A good universal weight for nonconventional ergodic averages in norm

Abstract

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$(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$and bounded functions$f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$, 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$x\in X_{f_{1},f_{2}}$, for every other measure-preserving system$(Y,{\mathcal{G}},\unicode[STIX]{x1D708},S)$and for each bounded and measurable function$g_{1},\ldots ,g_{k}\in L^{\infty }(\unicode[STIX]{x1D708})$, the averages$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)g_{1}\circ S^{n}g_{2}\circ S^{2n}\cdots g_{k}\circ S^{kn}\end{eqnarray}$$converge in$L^{2}(\unicode[STIX]{x1D708})$.