A good universal weight for nonconventional ergodic averages in norm

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system $(X, \mathcal{F}, μ, T)$ and bounded functions $f_1, f_2 \in L^\infty(μ)$, there exists a set of full-measure $X_{f_1, f_2}$ in $X$ that is independent of integers $a$ and $b$ and a positive integer $k$ such that for all $x \in X_{f_1, f_2}$ and for every other measure-preserving system $(Y, \mathcal{G}, ν, S)$, and each bounded and measurable function $g_1, \ldots, g_k \in L^\infty(ν)$, the averages $\frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)g_1 \circ S^n g_2 \circ S^{2n} \cdots g_k \circ S^{kn}$ converge in $L^2(ν)$.

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