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 $f\in L^p(X),1\le p<\mathrm{\infty }$, there exists a co-null set $X_f\subset X$ so that for all $\omega \in X_f$ $$\frac{1}{N}\sum _{n\le N}{e}^{2\pi iP(n)}f(T^n\omega )$$ converges for all polynomials $P$ which are either linear, or vanish to degree $2$ at the origin.