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

We provide a new proof of ``most" cases of the polynomial Wiener-Wintner theorem for $\sigma$-finite spaces, using hard-analytic methods. Specifically, we prove that whenever $(X,\mu,T)$ is a $\sigma$-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 $\omega \in X_f$ $$\frac{1}{N} \sum_{n \leq N} e^{2 \pi i P(n)} f(T^n \omega)$$ converges for all polynomials $P$ which are either linear, or vanish to degree $2$ at the origin.