Discrete Analogues in Harmonic Analysis: A Theorem of Stein-Wainger

For $d\ge 2,D\ge 1$, let ${\mathcal{P}}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calderón-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^{2πi P(m)} \Big|, \end{align*} is bounded on ${\mathrm{\ell }}^p({\mathbb{Z}}^D)$, for each $1<p<\mathrm{\infty }$. 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^{2πi P(t)} \ dt \Big|. \end{align*}