Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including the discrete ones.

Keith Conrad has written up an explicit construction of the set of all homomorphisms $\mathbb Q \to \mathbb T$ and this can be used to find the Bohr Compactification of $\mathbb R$ as follows. The dual group of $\mathbb R$ is just $\mathbb R$ again. So $b\mathbb R$ is the set of all group homomorphisms $\mathbb R \to \mathbb T$ under pointwise convergence. To see what theese look like, choose a Hamel basis $B$ and write $\mathbb R = \oplus_{b \in B} \mathbb Q$. Each group homomorphism factors into an homomorphism $\mathbb Q \to \mathbb T$ for each element of $B$.

However the construction given by Conrad is so complicated there is no chance I would have come up with it myself. So I doubt I will be able to describe $b \mathbb Z$ either.

Is there a known description?