Non-conventional ergodic averages for several commuting actions of an amenable group

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Let (X, μ) be a probability space, G a countable amenable group, and (Fn)n a left Folner sequence in G. This paper analyzes the non-conventional ergodic averages $$\frac{1}{{\left| {{F_n}} \right|}}\sum\limits_{g \in {F_n}} {\mathop \Pi \limits_{i = 1}^d } \left( {{f_i} \circ T_1^g \cdots T_i^g} \right)$$ associated to a commuting tuple of μ-preserving actions \({T_1}, \ldots {T_d}:G \curvearrowright X\) and f1,..., fd ∈ L∞(μ). We prove that these averages always converge in \({\left\| \cdot \right\|_2}\), 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.