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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 bycertain kinds of Cantor sets, for this to be a bounded sublinear operator. Thisresult is a discrete analogue of E. M. Stein's integral result, that themaximal operator below is bounded on $L^2(\mathbb R)$. \[ \mathcal{C}_2 f(x):=\sup_{\lambda \in \mathbb R} \left| \int f(x-y) \frac{e^{2\pi i \lambday^2}}{y} \ dy \right|.\] The proof of our result relies heavily on Bourgain'swork on arithmetic ergodic theorems, with novel complexity arising from theoscillatory nature of the question at hand, and difficulties arising from thethe parameter $\lambda$ above.

A Discrete Quadratic Carleson Theorem on $ \ell ^2 $ with a Restricted Supremum