A Hard-Analytic Proof of "Most" Polynomial Wiener-Wintner Theorems for Infinite Measure Spaces
Ben Krause
TL;DR
The paper proves a hard-analytic version of the polynomial Wiener-Wintner ergodic theorem for $\sigma$-finite measure-preserving systems, showing that for any $f\in L^p(X)$ there is a co-null set on which the averages $\frac{1}{N}\sum_{n\le N} e^{2\pi i P(n)} f(T^n\omega)$ converge for polynomials $P$ that are either linear or vanish to order two at $0$. The approach splits into two strands: linear modulations are handled via a lacunary-variation Carleson framework and time-frequency methods, while oscillatory polynomials that vanish at the origin are addressed through a Kohn–Selberg–Weyl (KSW) style orthogonality/decomposition that leverages Weyl sum estimates and a multilinear singular integral structure. The authors obtain sharp $\ell^2$ variation bounds $\|\mathcal{V}^r_d f\|_{\ell^2} \lesssim (\tfrac{r}{r-2})^2 \|f\|_{\ell^2}$ for $2<r<\infty$, and by interpolation extend these to $\ell^p$ bounds, yielding the convergence results on a co-null set for the specified polynomial classes. This work advances the understanding of polynomial Wiener-Wintner phenomena in infinite-measure contexts and clarifies the power and limits of hard-analytic, lacunary-variation techniques in ergodic theory.
Abstract
We provide a new proof of ``most" cases of the polynomial Wiener-Wintner theorem for $σ$-finite spaces, using hard-analytic methods. Specifically, we prove that whenever $(X,μ,T)$ is a $σ$-finite measure-preserving system, and $f \in L^p(X), \ 1 \leq p < \infty$, there exists a co-null set $X_f \subset X$ so that for all $ω\in X_f$ \[ \frac{1}{N} \sum_{n \leq N} e^{2 πi P(n)} f(T^n ω) \] converges for all polynomials $P$ which are either linear, or vanish to degree $2$ at the origin.