# On $L^{12}$ square root cancellation for exponential sums associated with
nondegenerate curves in ${\mathbb{R}}^{4}$

Ciprian Demeter  Department of Mathematics, Indiana University, 831 East 3rd
St., Bloomington IN 47405<EMAIL_ADDRESS>
###### Abstract.

We prove sharp $L^{12}$ estimates for exponential sums associated with
nondegenerate curves in ${\mathbb{R}}^{4}$. We place Bourgain’s seminal result
[2] in a larger framework that contains a continuum of estimates of different
flavor. We enlarge the spectrum of methods by combining decoupling with
quadratic Weyl sum estimates, to address new cases of interest. All results
are proved in the general framework of real analytic curves.

###### Key words and phrases:

decoupling, Weyl sums, square root cancellation

The author is partially supported by the NSF grant DMS-1800305

AMS subject classification: Primary 42A45, Secondary 11L07

##  1\. Introduction

Throughout this paper, $\phi_{3},\phi_{4}$ will be real analytic functions
defined on some open interval containing $[\frac{1}{2},1]$, and satisfying

$\|\phi_{k}^{\prime}\|_{C^{3}}=\sum_{1\leq n\leq 4}\max_{\frac{1}{2}\leq t\leq
1}|\phi_{k}^{(n)}(t)|\leq A_{1},\;\;k\in\\{3,4\\},$ (1)

$A_{2}\leq\left|\det\begin{bmatrix}\phi_{3}^{(3)}(t)&\phi_{3}^{(4)}(t)\\\
\phi_{4}^{(3)}(s)&\phi_{4}^{(4)}(s)\end{bmatrix}\right|\leq
A_{3},\;\;t,s\in[\frac{1}{2},1],$ (2) $|\phi_{3}^{(3)}(t)|\geq
A_{4},\;\;t\in[\frac{1}{2},1].$ (3)

$A_{1},\ldots,A_{4}$ are positive numbers that will determine the implicit
constants in various inequalities. While $\phi_{3},\phi_{4}$ being $C^{4}$
would suffice for our purposes, we choose to work with real analytic functions
for purely aesthetic reasons. The examples of most immediate interest are
power functions $\phi_{3}(t)=t^{a}$, $\phi_{4}(t)=t^{b}$, with the real
numbers $a$ and $b$ satisfying the restrictions $a\not=b$ and
$a,b\not\in\\{0,1,2\\}$.

For a finite interval $I\subset{\mathbb{Z}}$, we write (ignoring the
dependence on $\phi_{k}$)

${\mathcal{E}}_{I,N}(x)=\sum_{n\in
I}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}+\phi_{4}(\frac{n}{N})x_{4}).$

We make the following conjecture.

######  Conjecture 1.1.

Assume $\alpha\geq\beta\geq 0$ and $\alpha+\beta=3$. Let
$\omega_{3}=[0,N^{\alpha}]$ and $\omega_{4}=[0,N^{\beta}]$. Assume
$\phi_{3},\phi_{4}$ are real analytic on $(0,3)$ and satisfy (1), (2) and (3).
Then

$\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)|^{12}dx\lesssim_{\epsilon}N^{9+\epsilon}.$
(4)

For virtually all conceivable applications of (4), both $\phi_{3}$ and
$\phi_{4}$ will satisfy (3). Because of this symmetry, our restriction
$\alpha\geq\beta$ is essentially meaningless.

Let us write $A\lesssim B$ or $A=O(B)$ if $|A|\leq CB$ for some, possibly
large, but universal constant $C$. The notation $A\sim B$ will mean that
$A\lesssim B$ and $B\lesssim A$ at the same time. We will write $A\ll B$ or
$A=o(B)$ if $|A|\leq cB$, for some small enough, universal constant $c$. The
notation $A{\;\lessapprox}\;B$ will mean $A\lesssim(\log N)^{O(1)}B$, where
$N$ will be the key scale parameter.

Since $|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)|\sim N$ if
$x\in[0,o(\frac{1}{N})]\times[0,o(\frac{1}{N^{2}})]\times[0,o(1)]^{2}$, the
exponent 9 is optimal in (4). While the large value $N$ is attained by
$|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)|$ for a small subset of $x$, estimate
(4) implies that the average value of the exponential sum on the given domain
is $O(N^{\frac{1}{2}+\epsilon})$, when measured in $L^{12}$. We call this
$L^{12}$ square root cancellation. The relevance of the requirement
$\alpha+\beta=3$ is that it guarantees

$|[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}|(\sqrt{N})^{12}=N^{9}.$

The case $(\alpha,\beta)=(2,1)$ of the conjecture has been settled in [2].
This inequality was proposed by Huxley [8], with $\phi_{3}(t)=t^{3/2}$,
$\phi_{4}(t)=t^{1/2}$. In [2], it serves as the main ingredient in sharpening
the record on the Lindelöf hypothesis.

The only other known case prior to our work was $(\alpha,\beta)=(3,0)$. The
variable $x_{4}$ and the function $\phi_{4}$ play no role in this case, as
$\phi_{4}(\frac{n}{N})x_{4}=O(1)$. We treat $e(\phi_{4}(\frac{n}{N})x_{4})$ as
a coefficient $c_{n}=O(1)$. In this case, (4) follows from the inequality

$\int_{[0,1]^{2}\times[0,N^{3}]}|\sum_{n\in
I}c_{n}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}|^{12}dx_{1}dx_{2}dx_{3}\lesssim_{\epsilon}N^{9+\epsilon}\|c_{n}\|_{l^{\infty}}.$

Assuming $\phi_{3}$ satisfies (1) and (3), this was known as the Main
Conjecture in Vinogradov’s Mean Value Theorem, and was first solved in [10],
then in [5]. This reduction fails for all other values $\alpha<3$, since the
length of the interval $\omega_{3}$ becomes too small. The proofs in [2] and
[5] for $\alpha=2$ and $\alpha=3$ are fairly different, in spite of both
relying entirely on abstract decoupling.

Our main result here verifies Conjecture 1.1 in the range
$\frac{3}{2}\leq\alpha<2$.

######  Theorem 1.2.

Assume that $\frac{3}{2}\leq\alpha\leq 2$. Assume $\phi_{3},\phi_{4}$ are real
analytic on $(0,3)$ and satisfy (1), (2) and (3).

Then

$\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)|^{12}dx\lesssim_{\epsilon}N^{9+\epsilon}.$
(5)

If $\phi_{3}(t)=t^{a}$, $\phi_{4}(t)=t^{b}$ with $\alpha\leq a$, $\beta\leq
b$, $a+b\leq 9$, $a\not=b$ and $a,b\not\in\\{0,1,2\\}$, a very simple
rescaling argument for each member of the sum

${\mathcal{E}}_{[1,N],N}(x)=\sum_{1\leq M\leq N\atop{M\text{
dyadic}}}{\mathcal{E}}_{[\frac{M}{2},M],N}(x)$

shows that (5) holds with ${\mathcal{E}}_{[\frac{N}{2},N],N}$ replaced with
${\mathcal{E}}_{[1,N],N}(x)$. In particular, this is always the case for the
moment curve $\phi_{3}(t)=t^{3}$, $\phi_{4}(t)=t^{4}$. Other cases such as
$(\alpha,\beta)=(2,1)$, $(a,b)=(\frac{3}{2},\frac{1}{2})$ require slightly
more sophisticated arguments similar to the one in [2], but will not be
pursued here.

Theorem 1.2 will follow from its bilinear analog. We prove this reduction in
Section 7.

######  Theorem 1.3.

Let $I_{1},I_{2}$ be intervals of length $\sim N$ in $[\frac{N}{2},N]$, with
${\operatorname{dist}\,}(I_{1},I_{2})\sim{N}$. Assume $\phi_{3},\phi_{4}$ are
real analytic on $(0,2)$ and satisfy (1), (2) and (3). Assume that
$\frac{3}{2}\leq\alpha\leq 2$.

Then

$\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{I_{1},N}(x){\mathcal{E}}_{I_{2},N}(x)|^{6}dx\lesssim_{\epsilon}N^{9+\epsilon}.$

The implicit constant in this inequality is uniform over
$A_{1},\ldots,A_{4}\sim 1$.

The reason we prove Theorem 1.2 using bilinear, as opposed to linear or
trilinear methods, is rather delicate. As explained earlier, our results are
sharp, the exponent 9 in (5) cannot be lowered. Each of the decoupling
inequalities relevant to this paper has a certain critical exponent $p_{c}>2$.
Experience shows that to achieve sharp results via decoupling, this tool must
be used at the critical exponent $p_{c}$. When we apply decoupling on spatial
balls of radius $M$, we decouple the curve into arcs of length $M^{-1/2}$. It
seems very likely that the critical exponent for such a decoupling in the
linear setting is larger than 12. See [7] for a detailed discussion on this.
Because of this, using linear $L^{12}$ decoupling turns out to be inefficient.
Bilinearizing instead, gives us access to the $L^{6}$ decoupling of the
parabola. This is an ideal scenario, since 6 is precisely the critical
exponent in this setting.

The fact that $12=6\times 2$ turns out to be crucial to our argument. The
other factorization $12=4\times 3$ is also inefficient. An approach based on
trilinearity would use $L^{4}$ estimates for hypersurfaces in
${\mathbb{R}}^{4}$. But the critical exponent here is $10/3$, not $4$.

Most of the paper is devoted to proving Theorem 1.3. The proof will combine
abstract decoupling methods with quadratic Weyl sum estimates. The decoupling
techniques are introduced in Section 2. While these results are by now
standard, the observation that the superficially stronger $l^{2}(L^{12})$
decoupling holds true for nondegenerate curves in the bilinear setting appears
to be new. One of the key features in our argument is the use of this
inequality in places where the $l^{12}(L^{12})$ decoupling used in [2] becomes
inefficient. We combine this finer decoupling with estimates from number
theory. In short, here is how our approach works. The initial integral
involves quartic Weyl sums, for which sharp estimates are totally out of reach
at the moment. Decoupling is applied once or twice in order to lower the
complexity of the sums, to the level of manageable quadratic Weyl sums. These
sums will appear in various combinations, and need to be tackled with extreme
care, using various counting arguments such as Lemma 4.1 and Lemma 4.2.

