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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 $\Gamma\times \Gamma$?

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What is a Sidon set in the context of general compact abelian groups? I am used to Sidon sets being sets of integers. – JavaMan Oct 25 '11 at 21:09
@DJC: These are defined somewhere in Rudin's book on Fourier Analysis - I don't have my copy here right now, but it has to do with continuous functions on $G$ whose Fourier transforms are supported on $\Lambda$. – user16299 Oct 26 '11 at 1:26
I'm interested to know whether you found my answer acceptable. – Greg Martin Jan 24 '12 at 8:06
I updated the question. It seems that there exists several notions of "Sidon set". – Zouba Jan 25 '12 at 16:50

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 $\Gamma\times\Gamma$.

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