text:
\section{Introduction}

This paper will be concerned with a so-called discrete Carleson-type operator, namely the maximally modulated and maximally truncated discrete singular integral operator,
\begin{align}\label{e:disc0}
\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}
where $K : \mathbb{R}^D \to \mathbb{C}$ is a normalized Calder\'{o}n-Zygmund kernel: $K \in \mathcal{C}^1(\mathbb{R}^D \smallsetminus 0)$ with $\| K \|_{\text{CZ}(\mathbb{R}^D)} \leq 1$, i.e.\
\begin{align}\label{e:cznorm}
\sup_{0 < r < R} |\int_{r\leq |x| \leq R} K(x) \ dx|  + \sup_{x \neq 0} \ |x|^D \cdot |K(x)| + \sup_{x \neq 0} \ |x|^{D+1} \cdot |\nabla K(x)| \leq 1
\end{align}
with the left side of \eqref{e:cznorm} defining the norm $\| K \|_{\text{CZ}(\mathbb{R}^D)}$. For $d \geq 2, \ D \geq 1$,
\begin{align}\label{e:polysdD}
\mathscr{P}_{d,D} := \Big\{ \sum_{2 \leq |\alpha| \leq d} \lambda_\alpha x^{\alpha} \in \mathbb{R}[x_1,\dots,x_D] \Big\}
\end{align}
denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms; see \eqref{e:multiindex} below for multi-index notation.

\subsection{History}
The study of maximally polynomially modulated singular integrals has a long and rich history, most notably encompassing Carleson's celebrated theorem on convergence of Fourier series \cite{C}; recent work of Lie \cite{Lie} (also see Zorin-Kranich \cite{ZK}) have essentially concluded this line of research: for every $1 < p <\infty$, the operator
\begin{align}\label{e:fullcarl}
    \mathscr{C}_{d,D} f(x) := \sup_{P,R} |\int_{|t| > R} f(x-t) K(t) e^{2\pi i P(t)}  \ dt|
\end{align}
satisfies
\begin{align}\label{e:fullcarl}
    \| \mathscr{C}_{d,D} f \|_{L^p(\mathbb{R}^D)} \lesssim_{d,D,p} \| f \|_{L^p(\mathbb{R}^D)},
\end{align}
where the supremum is taken over all polynomials in $D$ variables, of degree at most $d$, and all $0 < R <\infty$.

The major challenge presented in developing this theory is that the operator \eqref{e:fullcarl} is invariant under modulation by any polynomial of degree $\leq d$, a point which propagates throughout the analysis, and limits the efficacy of the standard Calder\'{o}n-Zygmund/Littlewood-Paley approach, in which the zero frequency has a distinguished role in the analysis. Indeed, if $Q(t)$ is a polynomial of degree $\leq d$, then for any $P$ of degree $\leq d$
\begin{align*}
&\int_{|t| > R} \Big( f(x-t) e^{2 \pi i Q(x-t)} \Big) K(t) e^{2\pi i P(t)}  \ dt \\
&= e^{2 \pi i Q(x)} \cdot \int_{|t| > R} \Big( f(x-t) e^{2 \pi i \big( Q(x-t) - Q(x) \big)} \Big) K(t) e^{2\pi i P(t)}  \ dt \\
&\qquad  \qquad =: e^{2 \pi i Q(x) } \cdot \int_{|t| > R} f(x-t) K(t) e^{2\pi i P_x(t)} \ dt
\end{align*}
where $t \mapsto P_x(t)$ has degree $\leq d$, uniformly in $x$. Consequently,
\begin{align*}
\Big| \int_{|t| > R} \Big( f(x-t) e^{2 \pi i Q(x-t)} \Big) K(t) e^{2\pi i P(t)}  \ dt \Big| \leq \mathscr{C}_{d,D} f(x);
\end{align*}
taking suprema, we see that
\begin{align*}
\mathscr{C}_{d,D} \big( f \cdot e^{2 \pi i Q( \cdot) } \big)(x) = \mathscr{C}_{d,D} f(x).
\end{align*}

Earlier, Stein and Wainger \cite{SW} investigated the analogue of \eqref{e:fullcarl} when this modulation invariance is eliminated, and the role of the zero frequency remains appropriately distinguished. In particular, they were interested in understanding the following operator:
\begin{align}\label{e:SW00}
    C_{d,D} f(x) := \sup_{P \in \mathscr{P}_{d,D}} \Big| \int  f(x-t) K(t) e^{2\pi i P(t)} \ dt \Big|
\end{align}
where, now, $\mathscr{P}_{d,D}$ denotes \eqref{e:polysdD}, the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms, and $K$ is a normalized Calder\'{o}n-Zygmund kernel satisfying \eqref{e:cznorm}. They were able to establish full $L^p$ estimates, for $1 < p < \infty$, for $C_{d,D}$ without resorting to modulation-invariant tools, but rather by relying on oscillatory integral techniques. In recent years, significant attention has been devoted to exploring their arguments, and a number of papers studying oscillatory integrals without modulation invariance in a wide variety of contexts have been written, see for instance \cite{G+, GRY,KL,BK11,BK12,P, Ramos0, Ramos1}. Indeed, this work was explicitly discussed in \cite{F+}, a review of Stein's major mathematical contributions.

\subsection{Discrete Harmonic Analysis}
Recently, the study of maximally modulated singular integrals without modulation invariance has been conducted in the integer setting. This study was initiated in \cite{KL}, where the analogue of Stein's purely quadratic Carleson operator \cite{S} was introduced and studied:
\begin{align}\label{e:disc1}
    \sup_\lambda \Big| \sum_{m \neq 0} f(x-m) \frac{e^{2\pi i \lambda m^2}}{m} \Big|.
\end{align}
At this point, the theory of polynomial Radon transforms, initiated by Jean Bourgain \cite{B0,B2,B1}, had become well-developed, see \cite{MST1,MST2,MSW}; the operator \eqref{e:disc1} was the first example of a discrete analogue in harmonic analysis that was not of this form. Accordingly, it proved surprisingly resistant to early attempts to bound it, even on $\ell^2(\mathbb{Z})$, see \cite{KL}. These difficulties were later resolved in \cite{BK11}, where a full $\ell^2(\mathbb{Z})$ theory was developed, and more recently in \cite{BK12}, where \eqref{e:disc1} was shown to be bounded on  $\ell^p(\mathbb{Z})$ for each $1 < p < \infty$ as a special case of broader work concerning suprema over one parameter families of modulation parameters.

\subsection{Main Results}
In this paper, we establish a discrete analogue of Stein-Wainger's result:

\begin{mthm}\label{t:main}
For every $1 < p < \infty$, the operator \eqref{e:disc0} is bounded on $\ell^p$:
\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| \|_{\ell^p(\mathbb{Z}^D)} \lesssim_{d,D,p} \| f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align*}
\end{mthm}

An immediate application of our theorem is to variable coefficient singular integrals. For
\[ V = (v_1,\dots,v_k), \; \; \; v_1,\dots,v_k : \mathbb{Z}^D \to \mathbb{Z}^D,\]
 consider the variable coefficient singular integral operator
\begin{align*}
&T_V f(x_0,x_1,\dots,x_k) : \mathbb{Z}^{D\cdot (k+1)} \to \mathbb{C} \\
& \qquad := \sum_{m \neq 0, \ m \in \mathbb{Z}^D} f(x_0-m,x_1 - v_1(x_0) P_1(m),\dots,x_k - v_k(x_0) P_k(m) ) \cdot K(m),
\end{align*}
where $K$ is a normalized Calder\'{o}n-Zygmund kernel, see \eqref{e:cznorm}.

By taking a partial Fourier transform in the final $k$ variables, and applying Plancherel's theorem and the $p=2$ case of Theorem \ref{t:main}, we arrive at the following Corollary.
\begin{mcor}
Suppose that $f \in \ell^2(\mathbb{Z}^{D(k+1)})$ and that $\{ P_i(m)  \} \subset \mathbb{Z}[m_1,\dots,m_D]$ are a collection of polynomials without linear terms and maximal degree $d$. Then for each $d,D,k$,
\begin{align*}
\sup_{V = (v_1,\dots,v_k)} \ \| T_V f \|_{\ell^2(\mathbb{Z}^{D(k+1)})} \lesssim_{d,D,k} \| f\|_{\ell^2(\mathbb{Z}^{D(k+1)})}.
\end{align*}
\end{mcor}

Theorem \ref{t:main}, along with the recent preprint \cite{B++} is the first multi-parameter result in discrete analogues in harmonic analysis: one can express
\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| \\
& \qquad = \sup_{\lambda_\alpha : \alpha \in \Gamma, \ N} \Big| \sum_{0 < |m| \leq N} f(x-m) K(m) e^{2\pi i \cdot \sum_\alpha \lambda_\alpha m^{\alpha} }\Big|
\end{align*}
see \eqref{e:Gamma}; the multi-parameter nature of the supremum necessitated a different set of techniques than those used in \cite{KL,BK11,BK12}. The analysis here is much closer in spirit to that of Stein-Wainger, with the Kolmogorov-Seliverstov method of $TT^*$ playing a crucial role. The technical ingredient needed to address \eqref{e:SW00} was the oscillatory integral bound
\begin{align}\label{e:coeffnorm}
    |\int_{[0,1]^D} e^{2 \pi i P(t)} \ dt | \lesssim  (1 + \| P \|)^{-\theta}
\end{align}
and the corresponding sub-level estimate
\begin{align}\label{e:sublevel}
    |\{ t \in [0,1]^D : |P(t)| \leq \epsilon \}| \leq \big( \frac{\epsilon}{\|P\|} \big)^{\theta}
\end{align}
for real-valued polynomials
\begin{align*}
    P(t) = \sum_\alpha \lambda_\alpha t^\alpha
\end{align*}
equipped with the coefficient norm
\begin{align}\label{e:Ecn}
    \| P \| := \sum_{|\alpha| \neq 0} | \lambda_\alpha|.
\end{align}
\eqref{e:sublevel} is often known as a \emph{non-concentration estimate}, as it says that it is very hard for the image of a polynomial to cluster disproportiantely near a single value. The optimal bound $\theta = 1/d$ is established in \cite{SW}, but the existence of any $\theta > 0$ would still be sufficient to establish their main result.

In this work, we develop an analogous mechanism for estimating exponential sums and use it to establish appropriate non-concentration estimates.

Classical equidistribution theory dictates that any polynomial $P: \mathbb{Z}^D \to \mathbb{R}$ of degree $\leq d$, and any (large) integer ${A}$, there exists some $M \lesssim_{{A} ,d} 1$ so that on the interval
\begin{align*}
\{ 1,\dots,N-1,N \}
\end{align*}
 one may decompose
\begin{align}\label{e:taodec}
P = P_{\text{Smooth}} + P_{\text{Equi}} + P_{\text{Rat}},
\end{align}
see \cite[Proposition 1.1.17]{T0}, so that -- essentially --
\begin{align}\label{e:stop}
|\sum_{n=1}^N e^{2 \pi i P(n)} | \lesssim M^{2} \max_{\frac{N}{M^{{A}}} \leq T \leq N} \Big| \sum_{n=1}^T e^{2 \pi i P_{\text{Equi}}(n)} \Big|  + M^{-A}
\end{align}
(say),
where $P_{\text{Equi}}$ is $M^{{A}}$-\emph{equidistributed}, in that
\begin{align*}
\sup_{\frac{N}{M^{A}} \leq T \leq N} \Big| \frac{1}{T} \sum_{n=1}^T e^{2 \pi i P_{\text{Equi}}(n)} \Big| = o_{M}(1)
\end{align*}
while $P_{\text{Smooth}}$ is smooth,
\begin{align*}
\| P_{\text{Smooth}} \|_{\text{Lip}} \leq \frac{M}{N}
\end{align*}
and $P_{\text{Rat}}$ has rational coefficients of bounded denominator, in that there exists $Q \leq M$ so that
\begin{align*}
Q \cdot P_{\text{Rat}} \equiv 0 \mod 1.
\end{align*}

The equation \eqref{e:stop} speaks to a negotation: the faster that $P$ equidistributes $\mod 1$, the smaller that the pertaining exponential sums become, but the longer it takes to achieve \eqref{e:taodec}, the less control over \eqref{e:stop} we enjoy. With this in mind, we introduce the scale-dependent stopping time
\begin{align}\label{e:coeffnorm00}
N_{N}(P) := \min\{ M : P_{\text{Equi}} \equiv 0 \text{ vanishes entirely} \}
\end{align}
and note that it is well-defined (a trivial upper bound is $N^{\text{deg}(P)}$). The significance of this quantity is that we may use inverse theorems to bound
\begin{align}\label{e:keyquant}
\Big| \frac{1}{N} \sum_{n=1}^N e^{2 \pi i P(n)} \Big| \lesssim N_N(P)^{-\theta}
\end{align}
for some $\theta = \theta_{A,d} > 0$, which acts as a substitute to \eqref{e:coeffnorm}; we use \eqref{e:coeffnorm00} to deduce appropriate arithmetic analogues of the estimate \eqref{e:sublevel}, which contains the key analytical input behind our approach.

The key quantitative estimate \eqref{e:keyquant} essentially appears as \cite[Lemma 1.16]{T0}. We will need a higher-dimensional version, which first appeared in \cite{Tblog}; we provide an alternative proof for completeness in our Appendix \ref{s:app}.

\subsection{Structure}
The structure of the paper is as follows:\\
\medskip

We begin by reviewing \cite{SW} in \S \ref{s:SW}, with a focus on the central role of the estimates \eqref{e:coeffnorm} and \eqref{e:sublevel};

In \S \ref{s:exp}, we introduce and discuss \eqref{e:coeffnorm00}, our analogue of the Euclidean coefficient norm. We use this technology to deduce some non-concentration estimates for polynomials;

In \S \ref{s:carl}, we reduce the study of \eqref{e:disc1} to the special case where the coefficients of $P$ live very close to cyclic subgroups with small denominators;

In \S \ref{s:nthry}, we use the circle method to approximate our operators, subject to the constraint on the coefficients of the polynomials. The arguments here are similar to those in \cite[\S 5]{BK11};

In \S \ref{s:finish}, we complete the proof.

Our Appendix \ref{s:app} contains the proofs of our exponential sums estimates.

\subsection{Notation}
Throughout, we let $e(t) := e^{2\pi i t}$ denote the complex exponential. For real $K > 0$, let
\begin{align}\label{e:mu}
    \mu_K(s) := \frac{1}{K} \cdot (1 - \frac{|s|}{K})_+
\end{align}
 be the one-dimensional Fej\'{e}r kernel at scale $K$, so that
\begin{align*}
    \widehat{\mu_K}(\xi) = \big( \frac{\sin(\pi K \xi)}{\pi K \xi} \big)^2.
\end{align*}

We use the induced norm on the Torus
\begin{align*}
\| x \|_{\mathbb{T}} := \min\{ |x-n| : n \in \mathbb{Z} \}.
\end{align*}

For multi-indices
\begin{align*}
    \alpha = (\alpha_1,\dots,\alpha_D), \; \; \; \alpha_i \in \mathbb{Z}_{\geq 0}
\end{align*}
we define
\begin{align}\label{e:multiindex}
    x^\alpha := \prod_{i=1}^D x_i^{\alpha_i}.
\end{align}
We let
\begin{align}\label{e:coord}
 {e_j} := (0,\dots,0,1,0,\dots,0)
\end{align}
denote the coordinate vector with $1$ in the $j$th component.

We define the ordering on multi-indices $\beta \leq \alpha$ if $\beta_i \leq \alpha_i, \ 1 \leq i \leq D$. If at least one strict inequality holds, we use $\beta < \alpha$.

