Multilinear Wiener-Wintner type ergodic averages and its application

This paper extends the generalized Wiener–Wintner Theorem built by Host and Kra to the multilinear case under the hypothesis of pointwise convergence of multilinear ergodic averages. In particular, we have the following result:Let $ (X, {\mathcal B}, \mu, T) $ be a measure preserving system. Let $ a $ and $ b $ be two distinct non-zero integers. Then for any $ f_{1}, f_{2}\in L^{\infty}(\mu) $, there exists a full measure subset $ X(f_{1}, f_{2}) $ of $ X $ such that for any $ x\in X(f_{1}, f_{2}) $, and any nilsequence $ {\textbf b} = \{b_n\}_{n\in {\mathbb Z}} $,$ \lim\limits_{N\rightarrow \infty}\frac{1}{N}\sum\limits_{n = 0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x) $exists.