In Section 3, we start with a careful examination of Bourgain’s argument from
[2], for $\alpha=2$. In many ways, this case turns out to be the easiest, as
it works via just $l^{12}L^{12}$ decoupling, and without any input from number
theory. In Section 4 we introduce our new methodology, addressing the
symmetric case $\alpha=\beta=\frac{3}{2}.$ This ends up being the most
delicate case, since it captures the biggest region near the origin where
constructive (and near constructive) interference occurs. Also, it is in this
case that the curve looks most genuinely four dimensional, as both
$\omega_{3}$ and $\omega_{4}$ are large. For comparison, recall that when
$\alpha=3$ the curve degenerates to a three dimensional one. Sections 5 and 6
extend our method to the remaining cases, by successively building on each
other. The case $\frac{9}{5}\leq\alpha<2$ combines elements of both
approaches.

To the best of our knowledge, this paper represents the first systematic
effort to combine abstract decoupling with Weyl sum estimates. The results
proved here are part of the vast program initiated in [6], concerned with
proving sharp $L^{p}$ estimates for the moment curve on spatial domains
smaller than the torus. In [6] only the moment curve in ${\mathbb{R}}^{3}$ is
considered, and all estimates there rely solely on decoupling techniques.

There remain a lot of interesting related questions. One of them has to do
with proving Conjecture 1.1 in the range $2<\alpha<3$. We may speculate that
the solution would combine some of the tools from our paper with those used to
solve the case $\alpha=3$, see also Remark 3.1. Second, $L^{p}$ moments are
also worth investigating for smaller values $p<12$, in particular for $p=10$.
See for example [1] for some recent progress and some interesting
applications. Section 8 of our paper contains an example that describes some
of the enemies and limitations of square root cancellation in this setting. It
seems plausible that small cap decoupling for the parabola (see [6]) will be
the right tool to attack this problem. We hope to address some of these
questions in future work.

######  Acknowledgment.

The author is grateful to Hongki Jung and Zane Li for pointing out a few typos
in the earlier version of the manuscript.

##  2\. Decoupling for nondegenerate curves in ${\mathbb{R}}^{4}$

Let us start by recalling the decoupling for nondegenerate curves in
${\mathbb{R}}^{2}$.

######  Theorem 2.1 ([3]).

Let $\phi:[0,1]\to{\mathbb{R}}$ be a $C^{3}$ function, with
$\|\phi^{\prime}\|_{C^{2}}=A_{5}<\infty$ and $\min_{0\leq t\leq
1}|\phi^{\prime\prime}(t)|=A_{6}>0.$ Then for each $f:[0,1]\to{\mathbb{C}}$
and each ball $B_{N}\subset{\mathbb{R}}^{2}$ with radius $N$ we have

$\|\int_{[0,1]}f(t)e(tu+\phi(t)w)dt\|_{L_{u,w}^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J\subset[0,1]\atop{|J|=N^{-1/2}}}\|\int_{J}f(t)e(tu+\phi(t)w)dt\|_{L_{u,w}^{6}(B_{N})}^{2})^{1/2}.$

The implicit constant is uniform over $A_{5}\sim 1$, $A_{6}\sim 1$.

We use this result to illustrate in the simplest terms the reduction of cubic
terms used in the next section for $\alpha=2$. Namely, let us show that for
$1\leq\alpha<3$

$\int_{[0,1]^{2}}|\sum_{1\leq m\leq
M}e(mu+(m^{2}+\frac{m^{3}}{M^{\alpha}})w)|^{6}dudw\lesssim_{\epsilon}M^{3+\epsilon}.$
(6)

The term $\frac{m^{3}}{M^{\alpha}}w$ is not negligible (in the sense of Lemma
9.2), as it is not $O(1)$. However, after a change of variables and using
periodicity in $u$, we may rewrite the integral as

$\frac{1}{M^{4}}\int_{[0,M^{2}]^{2}}|\sum_{1\leq m\leq
M}e(\frac{m}{M}u+\phi(\frac{m}{M})w)|^{6}dudw,$

where $\phi(t)=t^{2}+t^{3}M^{1-\alpha}$. Note that $A_{5},A_{6}\sim 1$,
uniformly over $M$. Inequality (6) is now a standard consequence of Theorem
2.1 with $N=M^{2}$. The cubic term becomes a perturbation of the quadratic
term, and does not significantly affect the constant $A_{6}$. Theorem 2.4 will
formalize this approach in four dimensions.

Throughout this section, $\phi_{3},\phi_{4}$ are arbitrary functions
satisfying (1) and (2). We denote by $E$ the extension operator associated
with the curve $\Phi$

$\Phi(t)=(t,t^{2},\phi_{3}(t),\phi_{4}(t)),\;t\in[\frac{1}{2},1].$ (7)

More precisely, for $f:[\frac{1}{2},1]\to{\mathbb{C}}$ and
$I\subset[\frac{1}{2},1]$ we write

$E_{I}f(x)=\int_{I}f(t)e(tx_{1}+t^{2}x_{2}+\phi_{3}(t)x_{3}+\phi_{4}(t)x_{4})dt.$

The following $l^{6}(L^{6})$ decoupling was proved in [2], see also [4]. It is
in fact a bilinear version of the $l^{12}L^{12}$ decoupling for the curve (7).

######  Theorem 2.2.

Let $I_{1},I_{2}$ be two intervals of length $\sim 1$ in $[\frac{1}{2},1]$,
with $dist(I_{1},I_{2})\sim 1$. Let also
$f_{i}:[\frac{1}{2},1]\to{\mathbb{C}}$. Then for each ball $B_{N}$ of radius
$N$ in ${\mathbb{R}}^{4}$ we have

$\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\frac{1}{3}+\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{6})^{1/6}.$

The sum on the right is over intervals $J_{i}$ of length $N^{-1/2}$.

In [2], this result is used in conjunction with the following estimate, an
easy consequence of transversality.

$\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{6}\lesssim
N^{-4}\|E_{J_{1}}f_{1}\|_{L^{6}(B_{N})}^{6}\|E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{6}.$
(8)

This is inequality (13) in [4], and there is a detailed proof there. For
reader’s convenience, we sketch a somewhat informal argument below.

The Fourier transform of $|E_{J_{i}}f_{i}|^{6}$ is supported inside a
rectangular box with dimensions $\sim N^{-1}\times N^{-1}\times N^{-1}\times
N^{-1/2}$. We have the following wavepacket representation on $B_{N}$,
slightly simplified for exposition purposes

$|E_{J_{i}}f_{i}(x)|^{6}\approx\sum_{P_{i}\in{\mathcal{P}}_{i}}a_{P_{i}}1_{P_{i}}(x).$

The coefficients $a_{P_{i}}$ are nonnegative reals. The rectangular boxes
$P_{i}$ have dimensions $\sim N\times N\times N\times N^{1/2}$ and tile
$B_{N}$. They can be thought of as $N^{1/2}$-neighborhoods of cubes with
diameter $\sim N$ inside hyperplanes ${\mathcal{H}}_{i}$. Since
${\operatorname{dist}\,}(J_{1},J_{2})\sim 1$, the angle between the normal
vectors of ${\mathcal{H}}_{i}$ is $\sim 1$. Thus, we have that $|P_{1}\cap
P_{2}|\lesssim N^{3}$. We conclude by writing

$\displaystyle\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{6}$
$\displaystyle\sim\sum_{P_{1}\in{\mathcal{P}}_{1}}\sum_{P_{2}\in{\mathcal{P}}_{2}}a_{P_{1}}a_{P_{2}}|P_{1}\cap
P_{2}|$ $\displaystyle\lesssim
N^{-4}\sum_{P_{1}\in{\mathcal{P}}_{1}}\sum_{P_{2}\in{\mathcal{P}}_{2}}a_{P_{1}}a_{P_{2}}|P_{1}||P_{2}|$
$\displaystyle\approx
N^{-4}\|E_{J_{1}}f_{1}\|_{L^{6}(B_{N})}^{6}\|E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{6}.$

It is worth observing that $\lesssim$ in inequality (8) is essentially an
(approximate) similarity $\approx$, making (8) extremely efficient. Indeed,
since $P_{1}$ and $P_{2}$ intersect $B_{N}$, we have that $|P_{1}\cap
P_{2}|\sim N^{3}$.

To address new values of $\alpha$ in this paper, we will need the following
$l^{2}(L^{6})$ decoupling. This implies the previous $l^{6}(L^{6})$
decoupling, and provides a critical improvement in the cases when the terms in
the sum are of significantly different sizes.

######  Theorem 2.3.

Let $I_{1},I_{2}$ be two intervals of length $\sim 1$ in $[\frac{1}{2},1]$,
with $dist(I_{1},I_{2})\sim 1$. Let also
$f_{i}:[\frac{1}{2},1]\to{\mathbb{C}}$. Then for each ball $B_{N}$ of radius
$N$ in ${\mathbb{R}}^{4}$ we have

$\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{2})^{1/2}.$

The sum on the right is over intervals $J$ of length $N^{-1/2}$.