We let $|\alpha| := \sum_i \alpha_i$, and let
\begin{align}\label{e:Gamma}
 \Gamma = \Gamma_{d,D} := \{ \alpha : 2 \leq |\alpha| \leq d\}
\end{align}
so that
\[ P \in \mathscr{P}_{d,D}\]
precisely when $P$ is a linear combination of monomials with exponents in $\Gamma$.
Note the upper bound
\begin{align*}
    |\Gamma| \leq \binom{D+d}{D}.
\end{align*}

For $\lambda = \{ \lambda_\alpha : \alpha \in \Gamma \} \in \mathbb{R}^{|\Gamma|}$, we use the notation
\begin{align}\label{e:poly}
P_\lambda(x) := \sum_\alpha \lambda_\alpha x^\alpha.
\end{align}

We will let $[R] := [-R,R]$ and $(R) := (0,R]$, with context determining whether we restrict to integers or not.

We let
\begin{align*}
    \vec{R} = (R_1,\dots,R_D)
\end{align*}
and define
\begin{align*}
    \vec{R}^\alpha := \prod_{i=1}^D R_i^{\alpha_i},
\end{align*}
and
\begin{align*}
|\vec{R}| := \prod_{i=1}^D R_i.
\end{align*}
We also set
\begin{align}\label{e:box}
[\vec{R}] = [R_1] \times \dots \times [R_D].
\end{align}

We will make use of the modified Vinogradov notation. We use $X \lesssim Y$, or $Y \gtrsim X$ to denote the estimate $X \leq CY$ for an absolute constant $C$ and $X,Y \geq 0$. If we need $C$ to depend on a parameter,
we shall indicate this by subscripts, thus for instance $X \lesssim_p Y$ denotes
the estimate $X \leq C_p Y$ for some $C_p$ depending on $p$. We use $X \approx Y$ as
shorthand for $Y \lesssim X \lesssim Y $. We reserve the notation
\begin{align}\label{e:sim}
    X \sim Y :=  \Big( X \in [Y/2, Y) \Big)
\end{align}
to denote the inequality $Y/2 \leq X < Y.$

We also make use of big-O notation: we let $O(Y)$ denote a quantity that is $\lesssim Y$, and similarly $O_p(Y)$ a quantity that is $\lesssim_p Y$.

\section{A Reveiw of Stein-Wainger}\label{s:SW}
Recall the Stein-Wainger maximal operator,
\begin{align}\label{e:SW}
    C_{d,D} f(x) := \sup_{\mathscr{P}_{d,D}} \Big| \int  f(x-t) K(t) e(P(t)) \ dt \Big|.
\end{align}

In this section we review the argument of \cite{SW} to prove the following Theorem with the key analytic estimates \eqref{e:coeffnorm} and \eqref{e:sublevel} as the departure point.
\begin{thm}[Stein-Wainger]\label{t:SW0}
For each $1<p < \infty$,
\begin{align*}
\| C_{d,D} f \|_{L^p(\mathbb{R}^D)} \lesssim_{d,D,p} \| f\|_{L^p(\mathbb{R}^D)}.
\end{align*}
\end{thm}

\subsection{Using The Estimates }
Since $K$ is a normalized Calder\'{o}n-Zygmund kernel, by \cite[\S 13]{FatS}
there exist a collection of mean zero $\mathcal{C}^1$ functions $\{\psi_j\}$ so that
\begin{align}\label{e:psi}
|\psi_j(x) | \cdot 2^{Dj} + |\nabla \psi_j(x)| \cdot 2^{D(j+1)} \leq C
\end{align}
for each $j$, so that each $\psi_j$ is supported in $\{ 2^{j-2} \leq |x| \leq 2^j \}$, and so that
\begin{align*}
K(x) = \sum_j \psi_j(x), \; \; \; x \neq 0.
\end{align*}

For $P_\lambda(t) = \sum_\alpha \lambda_\alpha t^\alpha$, decompose
\begin{align*}
&\int  f(x-t) K(t) e(P_\lambda(t)) \ dt \\
& \qquad = \sum_{j > j(\lambda)} \int f(x-t) \psi_j(t) e(P_\lambda(t)) \ dt + \sum_{j \leq j(\lambda)} \int f(x-t) \psi_j(t) e(P_\lambda(t)) \ dt \\
& \qquad \qquad = \sum_{l \geq 1} \sum_{j > j(\lambda)} \int f(x-t) \psi_j(t) e(P_\lambda(t)) \ dt \cdot \mathbf{1}_{ \| P_\lambda(2^j \cdot) \| \sim 2^l } \\
& \qquad \qquad \qquad + \sum_{j \leq j(\lambda)} \int f(x-t) \psi_j(t) \ dt + \sum_{j \leq j(\lambda)} O\big( \int |f(x-t)| |\psi_j(t)| |P_\lambda(t)| \ dt \big),
\end{align*}
see \eqref{e:sim}. Above, $j(\lambda)$ is the maximal integer, $j$, so that
\begin{align}\label{e:comp}
\| P_\lambda(2^j\cdot )\| = \sum_\alpha 2^{j|\alpha}| \cdot |\lambda_\alpha| \leq 1,
\end{align}

In particular, we may bound
\begin{equation}\label{e:A}
\begin{split}
&|\int  f(x-t) K(t) e(P(t)) \ dt|\\
& \qquad \leq \sum_{l \geq 1} \ \sup_P |\int f(x-t) \psi_{j(\lambda,l)}(t) e(P(t)) \ dt| + T^* f(x)+ M_{HL} f(x) \\
& \qquad \qquad =: \sum_{l \geq 1} \mathcal{A}_l f(x) + T^* f(x) + M_{HL} f(x),
\end{split}
\end{equation}
where $T^*f$ is a maximally truncated singular integral, and $j(\lambda,l)$ is defined to be the unique $j$ (if it exists) so that
\begin{align*}
\| P(2^j \cdot) \| \sim 2^l,
\end{align*}
see \eqref{e:comp}; if no such $j$ exists, set $\psi_{j(\lambda,l)} = 0$.

We will prove
\begin{proposition}\label{p:SW0}
For each $1<p< \infty$, there exists an absolute $c_p>0$ so that
\begin{align*}
\| \mathcal{A}_l f \|_{L^p(\mathbb{R}^D)} \lesssim 2^{-c_p l} \|f\|_{L^p(\mathbb{R}^D)}.
\end{align*}
\end{proposition}

Theorem \ref{t:SW0} then follows from a summation over $l \geq 1$.

By interpolation, it suffices to establish Proposition \ref{p:SW0} at the $L^2$ level. If we linearize the supremum, we may express $\mathcal{A}_l$ as an integral operator,
\begin{align*}
\mathcal{A}_l f(x) = \int K(x,t) f(t) \ dt
\end{align*}
with kernel
\begin{align*}
K(x,t) = e(P_{\lambda}(x-t)) \psi_{j(\lambda,l)}(x-t)
\end{align*}
where importantly $\lambda = \lambda(x)$ varies measurably with $x$, see \eqref{e:poly}.

By the Kolmogorov-Seliverstov method of $TT^*$, it suffices to show that
\begin{align*}
\| \int \mathcal{K}(x,y) f(y) \ dy \|_{L^2(\mathbb{R}^D)} \lesssim 2^{-cl} \|f\|_{L^2(\mathbb{R}^D)},
\end{align*}
where
\begin{align*}
\mathcal{K}(x,y) &= \int K(x,t) \overline{K(y,t)} \ dt \\
& \qquad = \int e(P_\lambda(x-t) - P_\mu(y-t)) \psi_k( x -t) \psi_r(y-t) \ dt
\end{align*}
and $\lambda = \lambda(x), \mu = \mu(y)$, and $k = k(\lambda,l)$, $r = r(\mu,l)$. Indeed
\begin{equation}\label{e:TT*use}
\begin{split}
    \| \mathcal{A}_l f\|_{L^2(\mathbb{R}^D)} &= \| \int K(\cdot,t) f(t) \ dt \|_{L^2(\mathbb{R}^D)} \\
    & \qquad \leq \| K \|_{\text{Ker}(\mathbb{R}^D)} \cdot \| f\|_{L^2(\mathbb{R}^D)} \\
    & \qquad \qquad = \| \mathcal{K} \|_{\text{Ker}(\mathbb{R}^D)}^{1/2} \cdot \|f\|_{L^2(\mathbb{R}^D)} \\
    & \qquad \qquad \qquad \lesssim 2^{-c/2 l} \cdot \|f\|_{L^2(\mathbb{R}^D)}.
\end{split}
\end{equation}
Above, we have used the operator norm
\begin{align*}
    \| K \|_{\text{Ker}(\mathbb{R}^D)} := \sup_{\| f \|_{L^2(\mathbb{R}^D)} = 1} \| \int K(x,t) f(t) \ dt \|_{L^2(\mathbb{R}^D)}.
\end{align*}

The key properties of $\mathcal{K}(x,y)$ are contained in the following Lemma.
\begin{lemma}\label{l:SW0}
There exists an absolute constant $c > 0$ so that
\begin{align*}
|\mathcal{K}(x,y)| &\lesssim 2^{-k D - cl} \mathbf{1}_{|x-y| \lesssim 2^k} + 2^{-rD - cl} \mathbf{1}_{|x-y| \lesssim 2^r } \\
& \qquad  + 2^{-kD} \mathbf{1}_{E(x)}(y) + 2^{-rD} \mathbf{1}_{E(y)}(x),
\end{align*}
where $E(x) \subset \{ |y| \lesssim 2^k \}, \ E(y) \subset \{ |x| \lesssim 2^r \}$ depend only on the identified variables and have \[ 2^{kD} |E(x)| + 2^{rD} |E(y)| \lesssim 2^{-cl}.\]
\end{lemma}

In particular, if we set the \emph{$h$-small set maximal function}
\begin{align*}
M_h f(x) := \sup_k \sup_E \ \frac{1}{2^{kD}} \int_E |f(x-y)| \ dy,
\end{align*}
where the supremum is over
\begin{align*}
E \subset \{|x| \lesssim 2^k\} : |E| \lesssim  2^{kD-h},
\end{align*}
then by interpolating the pointwise inequality $M_h f \lesssim M_{HL} f$ against the $L^\infty$ bound,
\[ \| M_h f \|_{L^{\infty}(\mathbb{R}^D)} \lesssim 2^{-h} \| f\|_{L^\infty(\mathbb{R}^D)} \]
one deduces that $M_h$ is a contraction on each $L^p$ space, and we may bound, for an appropriate $\| g \|_{L^2(\mathbb{R}^D)} = 1$,
\begin{align*}
&\| \int \mathcal{K}(x,y) f(y) \ dy \|_{L^2(\mathbb{R}^D)} \leq \int |g(x)| |\mathcal{K}(x,y)| |f(y)| \ dx dy \\
& \qquad \lesssim 2^{-cl} \int M_{HL} f(x) |g(x)| \ dx + \int M_{cl} f(x) |g(x)| \ dx + \int |f(y)| M_{cl} g(y) \ dy \\
& \qquad \qquad \lesssim 2^{-c/2 l} \| f \|_{L^2(\mathbb{R}^D)}.
\end{align*}

It remains only to prove Lemma \ref{l:SW0}, which follows from the following estimate, after changing variables appropriately, $v = x-y$, and using symmetry to reduce to the case where $ r \leq k$,

\begin{equation}
\label{e:kerbound}
\begin{split}
&|\int e(P_\lambda(v-2^{-r_0} t) - P_\mu(-t)) \psi( v -2^{-r_0} t) \psi(-t) \ dt| \\
& \qquad \qquad \qquad \lesssim 2^{-cl} + \mathbf{1}_{E_\lambda}(v), \; \; \; \; \; \;  r_0 \geq 0
\end{split}
\end{equation}
where $E_\lambda \subset \{ |v| \lesssim 1\}$ depends only on $\lambda$ and has $|E_{\lambda}| \lesssim 2^{-cl}$, and we think of $r_0 = k-r$.

There are two cases. If $2^{-r_0} \leq \eta \ll 1$, then
\[ t \mapsto P_{\lambda}(v-2^{-r_0} t) - P_\mu(-t) = P_{\lambda}(v) + \big( (P_{\lambda}(v-2^{-r_0} t) - P_\lambda(v) ) - P_\mu(-t) \big)
\]
has a coefficient norm that is $\gtrsim 2^l$, as the coefficient norm of
\[ t \mapsto P_{\lambda}(v-2^{-r_0} t) - P_\lambda(v) \]
is $O(2^{l-r_0})$. By \eqref{e:coeffnorm}, the bound \eqref{e:kerbound} holds. In the other case, we can bound the coefficient norm of
\[ t \mapsto P_{\lambda}(v-2^{-r_0} t) - P_\lambda(-t) \]
from below by the coefficient norm of the linear terms in the above difference; the presence of these linear terms is due to the lack of linear terms in $P_\mu$. In particular
\begin{align*}
\| P_\lambda(v-2^{-r_0} t) - P_\mu(-t) \| &\geq \| \sum_{j=1}^D \Big( \sum_{\alpha > e_j} \lambda_\alpha \alpha_j v^{\alpha - e_j} \Big) t^{e_j} \| \\
& \qquad \geq \sum_{j=1}^D \Big| \sum_{\alpha > e_j} \lambda_\alpha \alpha_j v^{\alpha - e_j} \Big| =: \sum_{j=1}^D |P_j(v)|;
\end{align*}
by another application of \eqref{e:coeffnorm}, it suffices to show that
\begin{align*}
|\{ |v| \lesssim 1 : \sum_{j=1}^D |P_j(v)| \lesssim 2^{c_0 l} \}| \lesssim 2^{-c_0 l}
\end{align*}
for some $c_0 > 0$. But his just follows from \eqref{e:sublevel}:
\begin{align*}
|\{ |v| \lesssim 1 : |P_j(v)| \lesssim 2^{c_0 l} \}| \lesssim \big( \frac{2^{c_0 l}}{ \|P_j \|} \big)^\theta \lesssim 2^{\theta (c_0 -1)l}
\end{align*}
for some $\theta > 0$ (which can be taken to be $1/d$). \\
\medskip

\subsubsection{Stein-Wainger: Continuous Summary and Discrete Preliminaries}
Aside from \eqref{e:sublevel}, which follows directly from \eqref{e:coeffnorm}, the techniques needed to establish Theorem \ref{t:SW0} are fairly modest:
\begin{itemize}
    \item The Hardy-Littlewood maximal function, $M_{HL}$;
    \item Maximally truncated singular integrals;
    \item $TT^*$ arguments;
    \item Interpolation.
\end{itemize}

The clarity of this scheme suggests that a similar approach should extend to the discrete situation, namely to Theorem \ref{t:main}. As is characteristic of the field, our analysis requires more delicacy as the positive Hardy-Littlewood maximal function is insensitive to destructive interference arising from arithmetic considerations. This difficulty can be partially resolved by earlier work \cite{BK11,BK12}, and by applying further $TT^*$ arguments, matters to formulating an arithmetic version of the Stein-Wainger argument.

Accordingly, we apply a discrete analogue of \eqref{e:coeffnorm} to derive the analogue of the crucial sublevel bound \eqref{e:sublevel}.

\section{Exponential Sums and Sublevel Estimates}\label{s:exp}
In this section we introduce the relevant analogues of \eqref{e:coeffnorm} and \eqref{e:sublevel} in the discrete context. As this result has been appeared \cite{Tblog}, we defer its proof to Appendix \ref{s:app}.

