Lower bound in the Roth theorem for amenable groups

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Let T₁, T₂ 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∩ T₁ᵍ A∩ T₁ᵍ T₂ᵍ 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.