###### Proof.

We briefly sketch the argument, that follows closely the one in [2]. Let
$b(N)$ be the best constant such that

$\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\|_{L^{6}(B_{N})}\leq b(N)(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|_{L^{6}(B_{N})}^{2})^{1/2}$

holds for all functions and balls as above. Fix $B_{N}$ and let
${\mathcal{B}}$ be a finitely overlapping cover of $B_{N}$ with balls $\Delta$
of radius $N^{2/3}$. It will soon become clear that the exponent $2/3$ is
chosen in order to make cubic terms negligible. Applying this inequality on
each $\Delta$, then summing up using Minkowski’s inequality shows that

$\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\|_{L^{6}(B_{N})}\leq
b(N^{2/3})(\sum_{H_{1}\subset I_{1}}\sum_{H_{2}\subset
I_{2}}\|E_{H_{1}}f_{1}E_{H_{2}}f_{2}\|_{L^{6}(B_{N})}^{2})^{1/2}.$

The intervals $H$ have length $N^{-1/3}$. We next analyze each term in the
sum. Let $l_{i}$ be the left endpoint of $H_{i}$, and write a generic point
$t_{i}\in H_{i}$ as $t_{i}=l_{i}+s_{i}$, with $s_{i}\in[0,N^{-1/3}]$. We use
Taylor’s formula for $k\in\\{3,4\\}$

$\phi_{k}(t_{i})=\phi_{k}(l_{i})+\phi_{k}^{\prime}(l_{i})s_{i}+\frac{\phi_{k}^{\prime\prime}(l_{i})}{2}s_{i}^{2}+\psi_{k,i}(s_{i}),$

where, due to (1), $\|\psi_{k,i}\|_{L^{\infty}([0,N^{-1/3}])}=O(\frac{1}{N})$.
Let us write

$\begin{cases}y_{1}=x_{1}+2l_{1}x_{2}+\phi_{3}^{\prime}(l_{1})x_{3}+\phi_{4}^{\prime}(l_{1})x_{4}\\\
y_{2}=x_{1}+2l_{2}x_{2}+\phi_{3}^{\prime}(l_{2})x_{3}+\phi_{4}^{\prime}(l_{2})x_{4}\\\
y_{3}=x_{2}+\frac{\phi_{3}^{\prime\prime}(l_{1})}{2}x_{3}+\frac{\phi_{4}^{\prime\prime}(l_{1})}{2}x_{4}\\\
y_{4}=x_{2}+\frac{\phi_{3}^{\prime\prime}(l_{2})}{2}x_{3}+\frac{\phi_{4}^{\prime\prime}(l_{2})}{2}x_{4}\end{cases}.$

It follows that

$|E_{H_{1}}f_{1}(x)E_{H_{2}}f_{2}(x)|=$
$|\int_{[0,N^{-1/3}]^{2}}f_{1}(l_{1}+s_{1})e(s_{1}y_{1}+s_{1}^{2}y_{3})f_{2}(l_{2}+s_{2})e(s_{2}y_{2}+s_{2}^{2}y_{4})e(L(y,s_{1},s_{2}))ds_{1}ds_{2}|.$

Here

$L(y,s_{1},s_{2})=x_{3}(\psi_{3,1}(s_{1})+\psi_{3,2}(s_{2}))+x_{4}(\psi_{4,1}(s_{1})+\psi_{4,2}(s_{2})).$

Lemma 9.1 shows that $x_{3},x_{4}$ depend linearly on
$y_{1},y_{2},y_{3},y_{4}$ with coefficients $O(1)$. This allows us to write

$e(L(y,s_{1},s_{2}))=e(\sum_{i=1}^{4}y_{i}(g_{i}(s_{1})+h_{i}(s_{2})))$

with $\|g_{i}\|_{\infty},\|h_{i}\|_{\infty}=O(\frac{1}{N})$. Letting
$\bar{f}_{i}(s_{i})=f_{i}(l_{i}+s_{i})$ and

$\begin{cases}\eta_{1}(s_{1},s_{2})=s_{1}+g_{1}(s_{1})+h_{1}(s_{2})\\\
\eta_{3}(s_{1},s_{2})=s_{1}^{2}+g_{3}(s_{1})+h_{3}(s_{2})\\\
\eta_{2}(s_{1},s_{2})=s_{2}+g_{2}(s_{1})+h_{2}(s_{2})\\\
\eta_{4}(s_{1},s_{2})=s_{2}^{2}+g_{4}(s_{1})+h_{4}(s_{2})\end{cases}$

we write

$|E_{H_{1}}f_{1}(x)E_{H_{2}}f_{2}(x)|=$
$|\int_{[0,N^{-1/3}]^{2}}\bar{f}_{1}(s_{1})e(y_{1}\eta_{1}(s_{1},s_{2})+y_{3}\eta_{3}(s_{1},s_{2}))\bar{f}_{2}(s_{2})e(y_{2}\eta_{2}(s_{1},s_{2})+y_{4}\eta_{4}(s_{1},s_{2}))ds_{1}ds_{2}|.$

For $\bar{J}_{i}\subset[0,N^{-1/3}]$ we write

${\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}(y)=$
$|\int_{\bar{J}_{1}\times\bar{J}_{2}}\bar{f}_{1}(s_{1})e(y_{1}\eta_{1}(s_{1},s_{2})+y_{3}\eta_{3}(s_{1},s_{2}))\bar{f}_{2}(s_{2})e(y_{2}\eta_{2}(s_{1},s_{2})+y_{4}\eta_{4}(s_{1},s_{2}))ds_{1}ds_{2}|.$

This is the extension operator associated with the surface
$(\eta_{1},\ldots,\eta_{4})$, applied to the function $f_{1}\otimes f_{2}$.

We use Lemma 9.1 to write

$\int_{B_{N}}|E_{J_{1}}f_{1}(x)E_{J_{2}}f_{2}(x)|^{6}dx=\int_{\bar{B}_{N}}{\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}(y)^{6}dy.$

Here $\bar{B}_{N}$ is a ball of radius $\sim N$ and $J_{i}=\bar{J}_{i}+l_{i}$.
Note that the surface $(\eta_{1},\ldots,\eta_{4})$ is within $O(N^{-1})$ from
the surface

$(s_{1},s_{1}^{2},s_{2},s_{2}^{2}),\;\;s_{i}\in[0,N^{-1/3}],$

so -for decoupling purposes- the two surfaces are indistinguishable when
paired with spatial variables $y$ ranging through a ball of radius $N$. The
latter surface admits an $l^{2}(L^{6})$ decoupling, as can be easily seen by
using Theorem 2.1 twice. The same remains true for the surface
$(\eta_{1},\ldots,\eta_{4})$, and thus

$\|{\mathcal{I}}_{[0,N^{-1/3}],[0,N^{-1/3}]}\|_{L^{6}({\bar{B}_{N}})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{{\bar{J}_{1},\bar{J}_{2}}\subset[0,N^{-1/3}]}\|{\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}\|^{2}_{L^{6}({\bar{B}_{N}})})^{1/2}.$

The sum on the right is over intervals of length $N^{-1/2}$. If we undo the
change of variables we find

$\|E_{H_{1}}f_{1}E_{H_{2}}f_{2}\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
H_{1}}\sum_{J_{2}\subset
H_{2}}\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\|^{2}_{L^{6}(B_{N})})^{1/2}.$

Putting things together, we have proved the bootstrapping inequality

$b(N)\lesssim_{\epsilon}N^{\epsilon}b(N^{2/3}).$

We conclude that $b(N)\lesssim_{\epsilon}N^{\epsilon}$, as desired.

∎

We will also record the following close relative of Theorem 2.3, that will be
needed in the next sections.

######  Theorem 2.4.

Assume $\psi_{1},\ldots,\psi_{4}:[-1,1]\to{\mathbb{R}}$ have $C^{3}$ norm
$O(1)$, and in addition satisfy

$|\psi_{2}^{\prime\prime}(t)|,|\psi_{3}^{\prime\prime}(t)|\ll
1,\;\forall\;|t|\leq 1$

and

$|\psi_{1}^{\prime\prime}(t)|,|\psi_{4}^{\prime\prime}(t)|\sim
1,\;\forall\;|t|\leq 1.$

Let $E$ be the extension operator associated with the surface

$\Psi(\xi_{1},\xi_{2})=(\xi_{1},\xi_{2},\psi_{1}(\xi_{1})+\psi_{2}(\xi_{2}),\psi_{3}(\xi_{1})+\psi_{4}(\xi_{2})),\;\;|\xi_{1}|,|\xi_{2}|\leq
1.$

More precisely, for $F:[-1,1]^{2}\to{\mathbb{C}}$, $R\subset[-1,1]^{2}$ and
$x\in{\mathbb{R}}^{4}$ we write

$E_{R}F(x)=\int_{R}F(\xi_{1},\xi_{2})e(x\cdot\Psi(\xi_{1},\xi_{2}))d\xi_{1}d\xi_{2}.$

Then for each ball $B_{N}\subset{\mathbb{R}}^{4}$ with radius $N$ we have

$\|E_{[-1,1]^{2}}F\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{H_{1},H_{2}\subset[-1,1]}\|E_{H_{1}\times
H_{2}}F\|^{2}_{L^{6}(B_{N})})^{1/2},$ (9)

where the sum is taken over intervals of length $N^{-1/2}$.

In particular, for each constant coefficients $c_{m_{1},m_{2}}\in{\mathbb{C}}$
we have

$\|\sum_{m_{1}\leq N^{1/2}}\sum_{m_{2}\leq
N^{1/2}}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{N^{1/2}},\frac{m_{2}}{N^{1/2}}))\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}\|c_{m_{1},m_{2}}\|_{l^{2}}|B_{N}|^{1/6},$
(10)

while if $M\geq N^{1/2}$ we have

$\|\sum_{m_{1}\leq M}\sum_{m_{2}\leq
M}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{M},\frac{m_{2}}{M}))\|_{L^{6}(B_{N})}$
(11) $\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1},J_{2}}\|\sum_{m_{1}\in
J_{1}}\sum_{m_{2}\in
J_{2}}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{M},\frac{m_{2}}{M}))\|^{2}_{L^{6}(B_{N})})^{1/2},$

with $J_{i}$ intervals of length $\sim MN^{-1/2}$ partitioning $[1,M]$.