We begin by defining scale-dependent coefficient norms in full generality:

\begin{definition}
Suppose $P(x) = \sum_\alpha \lambda_\alpha x^\alpha \in \mathbb{R}[x_1,\dots,x_D]$, that $R_i \geq 1$, and set
\begin{align*}
\vec{R} = (R_1,\dots,R_D).
\end{align*}
Let $s_0$ denote the minimal $s$ so that either there exists some $Q \leq 2^s, \ s \geq 1$ so that
\begin{align*}
\sum_\alpha  \| Q \lambda_\alpha \|_{\mathbb{T}} \cdot \vec{R}^{\alpha} \leq 2^s,
\end{align*}
or
\begin{align*}
\sum_\alpha \| \lambda_\alpha \|_{\mathbb{T}} \cdot  \vec{R}^{\alpha}  \leq 2^s
\end{align*}
if $s \leq 0$.

Then, define the \emph{coefficient norm} of $P$ at scale $\vec{R}$,
\begin{align*}
N_{\vec{R}}(P) := 2^{s_0}.
\end{align*}
\end{definition}

If $\vec{R} = (R,\dots,R)$ is cubic, we will abbreviate
\begin{align*}
N_{\vec{R}}(P) = N_R(P).
\end{align*}

\begin{example}\label{ex:poly}
Suppose that $\lambda_\alpha = \frac{A_\alpha}{Q} \in \mathbb{Q}$ where $(\{ A_\alpha\}, Q) = 1$ are reduced. If $R_i \geq Q^{1+\delta}$ where $1 > \delta > 0$ is small, then \[ N_{\vec{R}}(P) \gtrsim Q^{\delta}.\]
\begin{proof}[Reason]
Suppose that $N_{\vec{R}}(P) \leq 2^t$, where $2^t < \frac{Q}{10}$ (say). This means that there exists $Q_1 \leq 2^t$ so that
\begin{align*}
\sum_\alpha \| \frac{A_\alpha}{Q} Q_1 \|_{\mathbb{T}} \cdot \vec{R}^{\alpha} \leq 2^t;
\end{align*}
since $(\{ A_\alpha\},Q)$ are reduced, there must be some $\alpha_0$ so that $\frac{A_{\alpha_0}}{Q} Q_1 \not \equiv 0 \mod 1$, leading to the lower bound
\begin{align*}
Q^{-1} \leq \| \frac{A_{\alpha_0}}{Q} Q_1 \|_{\mathbb{T}},
\end{align*}
and thus
\begin{align*}
\min_i \frac{R_i}{Q} \leq Q^{-1} \vec{R}^{\alpha_0} \leq \| \frac{A_{\alpha_0}}{Q} Q_1 \|_{\mathbb{T}} \cdot \vec{R}^{\alpha_0} \leq 2^t,
\end{align*}
so $2^t \geq Q^\delta$.
\end{proof}
\end{example}

The significance of the quantity $N_{\vec{R}}(P)$ is that it controls various exponential sums, analogous to the way the coefficient norms control oscillatory integrals in Euclidean space; note the trivial upper bound
\begin{align}\label{e:triv}
N_{\vec{R}}(P) \leq \sum_\alpha \vec{R}^{\alpha},
\end{align}
and the multiplicative property
\begin{align*}
N_{\vec{R}}(P) \leq 2^{\lceil \log_2 k \rceil} \cdot N_{\vec{R}}(k \cdot P)\; \; \; \; k \geq 1
\end{align*}
which is occasionally sharp, as can seen for polynomials with coefficients in $\mathbb{Z}/k \mathbb{Z}$. To see this, just observe that whenever $N_{\vec{R}}(kP) = 2^s$, there exists some multiple of $k$, $Qk$, with $Q \leq 2^s$, so that
\begin{align*}
\sum_\alpha \| Q k \lambda_\alpha \|_{\mathbb{T}} \cdot \vec{R}^{\alpha} \leq 2^s \leq 2^{\lceil \log_2 k \rceil} 2^s.
\end{align*}

Finally, we record the following convexity lemma concerning the coefficient norms.
\begin{lemma}\label{l:conv}
For any $P$, for each $s \geq 1$ the set
\begin{align*}
\{ k : N_{2^k}(P) = 2^s \}
\end{align*}
is an interval. When $s \leq 0$, there exists at most one $k$ so that $N_{2^k}(P) = 2^s$.
\end{lemma}
\begin{proof}
The case where $s \leq 0$ is clear, so we consider the more interesting case where $s \geq 1$. We need to show that if
\[ N_{2^k}(P) = N_{2^h}(P) = 2^s, \; \; \; k \leq h,\]
then $N_{2^{k'}}(P) = 2^s$ as well, for all $k \leq k' \leq h$. Since $N_{2^k}(P) = 2^s$, for any $Q \leq 2^{s-1}$,
\begin{align*}
2^{s-1} < \sum_{\alpha} \| Q \lambda_\alpha \|_{\mathbb{T}}  \cdot  2^{k |\alpha|} \leq \sum_{\alpha} \| Q \lambda_\alpha \|_{\mathbb{T}} \cdot  2^{k' |\alpha|},
\end{align*}
which says that $N_{2^{k'}}(P) \geq 2^s$; since $N_{2^h}(P) = 2^s$, there exist some $Q' \leq 2^s$ so that
\begin{align*}
2^s \geq \sum_\alpha \| Q' \lambda_\alpha \|_{\mathbb{T}}  \cdot  2^{h|\alpha|} \geq \sum_\alpha \| Q' \lambda_\alpha \|_{\mathbb{T}}  \cdot  2^{k'|\alpha|},
\end{align*}
for the reverse inequality.
\end{proof}

The key property of the coefficient norm is captured by the following Theorem, see \cite[Proposition 8]{Tblog}; a complete proof can be found in Appendix \ref{s:app} below.

\begin{thm}[Coefficient Norms Control Exponential Sums]\label{p:coeffnorm}
There exists an absolute $\theta = \theta_{d,D} > 0$ so that the following holds whenever $N_{\vec{R}}(P) = 2^s \geq 2$:

For every multi-dimensional arithmetic progression,
\[ \vec{\mathcal{P}} = \mathcal{P}_1 \times \dots \times \mathcal{P}_D, \; \; \; \mathcal{P}_i \subset [R_i] \]

\begin{align*}    \Big| \frac{1}{|\vec{R}|} \sum_{n \in \mathcal{P}} e(k \cdot P(n)) \phi(n) \Big| \lesssim \big( \frac{|k|}{2^s} \big)^{\theta} + \sum_i R_i^{-\theta} , \end{align*}
 whenever
\begin{align*}
    \| \phi \|_{\ell^\infty} + \sum_{j=1}^D R_j \cdot \| \phi - \phi(\cdot -  {e_j}) \|_{\ell^\infty} \leq 1.
\end{align*}

If $N_{\vec{R}}(k \cdot P) = 2^s \leq 1$, then
\begin{align*}
\frac{1}{|\vec{R}|} \sum_{n \in \mathcal{P}} e(k \cdot P(n)) \phi(n) = \frac{1}{|\vec{R}|} \sum_{n \in \mathcal{P}}  \phi(n) +O(N_{\vec{R}}(k \cdot P)).
\end{align*}
\end{thm}
The second point just follows from the mean-value theorem; the content of Theorem \ref{p:coeffnorm} concerns the case where the coefficient norm is large; the condition on the amplitudes, $\phi$, is standard, as one can quickly reduce to the case of constant amplitudes, see for instance \cite[\S A.1]{MSZK}.

\subsection{Sublevel Estimates}
In the Euclidean setting, the sublevel estimate \eqref{e:sublevel} follows directly from the coefficient-norm bound \eqref{e:coeffnorm}; in our present context, we can use Theorem \ref{p:coeffnorm} to derive appropriate sublevel estimates as well.

The following lemmas will be used to bound the percentage of time a polynomial can cluster extremely close to cyclic subgroups with small denominators; these fill the role of \eqref{e:sublevel}.

Our first lemma will be used in the case where the polynomial in question has a fairly large coefficient norm.

\begin{lemma}[Non-Concentration for Polynomials with Small Coefficient Norms] \label{c:nonconc}
Suppose $A \leq N_R(P)^{\theta}$, where $N_{R}(P) \geq 2$ and $\theta = \theta_{d,D} > 0$ is sufficiently small. There exists a $C= C_{d,D}$ so that whenever $B \geq 100$ (say)
\begin{align}\label{e:sub11}
    |\{ |v| \leq R : \min_{q \leq A} \|P(v) q\|_{\mathbb{T}} \leq B^{-1} \}| \lesssim R^D \cdot  A \cdot \Big( N_R(P)^{-\theta} + B^{-\theta} +R^{-\theta} \Big)
\end{align}
\end{lemma}
\begin{proof}
By the union bound, it suffices to consider a single $q \leq A$, but without the factor of $A$ on \eqref{e:sub11}. Suppose that $N_R(P) = 2^s$. Assume, as we may, that $B$ is an integer. We dominate the indicator function by a Fej\'{e}r kernel and bound
\begin{align*}
&\sum_{|v| \leq R} \mathbf{1}_{\| \beta \|_{\mathbb{T}} \leq B^{-1}}(P(v)q) \\
& \qquad \leq \sum_{|v| \leq R} \Big( \sum_k \mu_B(k) e(k \cdot P(v) q) \Big),
\end{align*}
see \eqref{e:mu}. In particular, if we set $\vec{R} = [R]^D \times [B]$, and define
\begin{align*}
P_0(v,k) := k \cdot P(v) q \in \mathbb{R}[v_1,\dots,v_D,k]
\end{align*}
to be a polynomial of $D+1$ many variables, it suffices to show that
\begin{align*}
\Big| \frac{1}{R^D} \sum_{|v| \leq R, \ k} \mu_B(k) e(P_0(v,k)) \Big| \lesssim \Big( N_R(P)^{-\theta} + B^{-\theta} +R^{-\theta} \Big).
\end{align*}

If $N_{\vec{R}}(P_0) \geq 2^{s-1}/A \gtrsim 2^{s(1-\theta)}$, the bound is clear, so assume otherwise.

This says that there exist $Q_1 <  2^{s-1}/A $ so that
\begin{align*}
\sum_\alpha \| Q_1 (q \lambda_\alpha) \|_{\mathbb{T}} \cdot R^{|\alpha|} \leq B^{-1} \cdot 2^{s-1};
\end{align*}
let $Q_0 := Q_1 q < 2^{s-1}$ be minimal subject to this constraint.

On the other hand, since $N_R(P) = 2^s$, we know that for every $Q \leq 2^{s-1}$
\begin{align*}
\sum_\alpha \| Q \lambda_\alpha \|_{\mathbb{T}} \cdot R^{|\alpha|} > 2^{s-1};
\end{align*}
specializing to $Q = Q_0$ yields the contradiction.
\end{proof}

The following sub-level set estimate will serve as a substitute to Lemma \ref{c:nonconc}
when $P$ has a very large coefficient norm: $B = N_R(P)^{O(1)}$.
\begin{lemma}[Non-Concentration Estimate for Polynomials with Large Coefficient Norms]\label{c:sublevel1}
Suppose that $N_R(P) \geq R^{\eta}$ where $0 < \eta \ll 1$ is bounded away from zero, and that $0 <\kappa \ll \eta$ is sufficiently small.

Then there exists some absolute $\kappa_0 = \kappa_0(\eta) > 0$ so that
\begin{align*}
|\{ |v| \leq R : \min_{q \leq R^\kappa} \|P(v) q\|_{\mathbb{T}} \leq R^{\kappa - 1} \}| \lesssim R^{D - \kappa_0}.
\end{align*}
\end{lemma}
\begin{proof}
We use a union bound; thus, it suffices exhibit $\kappa_0 > 0$ so that for each $q \leq R^{\kappa}$,
\begin{align*}
|\{ |v| \leq R : \|P(v) q\|_{\mathbb{T}} \leq R^{\kappa - 1} \}| \lesssim R^{D - \kappa_0 - \kappa}.
\end{align*}
By another Fej\'{e}r kernel argument, it suffices to exhibit a $\kappa_0 > 0$ so that
\begin{align}\label{e:sum00}
\left|
\frac{1}{R^D} \sum_{|v| \leq R, n} \mu_{R^{1-\kappa}}(n) \cdot e(P_0(v,n)) \right| \lesssim R^{-\kappa_0 - \kappa},
\end{align}
where
\begin{align*}
P_0(v,n) = \sum_\alpha (q \cdot \lambda_\alpha) \cdot ( n v^\alpha ) \in \mathbb{R}[v_1,\dots,v_D,n]
\end{align*}
is a polynomial of $D+1$ variables. Set $\vec{R} = (R,\dots,R,R^{1-\kappa})$; we claim that $N_{\vec{R}}(P_0) \geq R^{\eta/2}$; this allows us to bound
\begin{align*}
\eqref{e:sum00} \leq R^\kappa \cdot ( R^{-\theta \eta} + R^{-\theta} ),
\end{align*}
which would yield the result.

So, suppose otherwise, and extract a minimal $Q_1 \leq R^{\eta/2}$ so that
\begin{align*}
\sum_\alpha \| (Q_1 q) \lambda_\alpha \|_{\mathbb{T}} {R}^{|\alpha| + 1 -\kappa} \leq R^{\eta/2}.
\end{align*}
On the other hand, since $N_R(P) \geq R^{\eta}$, for every $Q \leq \frac{R^{\eta}}{2}$
\begin{align*}
\sum_\alpha \| Q \lambda_\alpha \|_{\mathbb{T}} R^{|\alpha|} > R^{\eta};
\end{align*}
the contradiction arises by specializing $Q = Q_1 q \leq R^{\eta/2 + \kappa} \ll R^{\eta}$.
\end{proof}

In what follows, we will use the machinery developed above to prove Theorem \ref{t:main}.

\section{The Discrete Stein-Wainger Operator}\label{s:carl}
Regarding $\| K \|_{\text{CZ}(\mathbb{R}^D)}$ as given, let $\{ \psi_j \}$ be as in \eqref{e:psi}.

We introduce a large real parameter, \begin{align}\label{e:A0}
    A_0 = A_0(d,D,p)
\end{align} which we are free to adjust upwards finitely many times as needed.

Define
\begin{align*}
\mathcal{C}_{d,D} f(x) := \sup_{P \in \mathscr{P}_{d,D}, \ k_0} | \sum_{m \in \mathbb{Z}^D} \sum_{k=1}^{k_0}  f(x-m) \psi_k(m) e(P(m)) |;
\end{align*}
Theorem \ref{t:main} follows directly from the following result after an argument with the Hardy-Littlewood maximal function.
\begin{thm}\label{t:key}
For each $1 < p < \infty$ and $d,D$,
\begin{align*}
    \| \mathcal{C}_{d,D} f\|_{\ell^p(\mathbb{Z}^D)} \lesssim \|f\|_{\ell^p(\mathbb{Z}^D)}
\end{align*}
\end{thm}

By Proposition \ref{p:triangle} below, our attention will be focused on the ``oscillatory" truncated singular integrals
\begin{equation}\label{e:As}
\begin{split}
&\mathscr{A}_s f(x) \\
&:= \sup_{k_0 \geq 2^{s/A_0}} \Big| \sum_{k = 2^{s/A_0}}^{k_0} \sum_{m} \psi_k(m) e(P_{\lambda(x)}(m)) f(x-m) \cdot \mathbf{1}_{n : N_{2^k}(P_{\lambda(n)}) = 2^{ s}}(x) \Big|.
\end{split}
\end{equation}
 for $A_0$ as in \eqref{e:A0} a sufficiently large absolute constant, and $\lambda:\mathbb{Z}^D \to [0,1]^{|\Gamma|}$ an appropriate linearizing function. Note that for each $s \geq 1$, the set
\begin{align*}
\{ k : N_{2^k}(P) = 2^s \}
\end{align*}
is an interval, see Lemma \ref{l:conv}.

