  Consider the maximal operator $$\mathscr{C} f(x) =
\sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}}
f(x-y) e(\lambda |y|^{2d}) K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a
positive integer, $K$ a Calder\'on-Zygmund kernel and $n\ge 1$. This is a
discrete analogue of a real-variable operator studied by Stein and Wainger. The
nonlinearity of the phase introduces a variety of new difficulties that are not
present in the real-variable setting. We prove $\ell^2(\mathbb{Z}^n)$-bounds
for $\mathscr{C}$, answering a question posed by Lillian Pierce.
