Linear topologies on $\mathbb{Z}$ are not Mackey topologies

In this article it is shown that to every non-discrete Hausdorff linear topology on $\mathbb{Z}$ other metrizable locally quasi-convex group topologies can be associated which are strictly finer than the linear topology and such that the character groups coincide. Applying this result to the $p$-adic topology on $\mathbb{Z}$, we give a negative answer to the question of Dikranjan, whether this topology is Mackey.