We present this reduction in the form of a proposition.

\begin{proposition}\label{p:triangle}
The following pointwise bound holds:
\begin{align}\label{e:sum}
\mathcal{C}_{d,D} f \lesssim \sum_{s \geq 1} \mathscr{A}_s f + \mathcal{E} f + H^*f + M_{HL} f,
\end{align}
where $H^*f$ is a maximally truncated singular integral, $\mathscr{A}_s$ are as in \eqref{e:As}, and
\begin{align*}
\mathcal{E} f = \sum_{k \geq 1} \mathcal{E}_k f
\end{align*}
is a sum of single scale operators with
\begin{align}\label{e:Ek}
\| \mathcal{E}_k f \|_{\ell^p(\mathbb{Z}^D)} \lesssim k^{-2 } \cdot \| f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align}
\end{proposition}

The proof of Proposition \ref{p:triangle} will take up the early part of this section, with \eqref{e:Ek} being the crucial point. We accordingly defer the estimate \eqref{e:Ek} to Proposition \ref{p:errors} below.
\begin{proof}[Proof of Proposition \ref{p:triangle} Assuming \eqref{e:Ek}]

With $\lambda(x)$ an appropriate linearizing function, foliate
\begin{align*}
&|\sum_{k=1}^{k_0} \ \sum_{s \in \mathbb{Z}} \sum_{m \in \mathbb{Z}^D} f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))| \\
& \qquad = |\sum_{k=1}^{k_0} \ \sum_{s \in \mathbb{Z}}  \sum_{m \in \mathbb{Z}^D}  f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s}(x) |
\end{align*}
according to the size of the pertaining coefficient norms.

We bound the foregoing above by the sum of two terms, which we will address individually:
the stationary component
\begin{align}\label{e:stat1}
|\sum_{k=1}^{k_0} \  \sum_{m \in \mathbb{Z}^D}  f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \leq 1}(x) |;
\end{align}
and
the oscillatory component
\begin{align}\label{e:osc1}
    |\sum_{k=1}^{k_0} \ \sum_{s \geq 1}  \sum_{m \in \mathbb{Z}^D}  f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s}(x) |.
\end{align}

We begin by bounding \eqref{e:stat1}. Since the coefficient norm of $P_{\lambda(x)}$ is small in this case, we just Taylor expand the phase:
\begin{align*}
\eqref{e:stat1}
&\leq
| \sum_{k=1}^{k_0} \ \sum_{m \in \mathbb{Z}^D} f(x-m) \psi_k(m)
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \leq 1}(x) | \\
& \qquad  +
|\sum_{k=1}^{k_0} \ \sum_{m \in \mathbb{Z}^D}   f(x-m) \psi_k(m) \big( e(P_{\lambda(x)}(m)) - 1 \big)
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \leq 1}(x) | \\
& =: H^*f(x) + O\Big(
| \sum_{k=1}^{\infty} \sum_{s \leq 0} \sum_{m \in \mathbb{Z}^D} |f(x-m)| |\psi_k(m)| |P_{\lambda(x)}(m)|
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s}(x) \Big) | \\
& \qquad \lesssim H^*f(x) + M_{HL} f(x) \cdot \Big( \sum_{k=1}^{\infty} \sum_{s \leq 0} 2^s  \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s}(x) \Big) \\
& \qquad \qquad \lesssim H^*f(x) + M_{HL} f(x)
\end{align*}
where $H^*$ is a truncated singular integral, as
\begin{align*}
\{ k : N_{2^k}(P) \leq 1 \}
\end{align*}
is an interval, see Lemma \ref{l:conv}, and we crucially used that for each $s\leq 0$, there is at most one $k$ so that
\begin{align*}
N_{2^{k}}(P) = 2^s,
\end{align*}
see Lemma \ref{l:conv}. This concludes the estimate of \eqref{e:stat1}.

We now address \eqref{e:osc1}. To do so, bound
\begin{align*}
\eqref{e:osc1}
& \leq
|\sum_{k=1}^{k_0} \ \sum_{s : 2^s \leq k^{A_0}} \sum_{m \in \mathbb{Z}^D} f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s }(x)| \\
& \qquad +
|\sum_{k=1}^{k_0} \ \sum_{s : 2^s > k^{A_0}} \sum_{m \in \mathbb{Z}^D } f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s }(x)| \\
& \qquad \qquad \leq
\sum_{s \geq 1} | \sum_{k \geq 2^{s/A_0} }^{k_0} \ \sum_{m \in \mathbb{Z}^D } f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(n)}) \sim 2^s }(x) | \\
& \qquad \qquad  \qquad + \sum_{k \geq 1} \
\Big| \sum_{m \in \mathbb{Z}^D} f(x-m) \psi_k(m) e(P_{\lambda(x)}(m))
\cdot \mathbf{1}_{ n :  N_{2^k}(P_{\lambda(x)}) \geq k^{A_0} }(x) \Big| \\
& \qquad \qquad \qquad \qquad \leq \sum_{s \geq 1} \mathscr{A}_s f(x) + \sum_{k \geq 1} \mathcal{E}_k f(x)
\end{align*}
where the scale $k$ error terms are defined, explicitly,:
\begin{align*}
    \mathcal{E}_k f(x) := \sup_{P : N_{2^k}(P) \geq k^{A_0}} |\sum_{m \neq 0} e(P(m))\psi_k(m) f(x-m) |.
\end{align*}
\end{proof}

In particular, we have reduced the proof of Proposition \ref{p:triangle} to establishing \eqref{e:Ek}. This will be the focus of the following Proposition.

\begin{proposition}\label{p:errors}
There exists an absolute $c = c_{d,D} > 0$ so that for each $1 < p < \infty$
\begin{align*}
\| \mathcal{E}_k f \|_{\ell^p(\mathbb{Z}^D)} \lesssim k^{-c \cdot A_0 \frac{2}{p^*}} \cdot \| f\|_{\ell^p(\mathbb{Z}^D)}, \; \; \; p^* = \max\{p,p'\}.
\end{align*}
\end{proposition}

We prove Proposition \ref{p:errors} by showing that
\begin{align}\label{e:errors}
    &\| \sup_{P : N_{2^k}(P) = 2^s} | \sum_{m} e(P(m)) \psi_k(m) f(x-m)| \|_{\ell^2(\mathbb{Z}^D)}\\
    & \qquad \lesssim ( 2^{-\theta s} + 2^{-\theta k} ) \cdot \| f\|_{\ell^2(\mathbb{Z}^D)} \\
& \qquad \qquad \lesssim 2^{-cs} \| f\|_{\ell^2(\mathbb{Z}^D)}
\end{align}
and interpolating against a trivial single-scale estimate, see \eqref{e:triv} for the final inequality.

We prove \eqref{e:errors} by the Kolmogorov-Seliverstov method of $TT^*$. Regarding $s$ and $k$ as fixed, subject to the constraint $k^{A_0} \leq 2^s \leq 2^{C_{d,D} k}$, see \eqref{e:triv}, it suffices to prove that, uniformly in measurable maps
\begin{align*}
\mathbb{Z}^D \to \{ P \in \mathscr{P}_{d,D} : N_{2^k}(P) = 2^s \}
\end{align*}
the kernel
\begin{align*}
    K(x,m) := e(P_{\lambda(x)}(x-m)) \psi_k(x-m),
\end{align*}
satisfies
\begin{align*}
    \| K \|_{\text{Ker}(\mathbb{Z}^D)} := \sup_{\| f \|_{\ell^2(\mathbb{Z}^D)} = 1} \| \sum_m K(\cdot,m) f(m) \|_{\ell^2(\mathbb{Z}^D)} \lesssim 2^{-cs}.
\end{align*}

By arguing as in \eqref{e:TT*use} above, it suffices to instead bound
\begin{align}\label{e:TT*}
    \| \mathcal{K} \|_{\text{Ker}(\mathbb{Z}^D)} \lesssim 2^{-2cs}
\end{align}
where
\begin{align}\label{e:kernel}
    \mathcal{K}(x,n) = \sum_m e(P_{\lambda(x)}(x-m) - P_{\mu(n)}(n-m)) \psi_k(x-m) \psi_k(n-m)
\end{align}
where
\[ N_{2^k}(P_{\lambda(x)} ) = N_{2^k}(P_{\mu(n)}) = 2^s.\]
The key point is the following Lemma.
\begin{lemma}\label{l:kernel1}
There exists an absolute $c_0 > 0$ so that the following inequality holds pointwise:
\begin{align*}
   |\mathcal{K}(x,n)| \lesssim 2^{-c_0s - Dk} \cdot \mathbf{1}_{|x-n| \lesssim 2^k} + 2^{-kD}  \cdot \mathbf{1}_{O(x)}(n)
\end{align*}
where
\begin{align*}
    O(x) \subset \{ |v| \lesssim 2^k\}, \; \; \; |O(x)| \lesssim 2^{-c_0s + kD}
\end{align*} depends only on the $x$ variable.
\end{lemma}
In particular,
\begin{align*}
\sup_x \sum_n |\mathcal{K}(x,n)| \lesssim 2^{-c_0s}, \; \; \; \sup_n \sum_x |\mathcal{K}(x,n)| \lesssim 1,
\end{align*}
so \eqref{e:TT*} follows from Schur's test, with $c = \frac{c_0}{2}$.

\subsection{The Proof of Lemma \ref{l:kernel1}}
With $v = x-n$, $\lambda = \lambda(x)$ and $\mu = \mu(n)$, we need to show that
\begin{align*}
    &    |\mathcal{K}(x,n)| = |\sum_{m} \psi_k(v-m) \psi_k(-m) e(P_\lambda(v-m) - P_\mu(-m)) |\\ & \qquad \lesssim 2^{-kD - \kappa s}  \cdot \mathbf{1}_{|v| \lesssim 2^k} + 2^{-k D} \cdot \mathbf{1}_{\mathcal{U}(\lambda)}(v)
\end{align*}
where $\mathcal{U}(\lambda) \subset [2^k]^D$ has $|\mathcal{U}(\lambda)| \leq 2^{kD - \kappa s}$, and is independent of $\mu$.

The coefficient norm of the phase
\begin{align*}
    m \mapsto P_\lambda(v-m) - P_\mu(-m)
\end{align*}
is bounded below by that of the linear terms in $m$:
\begin{align*}
    m \mapsto \sum_{j=1}^D \big( \sum_\alpha \lambda_\alpha \alpha_j v^{\alpha - e_j} \big) m^{e_j} =: \sum_{j=1}^D P_j(v) \cdot m^{e_j}
\end{align*}

We distinguish between two cases according to the relationship between $s$ and $k$. Below, we let $\eta = \eta_{d,D} \ll 1$ denote a sufficiently small constant.\\

\medskip

\textbf{Case One: $s \leq \eta k$.}\\
Let $\kappa_0 > 0$ be a very small constant, depending on $d,D$.

Collect
\begin{align*}
    \mathcal{G} := \{ |v| \lesssim 2^k : N_{2^k}\Big( \sum_{j=1}^D P_j(v) m^{e_j} \Big) \geq 2^{\kappa_0 s} \}
\end{align*}
and observe that for $v \in \mathcal{G}$,
\begin{align*}
    |\sum_{m} \psi_k(v-m) \psi_k(-m) e(P_\lambda(v-m) - P_\mu(-m)) | \lesssim 2^{- \theta \kappa_0 s} + 2^{- \theta k} \lesssim 2^{-\theta \kappa_0s}.
\end{align*}

We set
\begin{align*}
    \mathcal{U}(\lambda) := [2^k]^D \smallsetminus \mathcal{G} \subset \bigcap_{j=1}^D \mathcal{B}_j
    \end{align*}
where
\begin{align*}
    \mathcal{B}_j &:=
    \{ |v| \lesssim 2^k : \min_{q \leq 2^{\kappa_0 s}} \| q P_j(v) \|_{\mathbb{T}} \leq 2^{\kappa_0 s-k} \}
\end{align*}
In the language of Lemma \ref{c:nonconc}, $A= 2^{\kappa_0 s}$, $B = 2^{k-\kappa_0 s}$, and $R = 2^k$, so by that lemma
\begin{align*}
|\mathcal{B}_j| \lesssim 2^{kD} \cdot 2^{-cs}.
\end{align*}

\medskip

\textbf{Case Two: $\eta k < s  \lesssim_{d,D} k$.}\\
Collect
\begin{align*}
    \mathcal{G} := \{ |v| \lesssim 2^k : N_{2^k}\Big( \sum_{j=1}^D P_j(v) m^{e_j} \Big) \geq 2^{\kappa k} \}
\end{align*}
and observe that for $v \in \mathcal{G}$,
\begin{align*}
    |\sum_{m} \psi_k(v-m) \psi_k(-m) e(P_\lambda(v-m) - P_\mu(-m)) | \lesssim 2^{- Dk - \theta \kappa k} \lesssim 2^{-Dk - \theta' s}.
\end{align*}

Collect
\begin{align*}
    \mathcal{U}(\lambda) := [2^k]^D \smallsetminus \mathcal{G} \subset \bigcap_{j=1}^D \mathcal{B}_j
    \end{align*}
where
\begin{align*}
    \mathcal{B}_j &:=
    \{ |v| \lesssim 2^k : \min_{q \leq 2^{\kappa k}} \| q P_j(v) \|_{\mathbb{T}} \leq 2^{(\kappa -1)k} \}
\end{align*}
By our second sub-level set estimate, Lemma \ref{c:sublevel1},
\[ |\mathcal{B}_j| \lesssim 2^{k(D - \kappa_0)} \lesssim 2^{kD - \kappa_1 s},\]
 which completes the proof.

With Proposition \ref{p:triangle} in mind, in the following section we will apply the circle method to approximate our oscillatory operators $\{ \mathscr{A}_s \}$, see \eqref{e:As}, by more tractable family of analytically-defined operators.

