Extension of Wiener-Wintner double recurrence theorem to polynomials

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.