  We show that a $k$-linear pointwise ergodic theorem on an ergodic
measure-preserving system implies a uniform $k$-linear nilsequence
Wiener-Wintner theorem on that system. The assumption is known to hold for
arbitrary systems and $k=2$ (due to Bourgain) and for distal systems and
arbitrary $k$ (due to Huang, Shao, and Ye).

Dynamical Systems