Pointwise double recurrence and nilsequences

Abstract

Consider a system $(X,\mathcal{F},\mu ,T)$, bounded functions $f_1,f_2\in L^{\infty }(\mu )$ and $a,b \in \ZZ.$ We show that there exists a set of full measure $X_{f_1,f_2}$ in $X$ such that for all $x\in X_{f_1,f_2}$ and for every nilsequence $b_n$ , the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)b_n \] converge. We will show that this can be deduced from the classical Wiener-Wintner theorem for the double recurrence theorem. Together with the past work on this subject, we will show that several statements regarding the extension of the double recurrence theorem are equivalent.

Abstract modified - Equivalence between the Wiener Wintner double recurrence theorem, the Polynomial Wiener Wintner double recurrence theorem and the Nilsequence Double Recurrence theorem included in this revised version

References in corpus (2)

• A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages
• Pointwise convergence of averages along cubes

Cited by in corpus (5)

• A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages
• A good universal weight for nonconventional ergodic averages in norm
• On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights
• On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights
• A good universal weight for multiple recurrence averages with commuting transformations in norm