\section{Approximations}\label{s:nthry}

We now construct analytic approximates to the multipliers
\begin{align}\label{31-e:Mj}
m_{j,\lambda}(\beta) := \sum_m \psi_j(m) e(-P_\lambda(m) - \beta \cdot m),
\end{align}
where $\lambda \in [0,1]^{|\Gamma|}$,
\begin{align*}
    P_\lambda(m) = \sum_\alpha \lambda_\alpha m^\alpha,
\end{align*}
see \eqref{e:poly}, and $\beta \in [0,1]^D$.
\begin{align*}
    \beta = (\beta_1,\dots,\beta_D).
\end{align*}

For $(\frac{A}{Q},\frac{B}{Q}) \in \mathbb{Q}^{|\Gamma|} \times \mathbb{Q}^D$, define the complete Gauss sum
\begin{align*}
    S(A/Q,B/Q) = \frac{1}{Q^D} \sum_{r \in (Q)^D} e( - P_{A/Q}(r) - \frac{B}{Q} \cdot r)
\end{align*}
\begin{lemma}
Suppose that $(A,B,Q) = 1$, but $(A,Q) = v > 1$. Then
\[ S(A/Q,B/Q) = 0.\]
\end{lemma}
\begin{proof}
Express $\frac{A_\alpha}{Q} = \frac{a_\alpha}{R}$ and $\frac{B_i}{Q} = \frac{B_i}{Rv}$. Expressing
\[ r = pR + l\]
we have
\begin{align*}
    P_{A/Q}(r) + \frac{B}{Q} \cdot r \equiv P_{A/Q}(l) + \frac{B}{Rv} \cdot l + \big( \frac{B}{v} \cdot p \big)
\end{align*}
In particular, since $v$ does not divide at least one of the $B_i$,
\begin{align*}
    S(A/Q,B/Q) &= \frac{1}{R^D} \sum_{l \in (R)^D} e(- P_{A/Q}(l) - \frac{B}{Q} \cdot l) \times \Big( \frac{1}{v^D} \sum_{p \in (v)^D } e( - \frac{B}{v} \cdot p) \Big) \\
& \qquad = 0.
\end{align*}
\end{proof}

Next, define
\[ \Phi_{j,\nu}(\beta) := \int e(-P_\nu(t) - \beta \cdot t) \psi_j(t) \ dt,\]
and
\[ \Phi_{j,\nu}^*(\beta) = \Phi_{j,\nu}(\beta) \cdot \mathbf{1}_{|\nu_\alpha| \leq j^{A_0} 2^{-j |\alpha|}}.\]

With
\begin{align*}
    L_{j,\lambda}^s(\beta) = \sum_{ \frac{A}{Q} : Q \sim 2^s} \sum_{B \in (Q)^D} S(A/Q,B/Q) \Phi_{j, \lambda - A/Q}^*(\beta - B/Q) \chi_s(\beta - B/Q)
\end{align*}
for $\chi_s$ a Schwartz function which satisfies
\begin{align*}
    \mathbf{1}_{|\beta_i| \leq  2^{-2^{10 \rho s}}} \leq \chi_s(\beta) \leq  \mathbf{1}_{|\beta_i| \leq 10 \cdot 2^{-2^{10 \rho s}}}
\end{align*}
for $\rho  > 0$ an extremely small constant determined below, consolidate
\begin{align*}
    L_{j,\lambda}(\beta) := \sum_{s: 2^s \leq j^{A_0}} L_{j,\lambda}^s(\beta).
\end{align*}

By arguing similarly to \cite{BK11,BK12}, we show that
\[ \sup_{\lambda} |L_{j,\lambda}^{\vee} * f| \]
well approximates
\[ \sup_{\lambda} |m_{j,\lambda}^{\vee} * f| \]
provided that
\begin{align}\label{e:Xk}
\lambda \in X_j := \prod_{\alpha} \Big( \bigcup_{q \leq j^{A_0}} \mathbb{Z}/q \mathbb{Z} + O(j^{A_0} 2^{-j|\alpha|}) \Big)
\end{align}

\begin{lemma}
Let $2^{-j} \leq \delta \leq 1$ be a small constant, and suppose that $|\lambda_\alpha - \frac{A_\alpha}{Q}| \leq \delta \cdot 2^{-j(|\alpha| -1)}$ for each $\alpha \in \Gamma$, and that $|\beta_i - \frac{B_i}{Q}| \leq \delta$ for each $1 \leq i \leq D$.

Then
\begin{align*}
    m_{j,\lambda}(\beta) = S(A/Q,B/Q) \Phi_{j,\lambda-A/Q}(\beta - B/Q) + O(Q \delta).
\end{align*}
\end{lemma}
\begin{proof}
With $m = pQ + r$, express
\begin{align*}
    P_{\lambda}(pQ + r) + \beta \cdot (pQ + r) &\equiv P_{A/Q}(r) + B/Q \cdot r \\
& \qquad + \big( P_{\lambda - A/Q}(pQ) + (\beta - B/Q) \cdot pQ \big) + O(Q \delta) \mod 1
\end{align*}
Summing yields
\begin{align*}
    m_{j,\lambda}(\beta) &= \sum_{p,r} \psi_j(pQ) e(- P_\lambda(pQ + r ) - \beta \cdot (pQ + r)) + O(Q \cdot 2^{-j}) \\
    & \qquad = S(A/Q,B/Q) \cdot \sum_p Q^D \psi_j(pQ) \cdot e(-P_{\lambda-A/Q}(pQ) - (\beta -B/Q) \cdot pQ) + O(Q \delta) \\
    & \qquad \qquad = S(A/Q,B/Q) \cdot \Phi_{j,\lambda-A/Q}(\beta - B/Q) + O(Q\delta)
\end{align*}
by a Riemann sum approximation.
\end{proof}

\begin{lemma}
Suppose that $L_{j,\lambda}(\beta) \neq 0$. Then there exists precisely one $(A/Q,B/Q)$ with $Q \leq j^{A_0}$ so that
\begin{align*}
    L_{j,\lambda}(\beta) = L_{j,\lambda}^s(\beta) = S(A/Q,B/Q) \Phi_{j,\lambda - A/Q}(\beta - B/Q) \chi_s(\beta - B/Q)
\end{align*}
if $Q \sim 2^s$.
\end{lemma}
\begin{proof}
Suppose that there exists some $(A'/Q',B'/Q')$ so that
\[ |\lambda_\alpha - \frac{A_\alpha'}{Q'}| \leq j^{A_0} 2^{-j |\alpha|}\]
If $A'/Q' \neq A/Q$, we would have the following chain of inequalities.
\[ j^{-2A_0} \leq \frac{1}{Q Q'} \leq |\frac{A_\alpha}{Q} - \frac{A_\alpha'}{Q'} | \lesssim j^{A_0} 2^{-2j} \]
Now, if $Q' \sim 2^{s_0}$ with $s_0 > s$, then $S(A'/Q',B'/Q') = 0$, as $A'/Q'$ would not be in reduced form. The only case to check is when there exist $B'/Q' \neq B/Q$ so that
\begin{align*}
    |\beta_i - \frac{B_i}{Q}|, \ |\beta_i - \frac{B'_i}{Q'}| \lesssim 2^{-2^{10 \rho s}}.
\end{align*}
If $B/Q \neq B'/Q'$, then
\[ |\frac{B_i}{Q} - \frac{B_i'}{Q'}| \geq \frac{1}{QQ'} \approx 2^{-2s},\]
for the desired contradiction.
\end{proof}

\subsection{Major Arcs}
Let $\epsilon_0 = 2^{-10}$, and for $Q \leq 2^{\epsilon_0 j}$, define
\begin{align*}
    \mathfrak{M}_j(A/Q,B/Q) := \{ (\lambda,\beta) \in \mathbb{T}^{|\Gamma|} \times \mathbb{T}^D : |\lambda_\alpha - \frac{A_\alpha}{Q}| \leq 2^{(\epsilon_0 - |\alpha|)j}, \ |\beta_i - \frac{B_i}{Q}| \leq 2^{(\epsilon_0 - 1)j} \},
\end{align*}
and collect
\begin{align*}
    \mathfrak{M}_j = \bigcup_{Q \leq 2^{\epsilon_0 j}} \mathfrak{M}_j(A/Q,B/Q)
\end{align*}

\begin{proposition}
Suppose $\lambda \in X_j$. Then there exists some $c_0 = c_0(d,D)> 0$ so that
\begin{align*}
    |m_{j,\lambda}(\beta) - L_{j,\lambda}(\beta)| \leq 2^{-c_0 \epsilon_0 j}.
\end{align*}
\end{proposition}
\begin{proof}
First suppose that $(\lambda,\beta) \notin \mathfrak{M}_j$. Then for all $Q \leq 2^{\epsilon_0 j}$,
\begin{align*}
    \sum_{\alpha \in \Gamma} \| Q \lambda_\alpha \|_{\mathbb{T}} \cdot 2^{j|\alpha|} + \sum_{i=1}^D  \| Q \beta_i \|_{\mathbb{T}}  \cdot 2^j > 2^{\epsilon_0 j}.
\end{align*}
So, $|m_{j,\lambda}(\beta)| \lesssim 2^{-\epsilon' \theta j} + 2^{-\theta j}$ for some appropriate $\theta = \theta(d,D) > 0$.

We next observe that for each $Q \leq j^{A_0}$ the Euclidean coefficient norm of the phase
\begin{align}\label{e:coeffn}
    t \mapsto  P_{\lambda - A/Q}(2^j t) + (\beta - \frac{B}{Q}) \cdot 2^j t
\end{align}
is $\geq 2^{\epsilon_0 j}$, see \eqref{e:Ecn}. Consequently,
\begin{align*}
    |L_{j,\lambda}(\beta)| \lesssim 2^{-\epsilon_0 j \theta }, \; \; \; \theta = 1/d
\end{align*}
by stationary phase estimates, see \eqref{e:coeffnorm}. Above, we used the fact that $L_{j,\lambda}(\beta) = L_{j,\lambda}^s(\beta)$ for some unique $s = s(\lambda,\beta)$, and once again the fact that for each $Q \leq 2^{\epsilon_0 j}$
\begin{align*}
\| \eqref{e:coeffn} \| = \sum_\alpha |\lambda_\alpha - \frac{A_\alpha}{Q}| \cdot 2^{j|\alpha|} + \sum_i |\beta_i - \frac{B_i}{Q}|\cdot 2^j \geq 2^{\epsilon_0 j},
\end{align*}
see \eqref{e:Ecn}.

Next, suppose that $(\lambda,\beta) \in \mathfrak{M}_j(A/Q,B/Q)$ with $Q \sim 2^{s_0} \leq j^{A_0}$.

Since $\lambda \in X_j$, there exists some $\frac{A'}{Q'}$ so that
\begin{align*}
    |\lambda_\alpha - \frac{A'_\alpha}{Q'} | \leq j^{A_0} 2^{-j |\alpha|},
\end{align*}
which forces $A'/Q' = A/Q$, as otherwise one would arrive at the following chain of inequalities for some $\alpha$:
\begin{align*}
j^{-2A_0} \leq |\frac{A_\alpha}{Q} - \frac{A'_\alpha}{Q'}| \leq |\lambda_\alpha - \frac{A_\alpha}{Q}| + |\lambda_\alpha - \frac{A_\alpha'}{Q'}| \leq 2^{(\epsilon_0 -|\alpha|)j} + j^{A_0} 2^{-j|\alpha|},
\end{align*}
Since $\beta \in \mathfrak{M}_j(A/Q,B/Q)$
\begin{align*}
    |\beta_i - B_i/Q| \leq 2^{(\epsilon_0 - 1)j} \ll 2^{-2^{c_0 s}} \ll 2^{-2^{10 \rho s}}
\end{align*}
so
\begin{align*}
    L_{j,\lambda}^{s_0}(\beta) = S(A/Q,B/Q) \Phi_{j,\lambda - A/Q}^*(\beta - B/Q)
\end{align*}
while
\begin{align*}
    m_{j,\lambda}(\beta) = S(A/Q,B/Q) \Phi_{j,\lambda- A/Q}^*(\beta - B/Q) + O(2^{(2\epsilon_0 - 1)j})
\end{align*}
where we have recalled the bounds: $Q \leq 2^{\epsilon_0 j}$ and $\delta \leq 2^{(\epsilon_0 -1)j}$.

Finally if $(\lambda,\beta) \in \mathfrak{M}_j(A/Q,B/Q)$ with $j^{A_0} < Q \leq 2^{\epsilon_0 j}$, we would necessarily have $A/Q = A'/Q'$ for some $A'/Q' \in X_j$ (so $Q' \leq j^{A_0})$. This would force $(A,Q) > 1$, and thus
\begin{align*}
    L_{j,\lambda}(\beta), m_{j,\lambda}(\beta) = O(2^{(2\epsilon_0 - 1)j})
\end{align*}
\end{proof}

\begin{lemma}
Set
\begin{align*}
    \mathcal{E}_{j,\lambda}(\beta) := m_{j,\lambda}(\beta) - L_{j,\lambda}(\beta)
\end{align*}
Then there exists $c_0 > 0$ so that
\begin{align*}
    \sup_\lambda \| \partial_\lambda^{\alpha_1,\dots,\alpha_m} \mathcal{E}_{j,\lambda}^{\vee} * f \|_{\ell^p(\mathbb{Z}^D)} \lesssim_{d,D} 2^{-c_0 \epsilon_0 \frac{2}{p*} j} \cdot 2^{j(|\alpha_1|+ \dots + |\alpha_m|)} \| f\|_{\ell^p(\mathbb{Z}^D)}
\end{align*}
for each $\alpha_1,\dots,\alpha_m, m \leq |\Gamma|, \ 1 < p < \infty$.
\end{lemma}
\begin{proof}
The $\ell^2$ estimate without any derivatives was just proven. To handle derivatives on $\ell^2$, just observe that
\begin{align*}
\partial_\lambda^{\alpha_1,\dots,\alpha_m} m_{j,\lambda} = 2^{j(|\alpha_1|+\dots+|\alpha_m|)} m_{j,\lambda}^{\alpha_1,\dots,\alpha_m}
\end{align*}
where $m_{j,\lambda}^{\alpha_1,\dots,\alpha_m}$ is like $m_{j,\lambda}$, except the amplitude $\psi_j(t)$ is replaced by
\begin{align*}
\prod_{i=1}^m \frac{t^{\alpha_i}}{2^{j|\alpha_i|}} \psi_j(t),
\end{align*}
which satisfies all of the same differential inequalities as does $\psi_j$ up to an absolute constant depending on $d,D$; similarly for $L_{j,\lambda}$.

To handle the $\ell^p$ estimates, bound
\begin{align}\label{e:pointL}
    |\partial_\lambda^{\alpha_1,\dots,\alpha_m} L_{j,\lambda}^{\vee}(x)| \lesssim j^{O_{d,D}(A_0)} \cdot 2^{j(|\alpha_1|+\dots+ |\alpha_m|)} \cdot 2^{-j D} \cdot (1 + 2^{-j}|x|)^{-100},
\end{align}
and
\begin{align*}
    |\partial_\lambda^{\alpha_1,\dots,\alpha_m} m_{j,\lambda}^{\vee}(x)| \lesssim 2^{j(|\alpha_1|+\dots+ |\alpha_m|)} \cdot |\psi_j(x)|.
\end{align*}
To see \eqref{e:pointL}, we just bound
\begin{align*}
    |L_{j,\lambda}^{\vee}(x)| &\lesssim j^{\epsilon} \cdot \max_{2^s \lesssim j^{A_0}} |(L_{j,\lambda}^s)^{\vee}(x)| \\
    & \qquad \lesssim j^{O_{d,D}(A_0)} \max_{B/Q : Q \leq j^{A_0}, \ 2^s \lesssim j^{A_0}} | \big( \Phi_{j, \lambda - A/Q}^*(\beta - B/Q) \chi_s(\beta - B/Q) \big)^{\vee}(x)| \\
    & \qquad \lesssim j^{O_{d,D}(A_0)} \cdot 2^{-j D} \cdot (1 + 2^{-j}|x|)^{-100},
\end{align*}
where the final estimate follows since the spatial scale of $2^j \gg 2^{2^{10 \rho s}}$ is so large compared to that of $\chi_s$.
\end{proof}

\begin{proposition}
There exists some $c= c(d,D) > 0$ so small that
\begin{align*}
    \| \sup_{\lambda \in X_j} |\mathcal{E}_{j,\lambda}^{\vee}*f| \|_p \lesssim 2^{-c \epsilon_0 \cdot \frac{2}{p^*} \cdot j} \|f\|_p
\end{align*}
where $p^* := \max\{p,p'\}$.
\end{proposition}
\begin{proof}
By decomposing $X_j$ into $j^{2 \cdot A_0|\Gamma|}$ many boxes of dimensions
\begin{align*}
\{ [2^{-j|\alpha|} ] : \alpha \in \Gamma\},
\end{align*}
it suffices to prove the estimate for a single box.

