Campbell–Petersen Wiener–Wintner conjecture for measure-preserving systems

Let $(X,\mu,T)$ be a measure-preserving system, where $(X,\mu)$ is a probability space and $T:X\to X$ preserves $\mu$. For $f\in L^2(X)$, define the truncated modulated ergodic Hilbert sums by

$$S_Nf(x,\tau)=\sum_{0<|k|<N}f(T^kx)\frac{e^{i\tau k}}k.$$

Campbell–Petersen's conjecture. For every measure-preserving system $(X,\mu,T)$ and every $f\in L^2(X)$,

\mu\Biggl\\{x:\lim_{N\to\infty}S_Nf(x,\tau)\text{ exists for all }\tau\Biggr\\}=1.

The conjecture asks for almost-everywhere Wiener–Wintner convergence of the ergodic Hilbert transform. The maximal-function bound follows from Calderón's transference and Carleson's theorem, but identifying a dense class on which convergence holds is the missing ingredient described in the source.