  We extend our result on the convergence of double recurrence Wiener-Wintner
averages to the case where we have a polynomial exponent. We will show that
there exists a single set of full measure for which the averages \[ \frac{1}{N}
\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\phi(p(n)) \] converge for any polynomial
$p$ with real coefficients, and any continuous function $\phi$ from the torus
to the set of complex numbers . We also show that if either function belongs to
an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends
on the degree of the polynomial $p$, then the averages converge to zero
uniformly for all polynomials. This paper combines the authors' previously
announced work.