The proof is by induction on $|\Gamma|$. Thus, let
\begin{align*}
\mathfrak{P}(R)
\end{align*}
denote the statement that for all $F : Q \times \mathbb{Z}^{D} \to \mathbb{C}$ with $Q$ a box of side-lengths $\{ [L_i] : 1\leq i \leq R\}$ satisfying
\begin{align*}
\sup_\lambda \| \partial^\gamma_\lambda F(\lambda;x) \|_{\ell^p(X)} \leq 100^{|\gamma|} \cdot  \vec{L}^{-\gamma} \cdot a(p)
\end{align*}
where $\max_i \gamma_i \leq 1$, the following estimate holds for some absolute constant $C_R < \infty$:
\begin{align*}
\| \sup_\lambda | F(\lambda;x)|  \|_{\ell^p(X)} \leq C_R \cdot a(p).
\end{align*}
Note that once we have established $\mathfrak{P}(R)$ for $R = |\Gamma|$, we may specialize
\begin{align*}
F(\lambda;x) := \mathcal{E}_{j,\lambda}^{\vee}*f(x),
\end{align*}
where $a(p) = 2^{-c_0 \epsilon_0 \frac{2}{p*}\cdot j} \cdot \| f\|_{\ell^p(\mathbb{Z}^D)}$.

Note that since the statement is translation invariant, we can and will assume that each box is centered at the origin; by dilation invariance we may assume that $L_i = 1$ for each $i$.

When $R = 1$, so $Q = [1]$, the result follows from the pointwise bound
\begin{align*}
|F(\lambda;x)|^p &\lesssim |F(\mu;x)|^p + \Big| \int_{[\mu,\lambda]} \partial_t F(t;x) \ dt \Big|^p \\
& \qquad \leq |F(\mu;x)|^p + \int_{[\mu,\lambda]} |\partial_t F(t;x)|^p \ dt \\
& \qquad \qquad \leq |F(\mu;x)|^p + \int_{[1]} |\partial_t F(t;x)|^p \ dt
\end{align*}
where $\mu \in [1]$ is arbitrary and
\begin{align*}
\sup_{t \in [1]} \| \partial_t F(t;x) \|_{\ell^p(\mathbb{Z}^D)} \leq a(p).
\end{align*}

For the inductive statement, express $\lambda = (\lambda',\lambda_R)$, and pointwise bound
\begin{align*}
|F(\lambda',\lambda_R;x)|^p &\lesssim \sup_{\lambda'} |F(\lambda',\mu_R;x)|^p + \Big| \int_{[\mu_R,\lambda_R]} \partial_t F(\lambda', t;x) \ dt \Big|^p \\
& \qquad \leq \sup_{\lambda'} |F(\lambda',\mu_R;x)|^p + \int_{[1]} \sup_{\lambda'} |\partial_t F(\lambda',t;x)|^p \ dt \\
& \qquad \qquad \leq \sup_{\lambda'} |F(\lambda',\mu_R;x)|^p + \int_{[1]} \sup_{\lambda'} |\partial_t F(\lambda',t;x)|^p \ dt,
\end{align*}
where $\mu_R \in [1]$ is arbitrary.

But, $\mathfrak{P}(R-1)$ applies to both
\begin{align*}
\lambda' \mapsto F(\lambda',\mu_R;x), \; \; \; \lambda' \mapsto \partial_t F(\lambda',t;x)
\end{align*}
uniformly in $\mu_R,t$, with the same constant $a(p)$, which closes the induction.

\end{proof}

\section{Analytic Estimates}\label{s:finish}
In the previous section, we reduced matters to estimating
\begin{align*}
\sup_{j_0,\lambda} |\sum_{j=1}^{j_0} L_{j,\lambda}^{\vee}*f|.
\end{align*}
We will decompose this maximal function by pigeon-holing in the sizes of the denominators of our rational approximates to $\{\lambda_\alpha\}$.

Define the operator
\begin{align*}
    L_\lambda^{s;j_0} := \sum_{2^{s/A_0} \leq j \leq j_0} L_{j,\lambda}^s,
\end{align*}
and, for any given $\lambda,j_0$ majorize

\begin{align*}
|\sum_{j=1}^{j_0} L_{j,\lambda}^{\vee}*f| & \leq \sum_{s = 1}^{\infty} \Big|\sum^{j_0}_{j : j^{A_0} \geq 2^s} \big( L^s_{j,\lambda}\big)^{\vee}* f| \\
& \qquad \leq \sum_{s =1}^{\infty} \sup_{\mu, j_0} |\big(L^{s;j_0}_\mu\big)^{\vee}*f|.
\end{align*}

In particular, the proof will be complete once we have proven the following proposition.

\begin{proposition}\label{p:mainprop}
There exists an absolute $c = c(d,D,p)$ so that for each $s \geq 1$,
\begin{align*}
    \| \sup_{\lambda,j_0} |(L_\lambda^{s;j_0})^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-c s} \| f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align*}
\end{proposition}

We introduce some further notation to stream-line the proof.

For a bounded multiplier, $m$, define
\begin{align*}
    \mathscr{L}_{s,A/Q}[m](\beta) := \sum_{B \in (Q)^D} S(A/Q,B/Q) m(\beta - B/Q) \chi_s(\beta - B/Q)
\end{align*}
where $\chi_s$ is as above. We next recall the Ionescu-Wainger exhaustion of the rationals: there exists a function
\begin{align*}
    h: \mathbb{Q}^D \to 2^{\mathbb{N}}
\end{align*}
so that
\begin{align*}
    h(B/Q) \leq Q
\end{align*}
if $(B,Q) = 1$, and if
\begin{align*}
    \mathcal{U}_s := \{ B/Q : h(B/Q) = 2^s \}
\end{align*}
then the following holds:
\begin{itemize}
    \item $\mathcal{U}_s \subset \{ B'/Q' : Q' \leq 2^{2^{s \rho}} \}$, where $\rho > 0$ is chosen sufficiently small relative to all other parameters introduced; and
    \item If $\chi_s'$ is like $\chi_s$, but is one on its support, then multipliers
    \begin{align*}
    \Pi_{m,s}(\beta) = \sum_{\theta \in \mathcal{U}_s} m(\beta - \theta) \cdot \chi_s'(\beta - \theta)
\end{align*}
satisfy
\begin{align*}
  \| \Pi_{m,s} \|_{M_p(\mathbb{Z}^D)} \lesssim \| m \|_{M_{2r}(\mathbb{R}^D)}
\end{align*}
for any $(2r)' \leq p \leq 2r$, where $M_p(X)$ denotes the multiplier norm
\begin{align*}
    \| M \|_{M_p(X)} := \sup_{\| f\|_{L^p(X)} = 1} \| M^{\vee}*f \|_{L^p(X)}, \; \; \; X = \mathbb{Z}^D, \mathbb{R}^D.
\end{align*}
\end{itemize}
This construction is ultimately due to Tao, \cite{Tao}, building of breathrough work of Ionescu and Wainger \cite{IW} and subsequent refinements \cite{MST1,MSZK}.

Note that we can factor
\begin{align}\label{e:factor}
    \mathscr{L}_{s,A/Q}[m] = \mathscr{L}_{s,A/Q}[1] \cdot \Pi_{m,s},
\end{align}
which is a key point in establishing Lemma \ref{l:borrow}.

We now observe the following identity, which we capture in the following lemma.
\begin{lemma}\label{l:easymaj}
The following identity holds:
\begin{align*}
    \mathscr{L}_{s,A/Q}[m]^{\vee}(n) &= \sum_{B \in (Q)^D} S(A/Q,B/Q) e(B/Q n) (m \chi_s)^{\vee}(n) \\
    & \qquad = e(-P_{A/Q}(n)) \cdot (m \chi_s)^{\vee}(n)
\end{align*}
and in particular,
\begin{align*}
    \sup_{A/Q} |\mathscr{L}_{s,A/Q}[m]^{\vee}*f(n)| \leq |(m \chi_s)^{\vee}|*|f|(n)
\end{align*}
pointwise.
\end{lemma}
\begin{proof}
The proof is just computation:
\begin{align*}
&\sum_{B \in (Q)^D} S(A/Q,B/Q) \int m(\beta - B/Q) \chi_s(\beta - B/Q) e(\beta n) \ d\beta \\
& \qquad = \sum_{B \in (Q)^D} S(A/Q,B/Q) \cdot e(B/Q n) \cdot (m \chi_s)^{\vee}(n) \\
& \qquad \qquad = \sum_{r \in (Q)^D} e( - P_{A/Q}(r) ) \cdot \frac{1}{Q^D}  \sum_{B \in (Q)^D} e(-B/Q\cdot (r-n)) \cdot (m \chi_s)^{\vee}(n) \\
& \qquad \qquad \qquad = e(-P_{A/Q}(n)) \cdot (m \chi_s)^{\vee}(n),
\end{align*}
where we used the relationship
\begin{align*}
\frac{1}{Q^D} \sum_{B \in (Q)^D} e( B/Q \cdot x) = \begin{cases} 1 & \text{ if } x \equiv 0 \mod Q \\
0 & \text{ otherwise. }
\end{cases}
\end{align*}
\end{proof}

With this notation in mind, we may express
\begin{align*}
    L_\lambda^{s;j_0} = \mathscr{L}_{s,A/Q}[ \Phi_{\lambda-A/Q}^{s;j_0}]
\end{align*}
where
\begin{align*}
    \Phi_\lambda^{s;j_0} = \sum_{2^{s/A_0} \leq j \leq j_0} \Phi_{j,\lambda}^*
\end{align*}
and $A/Q$ is the unique element with $Q \sim 2^s$ so that
\begin{align*}
    |\lambda_\alpha - \frac{A_\alpha}{Q}| \leq 2^{-10s-10},
\end{align*}
or an arbitrary element of the complement otherwise (note that in this case $L_\lambda^s(\beta) = 0$).

\begin{lemma}\label{l:smallgs}
    There exists an absolute $c = c(d,D,p) > 0$ so that for every $s \geq 1$
    \begin{align*}
        \| \sup_{A/Q : Q \sim 2^s} |\mathscr{L}_{s,A/Q}[1]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-cs} \|f\|_{\ell^p(\mathbb{Z}^D)}.
    \end{align*}
\end{lemma}
\begin{proof}
By interpolation with Lemma \ref{l:easymaj}, it suffices to prove the estimate at $\ell^2$. We apply a $TT^*$ argument, and are left to consider the kernel,
\begin{align*}
    \mathcal{K}_0(x,n) &= \sum_{B \in (Q)^D, \ B' \in (Q')^D} S(A/Q,B/Q) \overline{S(A'/Q',B'/Q')} \\
    & \qquad \times \sum_m e(B/Q (x-m)) e(-B'/Q'(n-m)) (\chi_s')^{\vee}(x-m) \overline{(\chi_s')^{\vee}}(n-m)
\end{align*}
where $A/Q = A(x)/Q(x)$ and $A'/Q' = A'(n)/Q'(n)$; our job is to show that
\begin{align}\label{e:kern0}
    \| \mathcal{K}_0 \|_{\text{Ker}(\mathbb{Z}^D)} \lesssim 2^{-c_0 s}.
\end{align}

We will do so by bounding
\begin{align*}
    |\mathcal{K}_0(x,n)| \lesssim 2^{-c_0s} \rho_s(v) + (\mathbf{1}_{E_{\text{Per}}(A/Q)} \cdot \rho_s)(v)
\end{align*}
where
\[ \rho_s(x) := \sum_n (\chi_s')^{\vee}(x-n) \cdot (\chi_s')^{\vee}(n) \]
has spatial scale $2^{2^{10 \rho s}}$, and $E_{\text{Per}}(A/Q)$ is the $Q$-periodic extension of some subset, $E(A/Q) \subset (Q)^D$ depending only on $A/Q$ which has density $2^{-c_0s}$.

In particular, we bound
\begin{align*}
\sup_x \sum_n |\mathcal{K}_0(x,n)| \lesssim 2^{-c_0 s}, \; \; \; \sup_n \sum_x |\mathcal{K}_0(x,n)| \lesssim 1,
\end{align*}
so \eqref{e:kern0} follows from Schur's test.

Turning to $\mathcal{K}_0(x,n)$, since $\chi_s'$ has such a small Fourier support, the sum vanishes unless $B/Q = B'/Q'$, as can be seen by applying Poisson summation. The only way that can happen is if $Q|Q'$ or vice versa; in either event we would find $Q = Q'$, since both have size $\sim 2^s$. This leads to the diagonalization
\begin{align*}
    \mathcal{K}_0(x,n) &= \sum_{B \in (Q)^D} S(A/Q,B/Q) \overline{S(A'/Q,B/Q)} \cdot  e(B/Q(x-n)) \cdot \rho_s(x-n) \\
    & \qquad = \frac{1}{Q^D} \sum_{r \in (Q)^D} e( - P_{A/Q}(x-n + r) + P_{A'/Q}(r)) \cdot \rho_s(x-n).
\end{align*}
We claim that there exists an absolute $c_0>0 $ so that
\begin{align}\label{e:rare}
   |\frac{1}{Q^D} \sum_{r \in (Q)^D} e( - P_{A/Q}(v + r) + P_{A'/Q}(r))| \leq 2^{-c_0s}  + \mathbf{1}_{E(A/Q)}(v)
\end{align}
where $E(A/Q) \subset (Q)^D$ has density $2^{-c_0s}$; consequently
\begin{align*}
|\mathcal{K}_0(x,n)|  \leq 2^{-c_0s} \rho_s(x-n) + (\mathbf{1}_{E_{\text{Per}}(A/Q)} \cdot \rho_s)(x-n).
\end{align*}
where $E_{\text{Per}}(A/Q)$ is the $Q$-periodic extension of $E(A/Q)$. But, since $P$ has no linear terms
\begin{align*}
    N_Q(P_{A/Q}) \geq 2^{s-1}
\end{align*}
since for any $|\alpha| \geq 2$
\begin{align*}
    \min_{q \leq 2^{s-2}} Q^{|\alpha|} \cdot \| q \frac{A_\alpha}{Q} \|_{\mathbb{T}} \geq Q^{|\alpha| -1 } > 2^{s-2};
\end{align*}
similarly, $N_Q(P_{A'/Q}) \geq 2^{s-1}$. So, \eqref{e:rare} follows from Lemma \ref{c:sublevel1}. The details are as follows: with
\[ P_j(v) = \sum_\alpha \frac{A_\alpha}{Q} \alpha_j v^{\alpha - e_j},\]
a polynomial with coefficient norm $\gtrsim_d 2^s$, it suffices to show that
\begin{align}\label{e:rare1}
|\{ v \in (Q)^D : N_Q( \sum_{j=1}^D P_j(v) \cdot r^{e_j}) \leq Q^{\kappa} \}| \lesssim Q^{D - \kappa_0};
\end{align}
but the left-hand side of \eqref{e:rare1} is contained in
\begin{align*}
\bigcap_{j=1}^D \{ v \in (Q)^D : \min_{q \leq Q^{\kappa}} \| q \cdot P_j(v) \|_{\mathbb{T}} \leq Q^{\kappa - 1} \},
\end{align*}
which has measure bounded by $Q^{D - \kappa_0}$ by Lemma \ref{c:sublevel1}.
\end{proof}

We next recall the following Lemma from \cite{BK12}.
\begin{lemma}\label{l:borrow}
For $j \geq 1$ let $\mathcal{K}_j$ denote a mean-zero $\mathcal{C}^1$ function supported on $\{ |x| \approx 2^j\}$, with
\begin{align}
    2^{jD} |\mathcal{K}_j(x)| +     2^{j(D+1)} |\nabla \mathcal{K}_j(x)| \leq C
\end{align}
uniformly in $j \geq 1$. Let $\mathcal{K}^{a,b} := \sum_{a \leq j < b} \mathcal{K}_j$. Then there exists $c = c(d,D,p) > 0$ so that
\begin{align*}
    \| \sup_{A/Q,J \geq 1} |\mathscr{L}_{s,A/Q}[\widehat{\mathcal{K}^{0,J}}]^{\vee}* f | \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-cs} \|f\|_{\ell^p(\mathbb{Z}^D)}.
    \end{align*}
\end{lemma}
\begin{proof}
See \cite[Lemma 4.4]{BK12}, noting that the argument is invariant under rearrangement of the polynomial phase in the appropriate Gauss sums,
\[ S(A/Q,B/Q) = \frac{1}{Q^D} \sum_{r \in [Q]^D} e(- \sum_{\alpha} \frac{A_\alpha}{Q} r^\alpha - B/Q \cdot r),\]
subject to the estimate from Lemma \ref{l:smallgs} and the factorization \eqref{e:factor}.
\end{proof}

We now prove Proposition \ref{p:mainprop}.

