So if the dude asks me to talk about the duality $\mathbb{Z} \leftrightarrow \mathbb{R}/ \mathbb{Z}$ then he means $k \mapsto e^{2 \pi i k x_0}$, I suppose.
@MattN You need to flesh out the facts that a homomorphism $\mathbb{Z} \to S^1 = \mathbb{R/Z}$ is given by an element of $\mathbb{R/Z}$ (fairly clear) and that a homomorphism $S^1 \to S^1$ is given by $z \mapsto z^n$ with $z \in \mathbb{Z}$.