The implicit constants in both inequalities are independent of $N$, $M$ and of
$\psi_{i}$.

###### Proof.

The exponential sum estimates (10), (11) are standard consequences of (9), so
we will focus on proving the latter. When

$\Psi(\xi_{1},\xi_{2})=(\xi_{1},\xi_{2},C_{1}\xi_{1}^{2}+C_{2}\xi_{2}^{2},C_{3}\xi_{1}^{2}+C_{4}\xi_{2}^{2})$

with $|C_{1}|,|C_{4}|\sim 1$ and $|C_{2}|,|C_{3}|\ll 1$, the result follows by
applying $l^{2}(L^{6})$ Theorem 2.1 twice (after an initial affine change of
variables that reduces it to the case $C_{1}=C_{4}=1$, $C_{2}=C_{3}=0$).

We use a bootstrapping argument similar to the one in Theorem 2.3. Let $d(N)$
be the smallest constant in (9). We need to prove that
$d(N)\lesssim_{\epsilon}N^{\epsilon}$.

We first note that

$\|E_{[-1,1]^{2}}F\|_{L^{6}(B_{N})}\leq
d(N^{2/3})(\sum_{U_{1},U_{2}\subset[-1,1]}\|E_{U_{1}\times
U_{2}}F\|^{2}_{L^{6}(B_{N})})^{1/2},$

where $U_{i}$ are intervals of length $N^{-1/3}$ centered at $l_{i}$. When
$|t|<\frac{1}{2}N^{-1/3}$ we have

$\psi_{i}(t)=\psi_{i}(l_{i})+\psi_{i}^{\prime}(l_{i})t+C_{i}t^{2}+O(\frac{1}{N}),$

where $|C_{1}|,|C_{4}|\sim 1$ and $|C_{2}|,|C_{3}|\ll 1$. It follows that,
after an affine change of variables, the restriction of $\Psi$ to $U_{1}\times
U_{2}$ may be parametrized as

$(\xi_{1},\xi_{2},C_{1}\xi_{1}^{2}+C_{2}\xi_{2}^{2}+O(\frac{1}{N}),C_{3}\xi_{1}^{2}+C_{4}\xi_{2}^{2}+O(\frac{1}{N})),\;\;|\xi_{1}|,|\xi_{2}|=O(N^{-1/3}).$

We decouple this on $B_{N}$, using the observation at the beginning of the
proof. We find

$\|E_{U_{1}\times
U_{2}}F\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{H_{1}\subset
U_{1},H_{2}\subset U_{2}}\|E_{H_{1}\times
H_{2}}F\|^{2}_{L^{6}(B_{N})})^{1/2}.$

It follows that $d(N)\lesssim_{\epsilon}N^{\epsilon}d(N^{2/3})$, which forces
$d(N)\lesssim_{\epsilon}N^{\epsilon}$.

∎

##  3\. Bourgain’s argument for the case $\alpha=2$

For the remainder of the paper, we will use the following notation

${\mathcal{E}}_{I}(x)=\sum_{n\in I}e(N\Phi(\frac{n}{N})x)=\sum_{n\in
I}e(nx_{1}+\frac{n^{2}}{N}x_{2}+\phi_{3}(\frac{n}{N})Nx_{3}+\phi_{4}(\frac{n}{N})Nx_{4}).$

Note that, compared to ${\mathcal{E}}_{I,N}$, we dropped the subscript $N$ and
renormalized the variables $x_{2}$, $x_{3}$ and $x_{4}$. Letting

$\Omega=[0,N]^{3}\times[0,1]$

and using periodicity in $x_{1}$, we need to prove that

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{9+\epsilon}.$

Bourgain’s argument from [2] for the case $\alpha=2$ of Theorem 1.3 involves
three successive decouplings. We simplify it slightly it and reduce it to only
two decouplings.

Step 1. Cover $\Omega$ with cubes $B$ of side length $1$, apply $l^{6}(L^{6})$
decoupling (Theorem 2.2) on each $B$ (or rather $NB$, after rescaling), then
sum these estimates to get

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{2+\epsilon}\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}.$

Here $J_{1},J_{2}$ are intervals of length $N^{1/2}$.

The remaining part of the argument will show the uniform estimate
$O(N^{6+\epsilon})$ for each term
$\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}.$
Fix $J_{i}=[h_{i},h_{i}+N^{1/2}]$.

Step 2. Note that when $x\in\Omega$

$|{\mathcal{E}}_{J_{i}}(x)|=|\sum_{m\leq
N^{1/2}}c_{m}(x_{4})e(mu_{i}+m^{2}w_{i}+\eta_{i}(m)x_{3})|$

where

$\begin{cases}u_{i}=x_{1}+\frac{2h_{i}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{i}}{N})x_{3}\\\
w_{i}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{i}}{N})\frac{x_{3}}{2N}\\\
\eta_{i}(m)={m^{3}}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h_{i}}{N})}{3!N^{2}}+{m^{4}}\frac{\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{i}}{N})}{4!N^{3}}+\ldots\end{cases}.$
(12)

We hide the whole contribution from $x_{4}$ into the coefficient
$c_{m}(x_{4})$. Indeed, since

$\phi_{4}(\frac{h_{i}+m}{N})Nx_{4}=\phi_{4}(\frac{h_{i}}{N})Nx_{4}+m\phi_{4}^{\prime}(\frac{h_{i}}{N})x_{4}+O(1),$

$x_{4}$ does not contribute significantly with quadratic or higher order
terms, so it produces no cancellations. We will only use that
$c_{m}(x_{4})=O(1)$.

At this point, we seek a change of variables. We want the new domain of
integration to be a rectangular box, to allow us to separate the four-variable
integral $\int_{\Omega}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$ into
the product of two-variable integrals. Note that the ranges of
$x_{1},x_{2},x_{3}$ are the same, $[0,N]$, but $x_{4}$ is restricted to the
smaller interval $[0,1]$. We cannot use periodicity to extend the range of
$x_{4}$ to $[0,N]$, because the individual waves
$e(m\phi_{4}^{\prime}(\frac{h_{i}}{N})x_{4})$ have different periods with
respect to the variable $x_{4}$. Because of this, the variable $x_{4}$ is
practically useless from this point on, it will not generate oscillations. To
generate a fourth variable with range $[0,N]$ for the purpose of a change of
coordinates, Bourgain produces a piece of magic.

First, he applies (8) on each cube $NB$

$\displaystyle\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$
$\displaystyle=N^{-4}\int_{NB}|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N}){\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})|^{6}$
$\displaystyle\lesssim
N^{-8}\int_{NB}|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N})|^{6}\int_{NB}|{\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})|^{6}=\int_{B}|{\mathcal{E}}_{J_{1}}|^{6}\int_{B}|{\mathcal{E}}_{J_{2}}|^{6}.$

Second, he uses the following abstract inequality, that only relies on the
positivity of $|{\mathcal{E}}_{J_{i}}|^{6}$

$\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{J_{1}}|^{6}\int_{B}|{\mathcal{E}}_{J_{2}}|^{6}\lesssim\int_{\Omega}dx\int_{(y,z)\in[-1,1]^{4}\times[-1,1]^{4}}|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)|^{6}dydz.$

Using periodicity in the $y_{1},z_{1}$ variables, this is

$\lesssim\frac{1}{N^{2}}\int_{x_{1}\in[0,N]\atop{}_{x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}\in[-1,1]}}dx_{1}\ldots
dz_{4}\int_{y_{1},z_{1},x_{2},x_{3}\in[0,N]}|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)|^{6}dy_{1}dz_{1}dx_{2}dx_{3}.$

In short, the variable $x_{1}$ is now replaced with the new variables $y_{1}$
and $z_{1}$. It remains to prove that

$\int_{y_{1},z_{1},x_{2},x_{3}\in[0,N]}|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)|^{6}dy_{1}dz_{1}dx_{2}dx_{3}\lesssim_{\epsilon}N^{7+\epsilon},$
(13)

uniformly over $x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$. With these
variables fixed, we make the affine change of variables
$(y_{1},z_{1},x_{2},x_{3})\mapsto(u_{1},u_{2},w_{1},w_{2})$

$\begin{cases}u_{1}=(y_{1}+x_{1})+\frac{2h_{1}}{N}(x_{2}+y_{2})+\phi_{3}^{\prime}(\frac{h_{1}}{N})(x_{3}+y_{3})\\\
u_{2}=(z_{1}+x_{1})+\frac{2h_{2}}{N}(x_{2}+z_{2})+\phi_{3}^{\prime}(\frac{h_{2}}{N})(x_{3}+z_{3})\\\
w_{1}=\frac{x_{2}+y_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}+y_{3}}{2N}\\\
w_{2}=\frac{x_{2}+z_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}+z_{3}}{2N}\end{cases}.$
(14)

The Jacobian is $\sim\frac{1}{N^{2}}$, due to (3). Note that
$x_{3}+y_{3}=A(w_{1}-w_{2})$, where $A$ depends just on $h_{1},h_{2}$, and
$|A|\sim N$. Using (12) we may write the last integral as

$N^{2}\times$ $\int_{|u_{i}|\lesssim N\atop{}_{|w_{i}|\lesssim
1}}|\sum_{m_{i}\leq
N^{\frac{1}{2}}}c_{m_{1},m_{2}}e(m_{1}u_{1}+m_{1}^{2}w_{1}+m_{2}u_{2}+m_{2}^{2}w_{2}+(\eta_{1}(m_{1})+\eta_{2}(m_{2}))A(w_{1}-w_{2}))|^{6}.$
(15)