\subsection{The Proof of Proposition \ref{p:mainprop}}
For $s \geq 1$ fixed, let
\begin{align*}
    \mathcal{J}_{l,\mu} := \{ j : j^{A_0} \geq 2^s, \ \| P_\mu(2^j \cdot) \| \sim 2^l \},
\end{align*}
see \eqref{e:Ecn}. Note that $|\mathcal{J}_{l,\lambda}| = O_{d}(1)$, see \cite[Lemma 2.1]{LX} for details; the key point is that there are only $O(d^2 \log A) = O_{d}(1)$ many scales $j$ so that
\begin{align*}
    A^{-1} \leq \frac{ \sum_{|\alpha| = k} 2^{jk} |\lambda_\alpha| }{\sum_{|\alpha| = k'} 2^{jk'} |\lambda_\alpha|} \leq A
\end{align*}
for $2 \leq k \neq k' \leq d$. By sparsifying our scales into $O_{d}(1)$ many sub-families, we will assume that
\begin{align*}
    \sup_{l} |\mathcal{J}_{l,\lambda}| \leq 1,
    \end{align*}
which we will index
\begin{align*}
    \mathcal{J}_{l,\lambda} = \{ j_l\}.
\end{align*}

We collect
\begin{align*}
    \mathbb{L}(j_0) := \{ l \leq - C s : j_l \leq j_0 \},
\end{align*}
where $C = C_{d,D,p}$ is a sufficiently large constant,

It suffices to bound

\begin{align*}
    \| \sup_{A/Q,\mu,j_0} |\sum_{l \in \mathbb{L}(j_0)} \mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-c s} \|f\|_{\ell^p(\mathbb{Z}^D)},
\end{align*}
and
\begin{align*}
    \| \sup_{A/Q,\mu} |\mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-c s} \cdot \min\{ 1, 2^{-cl} \} \cdot  \|f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align*}

We begin with the low frequency case; by direct computation, for any $A/Q$, we can express
\begin{align*}
    &\sum_n f(x-n) \sum_{l \in \mathbb{L}(j_0)} \sum_{B \in [Q]^D} S(A/Q,B/Q) \cdot e(B/Q n) \cdot \int \chi_s^{\vee}(n-t) e(-P_\mu(t)) \psi_{j_l}(t) \ dt \\
& \qquad = \sum_n f(x-n) \sum_{B \in [Q]^D} S(A/Q,B/Q) \cdot e(B/Q n) \cdot \int \chi_s^{\vee}(n-t) \Big( \sum_{l \in \mathbb{L}(j_0)}  \psi_{j_l}(t) \Big) \ dt \\
    & \qquad \qquad + O\Big(\sum_{l \leq -Cs} Q^D \cdot \sum_n |f(x-n)| \int |\chi_s^{\vee}(n-t)| |\psi_{j_l}(t)| |P_\mu(t)| \ dt \Big) \\
    & \qquad = \mathscr{L}_{s,A/Q}[\widehat{{K}^{J_-,J_+(\mu)}}]^{\vee}*f(x) \\
    & \qquad \qquad + O \Big( Q^D \cdot \sum_{l \leq -Cs} 2^l \cdot \sum_n |f(x-n)| \cdot \int |\chi_s^{\vee}(n-t)| |\psi_{j_l}(t)| \ dt \Big) \\
    & \qquad = \mathscr{L}_{s,A/Q}[\widehat{{K}^{J_-,J_+(\mu)}}]^{\vee}*f(x)
    + O\Big(2^{sD} \cdot 2^{-C s} \cdot M_{HL} f(x) \Big),
\end{align*}
where $J_-:= \min\{ j : j^{A_0} \geq 2^s \}$ and
\begin{align*}
    J_+(\mu) := \min \big\{ j_0, \max\{ j : \| P_\mu(2^j \cdot) \| \leq 2^{-Cs} \} \big\},
\end{align*}
see \eqref{e:Ecn}.

By Lemma \ref{l:borrow}, we bound
\begin{align*}
    \| \sup_{A/Q,\mu, j_0} |\sum_{l \in \mathbb{L}(j_0)} \mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-c s} \|f\|_{\ell^p(\mathbb{Z}^D)},
\end{align*}
provided that $C$ is chosen sufficiently large.

We now prove that for each $l$,
\begin{align*}
    \| \sup_{A/Q,\mu} |\mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} \lesssim 2^{-c s} \cdot \|f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align*}

First, we observe that

\begin{align*}
    \sup_{A/Q,\mu} | \mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f(x)| \lesssim M_{HL} f(x)
\end{align*}
by Lemma \ref{l:easymaj}. So, it suffices to exhibit the decay at the $\ell^2$ level.

We use the method of $TT^*$:
for an appropriate choice of linearizing functions,
\begin{align*}
A(x)/Q(x) : \mathbb{Z}^D \to \{ A/Q : Q \sim 2^s \}
\end{align*}
and
\begin{align*}
    P_{\mu(x)} : \mathbb{Z}^D \to \mathscr{P}_{d,D}
\end{align*}
we may bound
\begin{align*}
    \sup_{A/Q,\mu} |\sum_{l \in \mathbb{L}} \mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f(x)| \lesssim |\sum_n K(x,n) f(n)|,
\end{align*}
where
\begin{align*}
    K(x,n) &= \sum_{B \in (Q)^D} S( A(x)/Q(x), B/Q) e(B/Q(x-n)) \\
    & \qquad \times \int \chi_s^{\vee}((x-n) -t) e(-P_{\mu(x)}(t)) \psi_{j_l}(t) \ dt
\end{align*}
We exhibit an absolute $c > 0$ so that
\begin{align}\label{e:kernelest0}
 \| K  \|_{\text{Ker}(\mathbb{Z}^D)} \lesssim 2^{-c  s}
\end{align}
by $TT^*$. In particular, we will show that the integral operator with kernel
\begin{align*}
    & \mathcal{K}(x,z) = \sum_n K(x,n) \overline{K(z,n)} \\
    & =
    \sum_{B \in (Q)^D, B' \in (Q')^D } S( A(x)/Q(x), B/Q) \overline{S(A'(z)/Q'(z),B'/Q')} e(B/Q x - B'/Q' z) \\
    & \qquad  \times  \int
    \Big( \sum_n e(- (B/Q- B'/Q') n) \chi_s^{\vee}(x-n-t) \overline{\chi_s^\vee}(z-n-u) \Big) \\
    & \qquad \qquad \times e(-P_{\mu(x)}(t) + P_{\mu'(z)}(u)) \psi_{j_l}(t) \overline{\psi_{j_l}}(u) \ dt du
\end{align*}
satisfies
\begin{align}\label{e:schur1}
    \sup_x \sum_z |\mathcal{K}(x,z)| \lesssim 2^{-c_0 s}, \; \; \; \sup_z \sum_x|\mathcal{K}(x,z)| \lesssim 1,
\end{align}
at which point we can bound $\| \mathcal{K}  \|_{\text{Ker}(\mathbb{Z}^D)} \lesssim 2^{-c_0 /2 s}$ by Schur's test, from which \eqref{e:kernelest0} follows with $c = \frac{c_0}{4}$.

But, since $\chi_s$ has such small support, $\mathcal{K}(x,z)$ diagonalizes:
\begin{align*}
    \mathcal{K}(x,z) &= \sum_{B \in (Q)^D} S(A/Q,B/Q) \overline{S(A'/Q,B/Q)} e(B/Q(x-z)) \\
    & \qquad \times \int \rho_s(x-z) e(-P_{\mu(x)}(t) + P_{\mu'(z)}(u)) \psi_{j_l}(t) \overline{ \psi_{j_l}(u) } \ dt du \\
& \qquad \qquad + O(2^{-2^{10 \rho s}} \cdot 2^{-j_l D} \mathbf{1}_{|x-z| \lesssim 2^{j_l}})
\end{align*}
where the error term arises from approximating
\[ \sum_n \chi_s^{\vee}(a - n) \chi_s^{\vee}(b-n) \]
by
\begin{align*}
    \rho_s(a-b) = \sum_n \chi_s^{\vee}(a-b - n) \chi_s^{\vee}(n),
\end{align*}
using the smoothness of $\chi_s^{\vee}$ at spatial scales of the order $2^{2^{10 \rho s}}$.
By \eqref{e:rare}, we may bound
\begin{align*}
|\sum_{B \in (Q)^D} S(A/Q,B/Q) \overline{S(A'/Q,B/Q)} e(B/Q(x-z))| \lesssim 2^{-c_0 s} + \mathbf{1}_{E_{\text{Per}}(A/Q)}(x-z)
\end{align*}
where $E_{\text{Per}}(A/Q)$ is the $Q$-periodic extension of a subset $E(A/Q) \subset (Q)^D$ which depends only on $A/Q$, and has relative density $2^{-cs}$. Consequently
\begin{align*}
    |\mathcal{K}(x,z)| &\leq 2^{-c_0s} \cdot \int |\rho_s((x-z) - (t-u))| |\psi_{j_l}(t) \psi_{j_l}(u)| \ dtdy\\
& \qquad  + \int |\rho_s((x-t) - (z-u))| |\psi_{j_l}(t) \psi_{j_l}(u)| \ dtdy \cdot \mathbf{1}_{E_{\text{Per}}(A/Q)}(x-z).
\end{align*}
which establishes \eqref{e:schur1}, given the small Lipschitz norm of $\lesssim 2^{-2^{10 \rho s}}$ of
\[ v \mapsto \int |\rho_s(v-t+u)| |\psi_{j_l}(t) \psi_{j_l}(u)| \ dt du. \]

Finally, we just observe that for each $\theta \in \mathbb{T}^D$
\begin{align*}
    &\| \sup_\mu| \int \Phi_{l,\mu}(\beta-\theta) \chi_s(\beta - \theta) \hat{f}(\beta) e(\beta x) | \|_{\ell^p(\mathbb{Z}^D)} \\
& \qquad = \| \sup_\mu| \int \Phi_{l,\mu}(\beta) \chi_s(\beta) \hat{f}(\beta+\theta) e(\beta x) | \|_{\ell^p(\mathbb{Z}^D)}  \\
& \qquad \qquad  \lesssim 2^{-c l} \|f\|_{\ell^p(\mathbb{Z}^D)}, \; \; \; c = c_{d,D,p} > 0
\end{align*}
by applying Magyar-Stein-Wainger transference \cite{MSW} and the continuous result of Stein-Wainger \cite{SW}. Summing appropriately
\begin{align*}
    &\| \sup_{A/Q,\mu} | \mathscr{L}_{s,A/Q}[\Phi_{l,\mu}]^{\vee}*f | \|_{\ell^p(\mathbb{Z}^D)} \\
&\qquad  \leq \sum_{A/Q : Q \sim 2^s}     \| \sup_{\mu} | \mathscr{L}_{s,A/Q}[\Phi_{l,\mu}]^{\vee}*f | \|_{\ell^p(\mathbb{Z}^D)} \\
    & \qquad  \qquad \leq 2^{s(|\Gamma|+D)} \cdot \sup_\theta
        \| \sup_\mu| \int \Phi_{l,\mu}(\beta-\theta) \chi_s(\beta - \theta) \hat{f}(\beta) e(\beta x) | \|_{\ell^p(\mathbb{Z}^D)} \\
        &  \qquad \qquad \qquad \lesssim 2^{s(|\Gamma| +D)} \cdot 2^{-cl} \cdot \| f\|_{\ell^p(\mathbb{Z}^D)}.
\end{align*}
In particular, for any $l \geq 1$, we may bound
\begin{align*}
\| \sup_{A/Q,\mu} |\mathscr{L}_{s,A/Q}[\Phi_{j_l,\mu}]^{\vee}*f| \|_{\ell^p(\mathbb{Z}^D)} &\lesssim \min \{ 2^{-c s}, 2^{s (|\Gamma|+ D)} \cdot 2^{-cl} \} \cdot \|f\|_{\ell^p(\mathbb{Z}^D)} \\
& \qquad \lesssim 2^{-c_0 (s+l)} \cdot \|f\|_{\ell^p(\mathbb{Z}^D)},
\end{align*}
after interpolating appropriately.

redpajama_set_name: RedPajamaArXiv


In the event of a nuclear attack on the UK, would the NHS cope or would it collapse? At the height of the Cold War, answering this question became increasingly urgent for the UK government and the medical community.
n 1958, an official government booklet, 'The Hydrogen Bomb', admitted that in the event of a nuclear attack on the UK, "there would be far more injured people than even the expanded hospital service could cope with".
By the 1980s, the picture hadn't changed much, and those in power had few answers. "Casualties could occur on a scale which would overwhelm surviving hospital resources in some areas, depending on the scale of attack," admitted the Home Office's confidential guidance to the police.
To members of the medical peace movement, the prospects seemed less bright. The Medical Campaign Against Nuclear Weapons (MCANW) – formed in 1980 by health professionals, including doctors, nurses and researchers – employed medical arguments to make the case against nuclear weapons and the concept of a 'winnable' nuclear war.
[LEFT]: 'The Medical Consequences of Nuclear Weapons' was produced by The Medical Campaign Against Nuclear Weapons (MCANW), which was formed in 1980 to highlight the medical implications of a nuclear attack and to call for the abolition of nuclear weapons.
[RIGHT]: This map from 'The Medical Consequences of Nuclear Weapons' shows the likely effects of a nuclear blast and the predicted casualties.
MCANW also produced 'The Nuclear Casebook: an illustrated guide', which analysed discrepancies in official predictions of burns casualties.
Official predictions of nuclear casualties in the early 1980s were later understood to be serious underestimates.
Professional associations representing ordinary doctors and nurses conducted their own assessments on behalf of their members, too. Based on a realistic understanding of how well prepared the medical community was, these reports are arguably the most impartial, and the most revealing.
A panel from the British Medical Association (BMA) conducted a comprehensive, 18-month study – "an objective and scientific account" – of the medical implications of a nuclear attack. The results were made public in early 1983, and they made for grim reading.
The BMA found the Home Office's predicted casualty figures were significantly more conservative than most other estimates (including those of the US Office of Technology Assessment). The BMA estimated, for example, that there would be 2 to 3 million deaths from those burnt by exposure to an atomic blast; the Home Office chose not to take this factor into account, despite other government departments warning of vast numbers of burn casualties.
Government estimates suggested that fewer than a million deaths would result from fallout, while others estimated as many as 11 million. The BMA erred on the higher side, as Britain's dense population meant strategic Soviet targets, such as military bases, were frequently found close to population centres.
A character from Raymond Briggs's 'When the Wind Blows' reacts to the BMA report. Briggs' famous graphic novel depicts an elderly couple trying to survive a nuclear attack.
The BMA believed that the yield of enemy weapons that would strike the UK could be three to four times the government's assumptions.
Even seemingly positive actions could have catastrophic health implications: survivors were likely to adopt a 'Blitz spirit' and cluster together, facilitating the spread of disease to their radiation-weakened immune systems. There would be little that doctors could do.
The panel's conclusion was stark: Britain's medical community could only be prepared for nuclear catastrophe by being on a permanent war footing. Otherwise, even the smallest Soviet attack "would cause the medical services in the country to collapse".
MCANW suggested that action to prevent a nuclear war was more valuable than preparedness.
The BMA's report caused concern within the UK government. In an attempt to head off its impact, the Home Office wrote to scientific advisers across England and Wales, claiming the report was "strongly influenced by CND-type propaganda", and that it could not be regarded as an objective scientific document.
The Daily Mail attacked the "BMA horror report" as spreading "quack Kremlinology". However, this view was not shared within much of the medical community. World Medicine magazine called the report "uncharacteristically brave" and "meticulously non-political", but did point out that the report's publication itself – with its conclusion that the NHS could not survive a nuclear attack – meant the BMA had essentially taken a political position.
The nurses' union, the Royal College of Nursing, wrote its own 1983 report on nuclear war, calling the government's plans "totally inadequate", "naïve" and "misleading".
In the RCN's view, a post-apocalyptic society wouldn't see a return to medieval conditions, but something even worse as, unlike in medieval times, there would be virtually no access to safe food, clean water and adequate shelter.
This entry was posted in Cold War, England, History, Nuclear, United Kingdom. Bookmark the permalink.


