Let (X, μ) be a probability space, G a countable amenable group, and (Fn)n a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages (Formula Presented) associated to a commuting tuple of μ-preserving actions T1, … Td: G↷ X and f1,.., fd ∈ L∞(μ). We prove that these averages always converge in ‖ ⋅ ‖ 2, and that they witness a multiple recurrence phenomenon when f1 =.. = fd = 1A for a non-negligible set A ⊆ X. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.
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