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Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. Then $H_{f,g}$ is a subgroup of $G^2$ and $G^2$ is a dense subgroup of $\Bbb T^2$.

Are there homomorphisms $f,g:G\to G$ such that $H_{f,g}$ is dense in $\Bbb T\times \Bbb T$ and is there a way to characterize all such homomorphisms?

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