Lower bound in the Roth theorem for amenable groups

Abstract

Let $T_1$, $T_2$ be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that $\mu(A\cap T_1^g A\cap T_1^g T_2^g A) > \mu(A)^4-{\epsilon}$ on a syndetic set for any measurable set $A$ and any $\epsilon>0$. The proof uses the concept of a sated system introduced by Austin.