $F \in \mathcal {P}(E)$, if for all $\Lambda$, finite subsets of $\Gamma ,0 \notin \Lambda ,\Lambda + F \cap F$ is finite. Having defined the Banach algebra, $\tilde A(E) = c(E) \cap B(E)$, we prove the following: (i) $E \subset \Gamma$ is a Sidon set if and only if every $F \in \mathcal {P}(E)$ is a Sidon set; (ii) $E \in \mathcal {P}(\Gamma )$ is a Sidon set if and only if $\tilde A(E) = A(E)$.