  For $d \geq 2, \ D \geq 1$, let $\mathscr{P}_{d,D}$ denote the set of all
degree $d$ polynomials in $D$ dimensions with real coefficients without linear
terms. We prove that for any Calder\'{o}n-Zygmund kernel, $K$, the maximally
modulated and maximally truncated discrete singular integral operator,
\begin{align*} \sup_{P \in \mathscr{P}_{d,D}, \ N} \Big| \sum_{0 < |m| \leq N}
f(x-m) K(m) e^{2\pi i P(m)} \Big|, \end{align*} is bounded on
$\ell^p(\mathbb{Z}^D)$, for each $1 < p < \infty$. Our proof introduces a
stopping time based off of equidistribution theory of polynomial orbits to
relate the analysis to its continuous analogue, introduced and studied by
Stein-Wainger: \begin{align*} \sup_{P \in \mathscr{P}_{d,D}} \Big|
\int_{\mathbb{R}^D} f(x-t) K(t) e^{2\pi i P(t)} \ dt \Big|. \end{align*}
