  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.
