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

TL;DR

The paper advances the theory of higher-order Wiener-Wintner (WW) systems by establishing a reverse Bourgain bound that ties WW averages $W_N^J(f)$ to uniform multiple recurrence averages $M_N^J(f)$, yielding sublinearity-like behavior and stability under sums, factors, and products. It shows that WW averages bound off-diagonal terms and conditional-expectation terms, enabling factor-compatibility and product stability, including for weak WW averages. The authors extend convergence results to polynomial phases and derive polynomial return-time theorems, culminating in a multilinear ergodic Hilbert transform framework with polynomial weights that yields almost-everywhere convergence along subsequences and under general polynomial phases. Overall, the work provides a robust stability and convergence theory for higher-order WW systems, including alternative WW constructions and their equivalence under natural growth conditions, with clear implications for dynamics with nilfactors and polynomial recurrence structures.

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.

Theorems & Definitions (71)

• Remark 2.1
• Lemma 2.2: Van der Corput's estimate
• Lemma 2.3
• proof
• Lemma 2.4: Hölder's inequality on averages
• Lemma 2.5: Maximal inequality
• Theorem 3.1: Reverse Bourgain bound
• proof : Proof of Theorem \ref{['T:reverse BB, simplest case']}, with $k=1$
• Theorem 3.2
• proof