  Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$
functions \[ \mathcal{C}_{\Lambda} f( n ) := \sup_{ \lambda \in \Lambda} \left|
\sum_{m \neq 0} f(n-m) \frac{e^{2 \pi i\lambda m^2}} {m} \right| \] where
$\Lambda \subset [0,1]$. We give sufficient conditions on $\Lambda$, met by
certain kinds of Cantor sets, for this to be a bounded sublinear operator. This
result is a discrete analogue of E. M. Stein's integral result, that the
maximal operator below is bounded on $L^2(\mathbb R)$. \[ \mathcal{C}_2 f(x):=
\sup_{\lambda \in \mathbb R} \left| \int f(x-y) \frac{e^{2\pi i \lambda
y^2}}{y} \ dy \right|.\] The proof of our result relies heavily on Bourgain's
work on arithmetic ergodic theorems, with novel complexity arising from the
oscillatory nature of the question at hand, and difficulties arising from the
the parameter $\lambda$ above.
