Translating compact sets in locally compact groups

Question

I wonder, when $G$ is a locally compact group (fix a Haar measure) what can be said about the set $$\{ g\in G\colon g^{-1}A\cap A \neq \emptyset \},$$ when $A$ is a compact subset of $G$.

Answer 1

If $a\in g^{-1}A\cap A$ then $A\ni a=g^{-1}b$ for some $b\in A$, hence $g=ba^{-1}$. Conversely, if $a,b\in A$ and $ba^{-1}=g$ then $a\in g^{-1}A\cap A$. So $$S=\{g:g^{-1}A\cap A\ne\emptyset\}=AA^{-1}.$$

You say you want to know "what can be said about" $S$. It's not clear from this exactly what question about $S$ you want an answer to, but it seems likely that whatever can be said about $S$ follows from knowing that $S=AA^{-1}$. (In particular, for example, it follows that if $A$ is compact then $S$ is compact...)