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 of a polynomial exponent. We show that there exists a unique set of full measure for which the averages $$\frac{1}{N}\sum\limits_{n = 1}^N {{f_1}\left( {{T^{an}}x} \right){f_2}\left( {{T^{bn}}x} \right)\phi \left( {p\left( n \right)} \right)} $$ converge for all polynomials p with real coefficients and all complex-valued continuous functions ϕ on the unit circle T. 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, the averages converge to 0 uniformly for all polynomials. This paper combines the authors’ previously announced work.