Extension of Wiener-Wintner double recurrence theorem to polynomials

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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 1/N ∑ₙ₌₁ⁿ f₁(Tanx)f₂(Tbnx)ϕ(p(n)) converge for any polynomial p with real coefficients, and any continuous function ϕ 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.