The coefficient $c_{m_{1},m_{2}}$ depends only on
$m_{1},m_{2},x_{4},y_{4},z_{4}$, but not on the variables of integration
$u_{i},w_{i}$. The argument of each exponential may be rewritten as

$\frac{m_{1}}{N^{1/2}}u_{1}N^{1/2}+(\psi_{1}(\frac{m_{1}}{N^{1/2}})+\psi_{2}(\frac{m_{2}}{N^{1/2}}))w_{1}N+$
$\frac{m_{1}}{N^{1/2}}u_{2}N^{1/2}+(\psi_{3}(\frac{m_{1}}{N^{1/2}})+\psi_{4}(\frac{m_{2}}{N^{1/2}}))w_{2}N$

where

$\begin{cases}\psi_{1}(\xi)=\xi^{2}+&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{1}}{N})}{3!N^{3/2}}+{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{1}}{N})}{4!N^{2}}+\ldots\\\
\psi_{2}(\xi)=&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{2}}{N})}{3!N^{3/2}}+{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{2}}{N})}{4!N^{2}}+\ldots\\\
\psi_{3}(\xi)=&{-\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{1}}{N})}{3!N^{3/2}}-{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{1}}{N})}{4!N^{2}}-\ldots\\\
\psi_{4}(\xi)=\xi^{2}-&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{2}}{N})}{3!N^{3/2}}-{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{2}}{N})}{4!N^{2}}-\ldots\end{cases}.$

These functions satisfy the requirements in Theorem 2.4. The integral (15) is
the same as

$N^{-3}\int_{u_{i}=O(N^{3/2})\atop{}_{w_{i}=O(N)}}|\sum_{m_{1}\leq
N^{1/2}}\sum_{m_{2}\leq
N^{1/2}}c_{m_{1},m_{2}}e((u_{1},u_{2},w_{1},w_{2})\cdot\Psi(\frac{m_{1}}{N^{1/2}},\frac{m_{2}}{N^{1/2}}))|^{6}du_{1}du_{2}dw_{1}dw_{2}.$

If we cover the domain of integration with $\sim N$ balls $B_{N}$ and apply
(10) on each of them, we may dominate the above expression by

$N^{-3}N(N^{\frac{1}{2}+\epsilon}N^{4/6})^{6}=N^{5+\epsilon}.$

This proves (13) and ends the proof. Note that this argument treats the cubic
and higher order terms as perturbations of quadratic factors, as explained in
the proof of (6).

In summary, what is special about the case $\alpha=2$ is that the range of
$x_{3}$ in our initial integral over $\Omega$ is $[0,N]$. This was needed in
producing the large spatial range $w_{i}=O(N)$ for our final variables,
crucial for the application of (10). This inequality provides decoupling into
point masses, reducing the initial exponential sum to individual waves. In
Section 6 we will see that when $\alpha$ is slightly smaller than 2,
inequality (11) will have to replace (10), leading to quadratic Weyl sums
whose handling demands number theory.

######  Remark 3.1.

It is not clear whether a version of Bourgain’s method could be made to work
in the range $2<\alpha<3$. If successful, this would potentially provide a new
argument for Vinogradov’s Mean Value Theorem in ${\mathbb{R}}^{3}$. Decoupling
on cubes with size $N^{\beta-1}$ and using (8) on balls $B_{N^{\beta}}$ leads
to variables $y_{1},z_{1}$ with associated period equal to 1, much bigger than
their range $N^{\beta-1}$. The change of variables (14) is no longer efficient
in this case.

##  4\. Proof of Theorem 1.3 in the case $\alpha=\frac{3}{2}$

This time we let $\Omega=[0,1]\times[0,N]\times[0,N^{1/2}]\times[0,N^{1/2}]$.
Recall that

${\mathcal{E}}_{I}(x)=\sum_{n\in
I}e(nx_{1}+\frac{n^{2}}{N}x_{2}+\phi_{3}(\frac{n}{N})Nx_{3}+\phi_{4}(\frac{n}{N})Nx_{4}).$

We need to prove

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}dx\lesssim_{\epsilon}N^{8+\epsilon}.$
(16)

In this case, we will only need assumptions (1) and (2), but not (3). We start
by presenting a general principle that will explain the subtleties of our
argument. See also Remark 4.3.

Consider two partitions of $\Omega$, one into cubes $B$ with side length $l$
and another one into cubes $\Delta$ with side length $L\geq l$. The intervals
$J_{i}$ have length $\sqrt{\frac{N}{l}}$ and partition $I_{i}$. The intervals
$U_{i}$ have length $\sqrt{\frac{N}{L}}$ and partition $I_{i}$. The following
holds, via two applications of Theorem 2.3 (on cubes $B$ and $\Delta$,
combined with Minkowski’s inequality)

$\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|_{L^{6}(\Omega)}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1},J_{2}}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|_{L^{6}(\Omega)}^{2})^{1/2}\lesssim_{\epsilon}N^{\epsilon}(\sum_{U_{1},U_{2}}\|{\mathcal{E}}_{U_{1}}{\mathcal{E}}_{U_{2}}\|_{L^{6}(\Omega)}^{2})^{1/2}.$
(17)

Also, combining the above inequalities with Hölder shows that

$\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|_{L^{6}(\Omega)}\lesssim_{\epsilon}$
$N^{\epsilon}(\sharp(J_{1},J_{2}))^{\frac{1}{3}}(\sum_{J_{1},J_{2}}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|_{L^{6}(\Omega)}^{6})^{1/6}\lesssim_{\epsilon}N^{\epsilon}(\sharp(U_{1},U_{2}))^{\frac{1}{3}}(\sum_{U_{1},U_{2}}\|{\mathcal{E}}_{U_{1}}{\mathcal{E}}_{U_{2}}\|_{L^{6}(\Omega)}^{6})^{1/6}.$
(18)

Invoking periodicity in $x_{1},x_{2}$ and the invariance of (1), (2), (3)
under the change of sign $\phi_{k}\mapsto-\phi_{k}$, (16) is equivalent with
proving that

$\int_{[-N^{1/2},N^{1/2}]\times[-N,N]\times[-N^{1/2},N^{1/2}]\times[-N^{1/2},N^{1/2}]}|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)|^{6}dx\lesssim_{\epsilon}N^{8+\frac{1}{2}+\epsilon}.$
(19)

We first demonstrate the inefficiency of $l^{6}L^{6}$ decoupling for this
case, by working with the smaller domain

$S=[-o(N^{1/2}),o(N^{1/2})]^{4}.$

We cover $S$ with unit cubes and apply decoupling into intervals $J_{1},J_{2}$
of length $N^{1/2}$ as in the previous section, to dominate

$\int_{S}|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)|^{6}dx\lesssim_{\epsilon}N^{\epsilon}N^{6(\frac{1}{2}-\frac{1}{6})}\sum_{J_{1},J_{2}}\int_{S}|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)|^{6}dx.$
(20)

We will next show that the right hand side is too big, thus leading to an
overestimate for our initial integral. When $J=[h,h+N^{1/2}]$ and $n=h+m\in J$
we write

$\phi_{k}(\frac{n}{N})=\phi_{k}(\frac{h}{N})+\phi_{k}^{\prime}(\frac{h}{N})\frac{m}{N}+\frac{1}{2}\phi_{k}^{\prime\prime}(\frac{h}{N})(\frac{m}{N})^{2}+O(\frac{m}{N})^{3}.$

If $|x_{3}|,|x_{4}|\ll N^{1/2}$ and $m\leq N^{1/2}$, we guarantee that the
contribution from higher order terms is small

$O(\frac{m}{N})^{3}N(|x_{3}|+|x_{4}|)\ll 1.$

If we collect the contributions from linear and quadratic terms we find

$|{\mathcal{E}}_{J}(x)|=|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+o(1))|$

where

$\begin{cases}u=x_{1}+\frac{h}{N}x_{2}+\phi_{3}^{\prime}(\frac{h}{N})x_{3}+\phi_{4}^{\prime}(\frac{h}{N})x_{4}\\\
w=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h}{N})\frac{x_{4}}{N}\end{cases}.$

Using Lemma 9.1 we write

$\int_{S}|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)|^{6}dx\gtrsim$
$N^{2}\int_{(u_{1},u_{2})\in[0,o(N^{1/2})]^{2}\atop{(w_{1},w_{2})\in[0,o(N^{-1/2})]^{2}}}|\sum_{m\leq
N^{1/2}}e(mu_{1}+m^{2}w_{1}+o(1))|^{6}|\sum_{m\leq
N^{1/2}}e(mu_{2}+m^{2}w_{2}+o(1))|^{6}.$

We now use the fact that we have constructive interference

$|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+o(1))|\sim N^{1/2}$

on the set of measure $\sim\frac{1}{N}$

$(u,w)\in(\bigcup_{l\in\\{0,1,\ldots,o(\sqrt{N})\\}}[l,l+\frac{1}{\sqrt{N}}])\times[0,\frac{1}{N}].$

It follows that

$\int_{S}|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)|^{6}dx\gtrsim
N^{2}N^{6}N^{-2}=N^{6}.$

It is not hard to prove that this lower bound is sharp, but this has no
relevance to us here. The point of working with the symmetric domain $S$ was
to make sure that $w_{1},w_{2}\sim\frac{1}{N}$ are in the new domain of
integration. Going back to (20), the $l^{6}(L^{6})$ decoupling method leads to
the upper bound

$\int_{S}|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)|^{6}dx\lesssim_{\epsilon}N^{9+\epsilon}.$

This falls short by the factor $N^{1/2}$ from proving (19).

The second inequality in (4) shows that using $l^{6}(L^{6})$ decoupling on
cubes $\Delta$ that are larger than $N$ will only worsen the upper bounds we
get. On the other hand, working with smaller cubes will render decoupling
inefficient. The resulting exponential sums will be very difficult to handle
using number theory, since the cubic terms are no longer $O(1)$ in this case.

