On Convergence of Oscillatory Ergodic Hilbert Transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, $(X,\Sigma,\mu,\tau)$, with $\tau$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform $$\sum_{n=1}^\infty \frac{e^{2\pi i p(n)}}{n} \tau^n f(x)$$ converges $\mu$-a.e. for each $f \in L^r(X), \ 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, $$\sum_{n=1}^\infty \frac{X_n}{n} \tau^n f(x),$$ where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.