Mixing actions of the rationals

Abstract

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 Schmidt. In contrast, algebraic actions of the much larger group ${\mathbb{Q}}^{\ast }$ are shown to behave quite differently, with finite order of mixing possible on connected groups.