Mixing actions of the rationals

We study mixing properties of algebraic actions of $\mathbb Q^d$, showing in particular that prime mixing $\mathbb Q^d$ actions on connected groups are mixing of all orders, as is the case for $\mathbb Z^d$-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 $\mathbb Q^*$ are shown to behave quite differently, with finite order of mixing possible on connected groups.