Extension of Wiener-Wintner double recurrence theorem to polynomials

Idris Assani; Ryo Moore

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arXiv · 1409.0463

Extension of Wiener-Wintner double recurrence theorem to polynomials

Idris Assani·Ryo Moore

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Abstract

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.

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Idris Assani, Ryo Moore. 2014-08-25. Extension of Wiener-Wintner double recurrence theorem to polynomials. https://arxiv.org/abs/1409.0463

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