Slow Entropy of Quasi-Unipotent Actions on S-Arithmetic Homogeneous Spaces

We study metric and topological slow entropy for translations on a class of finite-volume $S$-arithmetic homogeneous spaces with a real and $p$-adic factor. We consider translations whose real component is quasi-unipotent and whose $p$-adic component lies in a compact open subgroup. At polynomial scales, we show that the metric and topological slow entropies coincide and are determined by the chain structure of the real adjoint action. In particular, the non-Archimedean component contributes no additional polynomial slow entropy. Our result extends the slow entropy formula of Kanigowski, Vinhage, and Wei for quasi-unipotent flows on real homogeneous spaces to this $S$-arithmetic setting.