Almost mixing of all orders and CLT for some $\mathbb{Z}^d$-actions on subgroups of $\mathbb{F}\_p^{\mathbb{Z}^d}$

For N d-actions by algebraic endomorphisms on compact abelian groups, the existence of non-mixing configurations is related to "S-unit type" equations and plays a role in limit theorems for such actions. We consider a family of endomorphisms on shift-invariant subgroups of F Z d p and show that there are few solutions of the corresponding equations. This implies the validity of the Central Limit Theorem for different methods of summation.