Multilinear Wiener-Wintner type ergodic averages and its application

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