More stability and convergence results for higher-order Wiener-Wintner systems

TL;DR

This paper advances the understanding of higher-order Wiener-Wintner systems by establishing their sublinear properties and improved stability, leading to broader convergence results including polynomial return times and multilinear ergodic transforms.

Contribution

It introduces new sublinearity properties of higher-order Wiener-Wintner averages and demonstrates their implications for stability and convergence in these systems.

Findings

Higher-order Wiener-Wintner averages are sublinear.

These averages bound conditional expectations and products.

Established convergence of multilinear ergodic Hilbert transform with polynomial phase.

Abstract

"Higher-order Wiener-Wintner averages" were constructed by Assani, Folks, and Moore to quantitatively control multiple recurrence averages. Systems in which these averages converge at a polynomial rate for a sufficiently large subset are termed "higher-order Wiener-Wintner systems of power type", in which properties like pointwise convergence of multiple recurrence averages and multiple return times averages has been shown. We establish that these higher-order Wiener-Wintner averages satisfy a type of sublinearity, and that they bound conditional expectations and products, which transfers to improved stability results of higher-order Wiener-Wintner systems under sums, factors, and products. We also establish more general convergence results for such systems, which include a polynomial return times theorem and convergence of the multilinear one-side ergodic Hilbert transform with polynomial phase.