A Hard-Analytic Proof of "Most" Polynomial Wiener-Wintner Theorems for Infinite Measure Spaces
Authors
Ben Krause
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.