Let us now describe the correct approach, that will critically rely on
$l^{2}$, rather than $l^{6}$ decoupling. The following level set estimate will
play a key role in various counting arguments. The main strength of the lemma
is in the case when $|l_{1}|\sim|l_{2}|$.

Throughout the remainder of the paper, the letter $l$ will be used to denote
integers, and their relative proximity to powers of $2$ will be denoted using
the symbol $\sim$. We make the harmless convention to write $0\sim 2^{0}$.

######  Lemma 4.1.

Assume $\phi_{3}$, $\phi_{4}$ satisfy (1) and (2). Let $l_{1},l_{2}$ with
$\max\\{|l_{1}|,|l_{2}|\\}\sim 2^{j}$, $j\geq 0$, and let

$f(t)=l_{1}\phi_{3}^{\prime\prime}(t)+l_{2}\phi_{4}^{\prime\prime}(t).$

Then we can partition the range of $f$ into sets $R_{s}$ with $0\leq s\leq j$,
each of which is the union of at most two intervals of length $\sim 2^{s}$,
such that for each $v\in R_{s}$ we have

$|f^{-1}(v+[-O(1),O(1)])\cap[\frac{1}{2},1]|\lesssim\frac{1}{\sqrt{2^{j+s}}}.$

All implicit constants are universal over all pairs of such $\phi_{3}$,
$\phi_{4}$ and over $l_{1},l_{2},s$.

###### Proof.

The result is trivial if $l_{1}=l_{2}=0$, so we will next assume that
$\max\\{|l_{1}|,|l_{2}|\\}\geq 1$.

We restrict $f$ to the interval $[\frac{1}{2},1]$. Since

$\begin{bmatrix}f^{\prime}(t)\\\
f^{\prime\prime}(t)\end{bmatrix}=\begin{bmatrix}\phi_{3}^{(3)}(t)&\phi_{4}^{(3)}(t)\\\
\phi_{3}^{(4)}(t)&\phi_{4}^{(4)}(t)\end{bmatrix}\begin{bmatrix}l_{1}\\\
l_{2}\end{bmatrix},$

(2) implies that for each $t\in[\frac{1}{2},1]$ we have

$\max\\{|f^{\prime}(t)|,|f^{\prime\prime}(t)|\\}\sim 2^{j}.$ (21)

We let $t_{0}$ be a point in $[\frac{1}{2},1]$ where $|f^{\prime}|$ attains
its minimum. If $|f^{\prime}(t_{0})|\sim 2^{j}$, then we may take $R_{j}$ to
be the whole range of $f$, and all other $R_{s}$ to be empty. Indeed, the Mean
Value Theorem shows that

$|f(t_{1})-f(t_{2})|\gtrsim 1$

whenever $|t_{1}-t_{2}|\gtrsim 2^{-j}$. It is worth observing that if
$|l_{1}|\gg|l_{2}|$, then (3) would immediately guarantee that
$|f^{\prime}(t_{0})|\sim 2^{j}$.

We now assume that $|f^{\prime}(t_{0})|\ll 2^{j}$. Due to (21), we must have
that $|f^{\prime\prime}(t_{0})|\sim 2^{j}$. We write for $t\in[\frac{1}{2},1]$

$f(t)=f(t_{0})+f^{\prime}(t_{0})(t-t_{0})+\frac{f^{\prime\prime}(t_{0})(t-t_{0})^{2}}{2}+O(2^{j}(t-t_{0})^{3}).$
(22)

Case 1. Consider $s$ with $2^{j}\geq
2^{s}>C\max\\{\frac{|f^{\prime}(t_{0})|^{2}}{2^{j}},1\\}$, for some large
enough $C$ independent of $j$. Using this and (22), we see that

$|f(t)-f(t_{0})|\ll 2^{s}\;\text{ whenever }|t-t_{0}|\ll 2^{\frac{s-j}{2}}.$
(23)

Define

$R_{s}=\\{v:\;|v-f(t_{0})|\sim 2^{s}\\}.$

Let $v\in R_{s}$ and let $w=v+O(1)$. Thus, we also have $|w-f(t_{0})|\sim
2^{s}$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$, $f(t_{2})=w$. Using (23)
it follows that $|t_{1}-t_{0}|,|t_{2}-t_{0}|\gtrsim 2^{\frac{s-j}{2}}$. Our
assumption shows that $2^{\frac{s-j}{2}}\gg\frac{|f^{\prime}(t_{0})|}{2^{j}}$.
Thus, $|t_{1}-t_{0}|,|t_{2}-t_{0}|\gg\frac{|f^{\prime}(t_{0})|}{2^{j}}$, and
using (22) again we conclude that

$|f(t_{i})-f(t_{0})|\sim 2^{j}|t_{i}-t_{0}|^{2}.$

Thus, $|t_{i}-t_{0}|\sim 2^{\frac{s-j}{2}}$. Using again (22) we find that if
$t_{1},t_{2}$ are on the same side of $t_{0}$ then

$|f(t_{1})-f(t_{2})|\sim 2^{\frac{s+j}{2}}|t_{1}-t_{2}|.$

We conclude that $|t_{1}-t_{2}|\lesssim\frac{1}{\sqrt{2^{j+s}}}$, as desired.

Next, we define $R_{s}$ for smaller values of $s$. We distinguish two cases.

Case 2a. Assume now that $|f^{\prime}(t_{0})|\leq 2^{j/2}$. For $s$ such that
$2^{s}$ is the largest dyadic power $\leq
C\max\\{\frac{|f^{\prime}(t_{0})|^{2}}{2^{j}},1\\}=C$ we define

$R_{s}=\\{v:\;|v-f(t_{0})|\lesssim 2^{s}\\}.$

We also let $R_{s^{\prime}}=\emptyset$ for smaller values of $s^{\prime}$. Let
$v\in R_{s}$ and $w=v+O(1)$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$,
$f(t_{2})=w$. Since in fact $|f(t_{i})-f(t_{0})|\lesssim 1$, (22) forces
$|t_{i}-t_{0}|\lesssim 2^{-j/2}\sim\frac{1}{\sqrt{2^{j+s}}}$, as desired.

Case 2b. Assume now that $|f^{\prime}(t_{0})|>2^{j/2}$. For $s$ such that
$2^{s}$ is the largest dyadic power $\leq
C\max\\{\frac{|f^{\prime}(t_{0})|^{2}}{2^{j}},1\\}=C\frac{|f^{\prime}(t_{0})|^{2}}{2^{j}}$
we define

$R_{s}=\\{v:\;|v-f(t_{0})|\lesssim 2^{s}\\}.$

We also let $R_{s^{\prime}}=\emptyset$ for smaller values of $s^{\prime}$. Let
$v\in R_{s}$ and $w=v+O(1)$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$,
$f(t_{2})=w$. Using that $|f^{\prime}(t)|\geq|f^{\prime}(t_{0})|$ for all $t$,
we find that

$|f(t_{1})-f(t_{2})|\geq|t_{1}-t_{2}||f^{\prime}(t_{0})|.$

We conclude that

$|t_{1}-t_{2}|\lesssim\frac{1}{|f^{\prime}(t_{0})|}\sim\frac{1}{\sqrt{2^{j+s}}},$

as desired.

∎

From now on, we will implicitly assume that all Weyl sums are smooth, as in
Lemma 9.2. This can be easily arranged using partitions of unity, namely
working with smooth $\gamma$ satisfying

$\sum_{l\in{\mathbb{Z}}}\gamma(\cdot+l)=1_{\mathbb{R}}.$

To simplify notation, these weights will be ignored.

Cover $\Omega$ with unit cubes
$B=B_{p,l_{1},l_{2}}=[0,1]\times[p,p+1]\times[l_{1},l_{1}+1]\times[l_{2},l_{2}+1]$
with

$p\leq N,\;\;l_{1},l_{2}\leq N^{1/2}.$

We first write

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\sim\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}.$

We use $l^{2}$ decoupling (Theorem 2.3) on each $B$

$\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}(\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3}$

where $J_{i}$ is of the form $[h_{i},h_{i}+N^{1/2}]$. When $x\in B$ and
$J=[h,h+N^{1/2}]$

$|{\mathcal{E}}_{J}(x)|=|\sum_{m\leq
N^{1/2}}e(mu+m^{2}w+{m^{3}}v+O(N^{-1/4}))|$

where

$\begin{cases}u=x_{1}+\frac{2h}{N}x_{2}+\phi_{3}^{\prime}(\frac{h}{N})x_{3}+\phi_{4}^{\prime}(\frac{h}{N})x_{4}\\\
w=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h}{N})\frac{x_{4}}{N}\\\
v=\frac{\phi_{3}^{\prime\prime\prime}(\frac{h}{N}){x_{3}}+\phi_{4}^{\prime\prime\prime}(\frac{h}{N}){x_{4}}}{6N^{2}}\end{cases}.$
(24)

The term $O(N^{-1/4})$ can be dismissed as it produces tiny errors consistent
with square root cancellation. Note that since $v=O(N^{-3/2})$, we have

$|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+{m^{3}}v)|\approx|\sum_{m\leq
N^{1/2}}e(mu+m^{2}w)|.$

See Lemma 9.2 for a rigorous argument. The key point is that we may dismiss
the cubic terms.

