Lower bound in the Roth theorem for amenable groups

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 $μ(A\cap T_1^g A\cap T_1^g T_2^g A) > μ(A)^4-ε$ on a syndetic set for any measurable set $A$ and any $ε>0$. The proof uses the concept of a sated system introduced by Austin.

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