Pointwise double recurrence and nilsequences
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.