Multilinear Wiener-Wintner type ergodic averages and its application

In this paper, we extend 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 $f_1,f_2\in L^{\mathrm{\infty }}(\mu )$, 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 $\mathbf{\text{b}}=\{b_n{\}}_{n\in \mathbb{Z}}$, $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{n=0}^{N-1}{b}_n{f}_1(T^{an}x)f_2(T^{bn}x)$$ exists.