Write

$I(h_{1},h_{2},B)=$
$\int_{(u_{1},u_{2},w_{1},w_{2})\in[0,1]^{2}\times[\frac{a_{1}-O(1)}{N},\frac{a_{1}+O(1)}{N}]\times[\frac{a_{2}-O(1)}{N},\frac{a_{2}+O(1)}{N}]}|\prod_{i=1}^{2}\sum_{m_{i}\leq
N^{1/2}}e(m_{i}u_{i}+m_{i}^{2}w_{i})|^{6}du_{1}du_{2}dw_{1}dw_{2},$

where

$\begin{cases}a_{1}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\\\
a_{2}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\end{cases}.$
(25)

Via the change of variables with Jacobian $\sim\frac{1}{N^{2}}$ (Lemma 9.1)

$\begin{cases}u_{1}=x_{1}+\frac{2h_{1}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{1}}{N})x_{3}+\phi_{4}^{\prime}(\frac{h_{1}}{N})x_{4}\\\
w_{1}=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{4}}{N}\\\
u_{2}=x_{1}+\frac{2h_{2}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{2}}{N})x_{3}+\phi_{4}^{\prime}(\frac{h_{2}}{N})x_{4}\\\
w_{2}=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{4}}{N}\end{cases}$

we see that

$\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}\lesssim
N^{2}I(h_{1},h_{2},B).$

Writing

$I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{N},\frac{a+O(1)}{N}]}|\sum_{m\leq
N^{1/2}}e(mu+m^{2}w)|^{6}dudw$

we find that

$\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}\lesssim
N^{2}I_{a_{1}}I_{a_{2}}.$

Let us analyze (25). The question is, for fixed $B$, what are the values of
$a_{1},a_{2}$ that arise (modulo $O(1)$ error terms), and what is their
multiplicity, when $h_{1},h_{2}$ range through the multiples of $N^{1/2}$ in
$[1,N]$.

Assume $l_{1}\sim 2^{j_{1}}$, $l_{2}\sim 2^{j_{2}}$, with
$2^{j_{1}},2^{j_{2}}\leq N^{1/2}$. We may assume $j_{1}\leq j_{2}$, the other
case is completely similar. We apply Lemma 4.1 to
$f(t)=\frac{1}{2}(l_{1}\phi_{3}^{\prime\prime}(t)+l_{2}\phi_{4}^{\prime\prime}(t))$.
For each $0\leq s_{1},s_{2}\leq j_{2}$ and each $p$ we have
$O(2^{s_{1}+s_{2}})$ pairs $(a_{1},a_{2})$ of integers with $a_{1}-p\in
R_{s_{1}}(l_{1},l_{2})$ and $a_{2}-p\in R_{s_{2}}(l_{1},l_{2})$. Note that we
index the intervals $R_{s_{i}}$ from Lemma 4.1 by $l_{1},l_{2}$. For each such
pair $(a_{1},a_{2})$, (25) has
$O(\frac{N}{2^{j_{2}}2^{\frac{s_{1}+s_{2}}{2}}})$ solutions $(h_{1},h_{2})$.
When we count solutions, we tolerate error terms of size $O(1)$.

Thus

$\displaystyle\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}$
$\displaystyle\lesssim N^{2}\sum_{p\leq N}\sum_{2^{j_{2}}\lesssim
N^{1/2}}\sum_{2^{j_{1}}\lesssim 2^{j_{2}}}\sum_{l_{1}\sim
2^{j_{1}}}\sum_{l_{2}\sim 2^{j_{2}}}\sum_{s_{1},s_{2}\leq
j_{2}}(\frac{N}{2^{j_{2}+\frac{s_{1}+s_{2}}{2}}})^{3}(\sum_{a_{1}\in
p+R_{s_{1}}(l_{1},l_{2})}\sum_{a_{2}\in
p+R_{s_{2}}(l_{1},l_{2})}I_{a_{1}}^{1/3}I_{a_{2}}^{1/3})^{3}$
$\displaystyle\lesssim N^{2}\sum_{p\leq N}\sum_{2^{j_{2}}\lesssim
N^{1/2}}\sum_{2^{j_{1}}\lesssim
2^{j_{2}}}2^{j_{1}+j_{2}}(\frac{N}{2^{j_{2}}})^{3}\sum_{s_{1},s_{2}\leq
j_{2}}(\sum_{a_{1}\in
p+R_{s_{1}}(l_{1},l_{2})}I_{a_{1}}^{2/3})^{3/2}(\sum_{a_{2}\in
p+R_{s_{2}}(l_{1},l_{2})}I_{a_{2}}^{2/3})^{3/2}.$

The last inequality follows from Cauchy–Schwarz. Next, we observe that
$p+R_{s_{i}}(l_{1},l_{2})\subset[p-O(2^{j_{2}}),p+O(2^{j_{2}})]$. These
intervals are roughly the same for roughly $2^{j_{2}}$ values of $p$. We can
thus dominate the above by

$\displaystyle\begin{split}&{\;\lessapprox}\;N^{2}\sum_{2^{j_{2}}\lesssim
N^{1/2}}\sum_{2^{j_{1}}\lesssim
2^{j_{2}}}2^{j_{1}+j_{2}}(\frac{N}{2^{j_{2}}})^{3}2^{j_{2}}\sum_{H\subset[0,N]\atop{|H|={2^{j_{2}}}}}(\sum_{a\in
H}I_{a}^{2/3})^{3}\\\ &\sim N^{5}\sum_{2^{j}\lesssim
N^{1/2}}\sum_{H\subset[0,N]\atop{|H|={2^{j}}}}(\sum_{a\in
H}I_{a}^{2/3})^{3}.\end{split}$ (26)

The sum runs over pairwise disjoint intervals $H$. It is easily seen to be
$O(N^{8})$, by using the following lemma with $M=N^{1/2}$.

######  Lemma 4.2.

Let

$I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M^{2}},\frac{a+O(1)}{M^{2}}]}|\sum_{m\leq
M}e(mu+m^{2}w)|^{6}dudw$

For each $2^{j}\leq M^{2}$ we have

$\sum_{H\subset[0,M^{2}]\atop{|H|={2^{j}}}}(\sum_{a\in
H}I_{a}^{2/3})^{3}{\;\lessapprox}\;M^{4}2^{2j}+M^{6}.$

###### Proof.

The arcs
$\\{x\in[0,1):\;{\operatorname{dist}\,}(x-\frac{b}{q},{\mathbb{Z}})\leq\frac{1}{qM}\\}$,
with $1\leq b\leq q\leq M$ and $(b,q)=1$, cover $[0,1)$. They may overlap,
which leads to double counting in our argument, but this will be harmless.

We consider the contribution from those $I_{a}$ with $\frac{a}{M^{2}}$ in some
arc with $q\sim Q$. Here $Q$ is dyadic and $Q\lesssim M$. We separate the
proof into two cases. Note that $H/M^{2}\subset[0,1]$ and has length
$2^{j}/M^{2}$. Also, $|b/q-b^{\prime}/q^{\prime}|\geq 1/qq^{\prime}$.

Case 1. $Q^{2}>\frac{M^{2}}{2^{j}}$. Each $H/M^{2}$ intersects
$\lesssim\frac{2^{j}Q^{2}}{M^{2}}$ arcs with $q\sim Q$. For each such $b/q$,
and each $1\leq 2^{k}\leq\frac{M}{q}$ there are $\sim\frac{M}{2^{k}Q}$ values
of $a$ with

$|\frac{a}{M^{2}}-\frac{b}{q}|\sim\frac{1}{qM2^{k}}.$

Call ${\mathbb{A}}(Q,k)$ the collection of all these $a$. For each
$a\in{\mathbb{A}}(Q,k)$, Lemma 9.2 gives

$I_{a}\lesssim\frac{1}{2^{k}M^{2}}(M^{1/2}2^{k/2})^{6}=2^{2k}M.$

The contribution from $a\in{\mathbb{A}}(Q,k)$ is

$\sum_{H\subset[0,M^{2}]\atop{|H|={2^{j}}}}(\sum_{a\in
H\cap{\mathbb{A}}(Q,k)}I_{a}^{2/3})^{3}\lesssim\frac{M^{2}}{2^{j}}(\frac{2^{j}Q^{2}}{M^{2}}\frac{M}{2^{k}Q})^{3}(M2^{2k})^{2}=M2^{k}2^{2j}Q^{3}.$

This is easily seen to be $O(2^{2j}M^{4})$, since $2^{k}=O(MQ^{-1})$ and
$Q=O(M)$. The contribution to the full sum is acceptable, since there are
${\;\lessapprox}\;1$ values of $Q$ and $k$.

Case 2. $Q^{2}<\frac{M^{2}}{2^{j}}$. There are $\lesssim Q^{2}$ arcs with
$q\sim Q$. Essentially, each $H$ is either disjoint from all these (so not
contributing at this stage) or (essentially) contained inside one of them. We
distinguish two subcases.

(a) If $\frac{2^{j}}{M^{2}}<\frac{1}{QM2^{k}}$ (this is stronger than
$Q^{2}<\frac{M^{2}}{2^{j}}$), there are $\frac{1}{QM2^{k}}\frac{M^{2}}{2^{j}}$
intervals $H/M^{2}$ contained in
$[\frac{b}{q}-\frac{1}{QM2^{k}},\frac{b}{q}+\frac{1}{QM2^{k}}]$. Their
contribution is

$\displaystyle\sum_{b<q\sim
Q}\sum_{H/M^{2}\subset[\frac{b}{q}-\frac{1}{QM2^{k}},\frac{b}{q}+\frac{1}{QM2^{k}}]\atop{|H|={2^{j}}}}(\sum_{a\in
H}I_{a}^{2/3})^{3}$ $\displaystyle\lesssim
Q^{2}\frac{M}{Q2^{k}2^{j}}2^{3j}(2^{2k}M)^{2}$
$\displaystyle=2^{2j}M^{3}Q2^{3k}.$

Using our assumption, this is $O(2^{-j}M^{6})$.