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The Rideout Family
Family Collage Here
The Retreat Center
Our first few weeks in Bombard
The past few months have been a whirlwind for us. Matt has made several trips from Montrouis to Bombard ( 7+ hours each way!) to complete their transition. On the first trip in early February, Sarah drove a pickup with all 3 kids and a full load by herself. She handled the roads well and tackled them with confidence! Matt followed her with a van load of supplies and Bono, their 150lb Mastiff!
Matt, Olive and Bono on the way north
Sarah's drove 12 hours with all three kids and a full load of supplies
Matt's wasted no time helping Steve with improvements to the property. The first project they completed was the installation of the shop door. What a blessing it is to be able to use the shop to keep our vehicles maintained. Deli and Astrel were a big help. It was a great surprise to Steve when he pulled in the drive to see the shiny new door up in place of old rotted plywood. The guys also painted the roof of the Leaches house in preparation for new solar panels.
Shop before
Shop after
Painting the roof of the Leaches house
Ready for solar panels
Sarah has been busy with Faith learning at the Hospital. She has worked with Faith on Sonography cases. Sarah has also gotten to help with a couple of deliveries. During the second delivery, she was able to help the mother with techniques to finish strong and birth a beautiful little girl. Sarah has always dreamt of becoming a licensed Nurse Midwife, now she is getting plenty of practice!
The first baby girl Sarah helped deliver
Matt's first patients to be driven home in the ambulance
Big news for our family
Two years ago, Matt had the opportunity to go on an atv trip around remote Northwest Haiti with a friend. During his trip, they visited 5 missionary families who live and serve in the NW. One of the stops on the journey was to Hospital Evangelique Bombardpolis, run by Steve and Faith Leach (www.haitihospital.org).
Front gate of Hospital Evangelique Bombardpolis
The impression this place had on him never left, He often caught himself thinking of Bombard and wondered what was going on with the mission. In August, Matt heard that Steve and Faith were planning to leave Haiti due to personal health concerns after having been in Bombard for 21 years.
For months, we wrestled with the idea of contacting them. Finally, Matt emailed Steve and he confirmed that they were indeed leaving. The Leachs had begun tentative plans to turn the hospital over to the Dept of Health. This would have been disastrous. Not only would the hospital lose its evangelical puporse, it would suffer a drastic funding crisis, and a suffer from extremely poor dis-nterested management.
We decided we could not let this happen, and we began praying intentionally for the hospital. Matt phoned Steve and asked if we could meet face to face to discuss their plans in order to find out if we could play some role in them.
We were invited to make the 8 hour drive from our home in Montrouis to visit with Steve and Faith for a week and allow Sarah to see the hospital for the first time. Sarah and the kids were able to see more of Haiti on our trip to Bombard than they have seen in the 5 years we have lived in Haiti.
Our family with Steve & Faith Leach
We immediately fell in love with Bombard, her people, and fully realized how critically important it was that this hospital remain open. Over the course of four decades, God has expanded the hospital's capabilities and reach. The hospital/clinic today delivers healthcare to a population of at least 50,000. The remote location of the hospital heightens its local impact as there is no facility offering equivalent services within several hours' hard travel, and, arguably, none within a day's journey that offers these services at such minimal cost.
We are partnering with International Missionary Fellowship and moving to Bombardpolis with our family. We hope to be fully moved and settled by the end of January. This will give us the maximum amount of time to work along side Steve and Faith before their planned departure in November of 2017.
As a family, we are beginning to realize the impact this change in roles will have on us. Sarah will begin working more, helping to improve the nursing care offered to patients, as well as helping with midwifery in the Hospital. Matt will begin overseeing the physical plant of the mission and will have an administrative/ accounting focus.
Willow and Olive had a ball playing in the shade house!
What are our immediate prayer requests?
Our families physical needs will change as we will live in a significantly more remote area. The remote nature of our new location requires us to purchase significantly more supplies every 2 months when we travel to Port au Prince. We will need better evacuation insurance and more reliable transportation for our family.
Toyota Landcruiser designed for the rugged rural roads.
We are desperately praying for God to bring another family to come alongside us in this ministry. There is far more work to do than Sarah and I are able to do while still raising young children.
Lastly and most importantly, pray against spiritual warfare. The enemy will surely attack and attempt to rob us of joy, target our marriage, attempt to destroy relationships within the mission and community of Bombard.
What will happen with the Retreat Center?
We are willing to manage the RC until Nov 2017 (the date at which the Leaches will leave Haiti) in the same ways we do as when we are on furlough. Chadasha is working to identify new missionaries as soon as possible to begin the leadership transition.
We are overwhelmed at how God has orchestrated the past few weeks, and are excited to continue the work that IMF has been doing in Bombard.
If God wasn't undeniably calling us to a new position serving the poor of rural NW Haiti, we would still be fighting for the RC to continue. We believe in the vision and purpose of the RC with all of our hearts. From the beginning, we prayed that God would make it abundantly clear when our involvement with the RC would change. While we didn't anticipate that change coming this soon, we believe the time has arrived. Thanks for the privilege of serving with you, and for your sacrifice on our behalf to live and work in Haiti.
Help Support Our Work
Option 1 of 4: $25.00 USD - monthly Option 2 of 4: $50.00 USD - monthly Option 3 of 4: $75.00 USD - monthly Option 4 of 4: $100.00 USD - monthly
Give a single gift
Isaiah 11:6 "...and a little child will lead them". Watermark theme. Powered by Blogger.

redpajama_set_name: RedPajamaCommonCrawl


The biggest misconception in outdoor photography is the use of fill flash. Fill flash confused me more than anything when I first started trying to understand photography. It is like accounting. When you think you should debit, you credit. To us (unprofessional/photographers) the idea of when to use the flash was a bit baffling. For example, you are standing in the bright sun and common sense tells you that there is already too much light so why would you need to provide additional light. Most people know that if they are indoors, they use a flash. Most point and shoot camera flashes default to auto flash, meaning, if the camera detects there is not enough light for a good picture, the auto flash will kick-in, in low light. What most people don't know is when you are outside and your camera is set to auto flash, it will automatically shut down your flash. This is perhaps when you need your flash the most. For example, you are out of the beach and it is 5pm, shadows are heavy and your subjects faces are shaded. This is where your fill flash is essential.
Go to your settings, override the auto flash(make sure that the flash icon does not have a line through the lightning bolt or flash symbol) and force your camera to flash. This will fill in the shadows (often cast by hats, glasses, noses etc) and light up the faces of your subjects. On the other hand, if your subjects look overexposed and washed out, you can't decrease the flash strength try moving back a little from your subject and using your zoom to get a tighter framing as this will decrease the impact of the flash. Also remember, with most cameras, about 6 or 7 feet is a normal flash range. Experimenting is the key. Sample images above show the difference between an image outdoors with the flash on and the flash off.
Get out there and flash away. Find out which settings work for you. We will talk about creating mood with lighting later which we will play with not using flash to create different moods. But for now, when you are out in that bright light and you see shadows being cast, remember that the flash in bright conditions is your friend.
Upload your images so we can see what you have learned.


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Please visit http://code.google.com/p/luadbi/w/list for details on using LuaDBI

Changes to this repository include:

- insert_id in connection (MySQL only)
- fix for string lengths (from sergeyzavadski)

redpajama_set_name: RedPajamaGithub


Today I'm returning to an old love – Kraftin' Kimmie Stamps! I have been completely smitten with their recent releases and particularly the stamp I used on today's card. Annie Rodrigue is a firm favourite of mine, and I love her darker illustrations the best.
I decided to take some progress photos while colouring Dragon Queen, to illustrate my colouring process when working on an image using the "No Line Colouring" technique. I've put together a step by step guide which you will find below, along with the marker colour palette. I hope that you find these resources useful in your own colouring adventures. If you like them, let me know and I will do my best to bring you more in future.
NOTE: this blog post contains some large images.
Kraftin' Kimmie Stamps Dragon Queen and her friend Vampire Queen are available in my online store Quixotic Paperie along with a range of other fantastic stamps.
For readers in Canada/US you can of course find the stamps at Kraftin' Kimmie Stamps!
Kraftin' Kimmie Stamps Dragon Queen was stamped with Memento London Fog ink on my favourite Make it Colour Blending Card before being coloured using Copic markers and Faber-Castell Polychromos pencils. The card is A5 (8″x6″ approx.) size and is top folding. The background was coloured with Copic markers. The sentiment comes in the Dragon Queen stamp set.
I framed the image using Spellbinders Nestabilities Grand Labels Four dies. I embossed the frame with Stampendous Tinsel Red embossing powder. The spider webs poking out from under the frame were die cut using a La-La Land Crafts die. The designer paper came from my scrap drawer, I think it is from the Echo Park Chillingsworth Manor collection.
This entry was posted in Colouring, Marker Palettes, Tutorials & Videos and tagged Colouring, Copic, Kraftin' Kimmie Stamps, Markers, No Line Colouring, No Lines on October 6, 2014 by Elaine.
I love, love, LOVE your creativity and talent. I do my best to follow along with Spectrum Noir markers. Thanks for sharing!


Over the past 10 years iOra has built its reputation as the industry standard for replicating and compressing information over challenging and disrupted networks throughout the defence environment. We are entrusted to move terabytes of data to and from naval ships at sea.
Naval Forces are involved in a number of key activities from readiness to fight to preventing conflict and protecting essential trade routes. Many of these activities are performed in the midst of vast oceans where network connections are limited at best.
We enable Commanders to communicate effectively and reliably with other Forces or HQ in remote and hostile environments where communications are challenged by limited network capacity or the need for security. We guarantee access to mission-critical information such as orders and mission plans, no matter what the state of the network. Our technology operates across networks even as slow as 2 kilobytes per second.
Our patented differencing software reduces substantially the amount of information sent thus ensuring that limited network capacities are available for other critical communication and information transfer requirements such as voice and video to and from widely dispersed units and their Headquarters.
Our solution also offers secure, timely and efficient information sharing with allies and friendly forces enabling efficient and timely collaboration wherever and whenever, across the globe.
Because we can deliver information effectively and efficiently over challenging networks, we are the trusted partners for the US Marine Corps, US Navy and the UK Ministry of Defence as well as highly sensitive national collaborations and other key global security organisations.


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A Flutter plugin for a terminating a [package:gql_link][] `Link` with a Go endpoint.

[package:gql_link]: https://pub.dev/packages/gql_link

This is not an officially supported Google product.

## Building

1.  Install [Go][].
1.  `cd nm_gql_go_link`.
1.  Compile the Go code for your target platform. Any combination of:
    *   `go generate -tags android`
    *   `go generate -tags ios`.
    *   `go generate -tags macos`.

After GraphQL schema changes,

1.  `cd nm_gql_go_link`
1.  `flutter pub run build_runner build`

[Go]: https://golang.org


This 2011 work is an interesting novel, and extremely well written. But it is too long. Eugenides delves too much into the back history of his main characters, and too much into these characters' area of study. His narrative approach is interesting, as he advances his triangular story of two male students and one female, but then he often backtracks to explain why they are acting as they are in the basic narrative.
The main character is Madeline Hanna. We meet her in her graduation year at Brown University. She is fascinated by the brilliant student Leonard Bankhead, even after she learns he is a manic depressive and his conduct varies enormously. Meanwhile, the steady Mitchell Grammaticus, another Brown student, has fallen for her but is hesitant to reveal his true feelings.
We spend a long time at Brown, as these three meet and interact, before following them out into the adult world. We also meet Madeline's parents, Alton and Phyllida, who do not want to see their daughter tied down with a manic depressive. But they do not understand their daughter, and do become unsympathetic characters.
Eugenides makes palpable the years at Brown by having Madeline, an English major, probe deeply into semiotics and deconstruction. She becomes a feminist evaluating the Victorian novelists. One cannot help but think the author is taking advantage of his own education, and I do think he probes more deeply here than is necessary.
We see most of this story through Madeline, and some through Leonard, but there is a significant portion devoted to Mitchell as he travels to Europe and then to India, hoping that Madeline will get Leonard out of her system by the time he returns. In India, he volunteers to serve with Mother Teresa. (Note that Eugenides also volunteered with Mother Teresa.) It is not a successful experience, but expresses the religious yearnings that have long motivated Mitchell's life.
Madeline, meanwhile, is living with Leonard on Cape Cod, where he has a basic science job in the field of genetics. But the science investigations he is involved with never attain the credibility of Madeline's English studies.
I also did not sense the atmosphere of the Cape, or of the Boston they visit, but the work did earn some credibility from me when the author has Madeline's married older sister living in Beverly, where I grew up. But, alas, Madeline never visits there.
This work did not receive the enthusiastic reviews that Middlesex and Virgin Suicides received. And I can understand this, for, as I indicated, it is overwritten. It should have been a shorter work. It bogs down too much in details. But with my own prejudice, I also wonder how much this poor reception was influenced by Mitchell's (and the author's) interest in religion. How much did that turn off some reviewers? Did it make Mitchell unworthy of either Madeline, or of their own interest in the outcome of this triangle?
The treatment of Leonard's problem is also excessive. Eugenides wants the reader to understand Leonard's problem, but he goes too far in explaining manic depression. But, on the other hand, he turns around the reader's reaction to this character that Madeline is devoting her life to—unwisely, Mitchell believes, in behalf of the reader. For at the end, we have considerable sympathy for Leonard, based primarily on his awareness of his own condition and its impact on others.
This acknowledgement is the key to the ending. It is an ending which I fully accept, even if for some it might not be a real ending, not one readers expect from a story today. And yet it is a literate ending—a, for me, satisfying ending. Note that it is also an ending that satisfies an author's ideal goal, for it depends not on the final paragraph, not on the final line, but on the novel's final word.
The title The Marriage Plot, refers to an English thesis Madeline is writing at Brown. It analyzes the endings of old-fashioned Victorian novels that normally finds the heroine ending up happily married. And, indeed, Madeline's own life, and this very novel, will also refer to the validity of that kind of ending. Note that when she has this thesis published, she brings it proudly to Mitchell—an acknowledgement that it is he who matches her temperament and her needs most closely.
One does wonder how much of Eugenides himself is written into his character Mitchell. Did he yearn for a fellow student while at Brown? Is some of Mitchell's shyness and reticence his own? Was this the germ of his story? And did he put all his feelings for that woman into the character of Madeline? It is not impossible. And yet it might also explain why this work is not a complete success. That he was too close to the actual experience, that he did not have the proper distance. All, I grant, to be mere speculation.