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 $(X,\mathcal{F},\mu ,T)$ and bounded functions $f_1,f_2\in L^{\infty }(\mu )$, 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 $(Y,\mathcal{G},\nu ,S)$, and each bounded and measurable function $g_1,\dots ,g_k\in L^{\infty }(\nu )$, 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 $L^2(\nu )$.

This is the final version to appear in Erg. Th.and Dyn.Syst. taking into account the referee comments

References in corpus (3)

• Uniformity in the Wiener-Wintner theorem for nilsequences
• Pointwise double recurrence and nilsequences
• A good universal weight for nonconventional ergodic averages in norm

Cited by in corpus (3)

• Weighted multiple ergodic averages and correlation sequences
• A good universal weight for nonconventional ergodic averages in norm
• Multilinear Wiener-Wintner type ergodic averages and its application