Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition).
Consider the set $\Lambda\times\Lambda$.
Is it a Sidon set of $\Gamma\times \Gamma$?
I assume that a Sidon set in $\Gamma$ is a set $S$ with the property that the pairwise sums $x+y$, with $x,y\in S$, are all distinct modulo commutativity $x+y=y+x$. In this case, it's irrelevant that $\Gamma$ is the dual of $G$; it's only important that $\Gamma$ is an abelian group.
In general, if $\Gamma,\Gamma'$ are abelian groups and $S\subset \Gamma$ and $S'\subset \Gamma'$ are subsets with more than one element each, then $S\times S'$ is never a Sidon subset of $\Gamma\times\Gamma'$: choosing $s_1\ne s_2$ from $S$ and $s_1'\ne s_2'$ from $S'$, we have the condemning coincidence $$ (s_1,s_1') + (s_2,s_2') = (s_1,s_2') + (s_2,s_1') $$ of pairwise sums. In particular, $\Lambda\times\Lambda$ is not a Sidon set in $\Gamma\times\Gamma$.