Rectangular shrinking targets for $\mathbb Z^m$ actions on tori: well and badly approximable systems

We investigate the shrinking target property for irrational rotations. This was first studied by Kurzweil (1951) and has received considerable interest as of late. Using a new approach, we generalize results of Kim (2007) and Shapira (2013) by proving a weighted effective analogue of the shrinking target property. Furthermore, our results are established in the much wider $S$-arithmetic setting.