Mixing actions of the rationals

We study mixing properties of algebraic actions of Qd, showing in particular that prime mixing Qd-actions on connected groups are mixing of all orders, as is the case for Zd-actions. This is shown using a uniform result on the solution of S-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group Q* are shown to behave quite differently, with finite order of mixing possible on connected groups.