(b) If $\frac{2^{j}}{M^{2}}>\frac{1}{QM2^{k}}$, for each $b/q$ with $q\sim Q$
there is only one $H/M^{2}$ that intersects

$|t-\frac{b}{q}|\sim\frac{1}{qM2^{k}}$

with at most $M^{2}\frac{1}{qM2^{k}}$ values of $a$ contributing from $H$. The
contribution from the $O(Q^{2})$ arcs with denominator $\sim Q$ is

$\lesssim Q^{2}(\frac{M}{Q2^{k}})^{3}(2^{2k}M)^{2}=\frac{2^{k}M^{5}}{Q}.$

Since $2^{k}\lesssim M$, this term is $O(M^{6})$.

∎

######  Remark 4.3.

One may wonder whether there is a clever way to estimate the sum

$\sum_{B\subset\Omega}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset
I_{2}}(\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3},$

without using number theory. To this end, the most natural thing to try is to
use Minkowski’s inequality and to bound this expression by

$(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset
I_{2}}(\int_{\Omega}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3}.$
(27)

However, a change of variables as before shows that for each $J_{1},J_{2}$

$\displaystyle\int_{\Omega}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$
$\displaystyle\geq
N^{-1/2}\int_{[0,N^{1/2}]^{4}}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$
$\displaystyle\sim
N^{-1/2}N^{2}[\int_{(u,w)\in[0,N^{1/2}]\times[0,N^{-1/2}]}|\sum_{m=1}^{N^{1/2}}e(mu+m^{2}w)|^{6}dudw]^{2}$
$\displaystyle\sim N^{11/2}.$

Using this approach, the upper bound we get for (27) is $N^{8+\frac{1}{2}}$.
As in our earlier attempt to use $l^{6}$ rather than $l^{2}$ decoupling, this
estimate falls short by a factor of $N^{1/2}$ from the sharp upper bound
$N^{8}$.

Also, due to (17), the expression (27) is only getting larger if $J_{i}$ are
replaced with smaller intervals. Thus, decoupling on cubes larger than $B$
(such as $N^{1/2}$-cubes) only worsens our upper bound.

A similar computation shows that the only case of Conjecture 1.1 that can be
approached with $l^{6}L^{6}$ decoupling is the case $\alpha=2$ discussed in
the previous section.

##  5\. The case $\frac{3}{2}<\alpha\leq\frac{9}{5}$

Let
$\Omega=[0,N^{\frac{2\alpha}{3}-1}]\times[0,N]\times[N^{\frac{1}{2}},N^{\alpha-1}]\times[0,N^{\beta-1}]$.
Using 1-periodicity in $x_{1}$, we need to prove that

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{7+\frac{2\alpha}{3}+\epsilon}.$

We cover $\Omega$ with cubes $B$ of side length $N^{\frac{2\alpha}{3}-1}$ and
write

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\sim\sum_{B\subset\Omega}\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}.$

The size of these cubes is the smallest that will make cubic terms negligible
after the decoupling. Since we need $\frac{2\alpha}{3}-1\leq\beta-1$ in order
to not exceed $\Omega$, this leads to the restriction $\alpha\leq\frac{9}{5}$.
Note also that the $x_{3}$ coordinate is $\geq N^{1/2}$. We can afford this
omission because of the $\alpha=\frac{3}{2}$ case discussed in the previous
section. Since we are about to decouple on cubes $B$ with size larger than 1,
Remark 4.3 tells us that applying this method for $x$ near the origin leads to
losses. Our next argument will make explicit use of the fact that $x_{3}$ is
away from the origin.

We use $l^{2}$ decoupling (Theorem 2.3) on each $B$ (or rather $NB$, after
rescaling)

$\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}(\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3}$

where $J_{i}$ is of the form $[h_{i},h_{i}+M]$, with
$M=N^{1-\frac{\alpha}{3}}$. We write $B$ as

$[0,N^{\frac{2\alpha}{3}-1}]\times[pN^{\frac{2\alpha}{3}-1},(p+1)N^{\frac{2\alpha}{3}-1}]\times[l_{1}N^{\frac{2\alpha}{3}-1},(l_{1}+1)N^{\frac{2\alpha}{3}-1}]\times[l_{2}N^{\frac{2\alpha}{3}-1},(l_{2}+1)N^{\frac{2\alpha}{3}-1}]$

with the integers $0\leq p\leq M^{2}$, $N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq
l_{1}\leq N^{\frac{\alpha}{3}}$ and $0\leq l_{2}\leq N^{3-\frac{5\alpha}{3}}$.
Since $\alpha>\frac{3}{2}$, we have that $l_{1}\gg l_{2}$.

We let as before

$I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M^{2}},\frac{a+O(1)}{M^{2}}]}|\sum_{m\leq
M}e(mu+m^{2}w)|^{6}dudw$

With a change of variables as in the previous section, we have

$\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}\sim
N^{2}(N^{\frac{2\alpha}{3}-1})^{2}I_{a_{1}}I_{a_{2}}$

where

$\begin{cases}a_{1}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\\\
a_{2}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\end{cases}.$
(28)

It is crucial that the cubic (and also the higher order) term is $O(1)$, cf.
(24)

$m^{3}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h}{N}){x_{3}}+\phi_{4}^{\prime\prime\prime}(\frac{h}{N}){x_{4}}}{N^{2}}=O(1),\;\;\forall
x\in\Omega,$

so it may be neglected according to Lemma 9.2.

If $l_{1}\sim 2^{j_{1}}$, it is immediate that $|a_{1}-p|\lesssim 2^{j_{1}}$,
$|a_{2}-p|\lesssim 2^{j_{1}}$. Also, for fixed $a_{1},a_{2},p,l_{1},l_{2}$,
(28) has $O((\frac{N}{M2^{j_{1}}})^{2})$ solutions $(h_{1},h_{2})$, modulo
$O(1)$. We do not need Lemma 4.1 here, since this time $l_{1}$ is much larger
than $l_{2}$. We now dominate
$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}$ as before by

$\lesssim_{\epsilon}N^{\epsilon}\sum_{j_{1}:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq
2^{j_{1}}\leq N^{\frac{\alpha}{3}}}\sum_{l_{1}\sim
2^{j_{1}}}\sum_{j_{2}:\;2^{j_{2}}\leq N^{3-\frac{5\alpha}{3}}}\sum_{l_{2}\sim
2^{j_{2}}}(\frac{N}{M2^{j_{1}}})^{6}N^{2}(N^{\frac{2\alpha}{3}-1})^{2}\sum_{p\leq
M^{2}}(\sum_{a:\;|a-p|\lesssim 2^{j_{1}}}I_{a}^{1/3})^{6}.$

We use Cauchy–Schwarz for the last expression to dominate the above by

$N^{\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq 2^{j}\leq
N^{\frac{\alpha}{3}}}2^{j}N^{3-\frac{5\alpha}{3}}2^{-6j}N^{2\alpha}N^{2}(N^{\frac{2\alpha}{3}-1})^{2}2^{j}2^{3j}\sum_{|H|\sim
2^{j}}(\sum_{a\in H}I_{a}^{2/3})^{3}$
$=N^{3+\frac{5\alpha}{3}+\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq
2^{j}\leq N^{\frac{\alpha}{3}}}2^{-j}\sum_{|H|\sim 2^{j}}(\sum_{a\in
H}I_{a}^{2/3})^{3}.$

Using Lemma 4.2, this is dominated by

$N^{3+\frac{5\alpha}{3}+\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq
2^{j}\leq N^{\frac{\alpha}{3}}}(M^{4}2^{j}+M^{6}2^{-j})\lesssim
N^{\epsilon}(N^{7+\frac{2\alpha}{3}}+N^{\frac{15}{2}+\frac{\alpha}{3}}).$

This is $O(N^{7+\frac{2\alpha}{3}+\epsilon})$, as desired, since
$\alpha>\frac{3}{2}$.

##  6\. The case $\frac{9}{5}\leq\alpha<2$

Let

$\Omega=[0,N^{\delta}]\times[0,N]\times[N^{\frac{4}{5}},N^{\alpha-1}]\times[0,N^{\beta-1}].$

Because of the case addressed in the previous section, we may assume
$x_{3}\geq N^{\frac{4}{5}}$. This will buy us some extra flexibility in
choosing $\delta$. In fact, we can work with any $\delta$ satisfying

$2-\frac{3\beta}{2}\leq\delta\leq\frac{9}{5}-\beta.$ (29)

We need to prove that

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{8+\delta+\epsilon}.$

We will first decouple on cubes $B$ with side length $N^{\beta-1}$. This is
the largest size that is available to us, due to the range in the $x_{4}$
variable. Unlike the case from the previous section, the resulting intervals
are not small enough to make the cubic terms negligible, to allow us to use
estimates for quadratic Weyl sums. We will accomplish that by means of a
further decoupling, on cubes of side length $N^{\delta}$, similar to the case
$\alpha=2$ described earlier.

To get started, we use $l^{2}$ decoupling (Theorem 2.3) on each cube $B$ of
side length $N^{\beta-1}$

$\int_{B}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}(\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3},$

where $J_{i}=[h_{i},h_{i}+M]$ has length $M=N^{1-\frac{\beta}{2}}$.

Next, we cover $\Omega$ with boxes

$\Delta=[0,N^{\delta}]\times[pN^{\delta},(p+1)N^{\delta}]\times[lN^{\delta},(l+1)N^{\delta}]\times[0,N^{\beta-1}],$

with $p\leq N^{1-\delta}$, $N^{\frac{4}{5}-\delta}\leq l\leq
N^{\alpha-1-\delta}$. If we sum up the above inequality over cubes
$B\subset\Delta$ and use Minkowski’s inequality, we find

$\int_{\Delta}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset
I_{1}}\sum_{J_{2}\subset
I_{2}}(\sum_{B\subset\Delta}\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6})^{1/3})^{3}.$
(30)

Next, we fix $J_{1},J_{2}$ and perform a second decoupling for the term
$\int_{\Delta}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$. We proceed as
in Section 3

$\displaystyle\int_{B}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}$
$\displaystyle=N^{-4}\int_{NB}|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N}){\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})|^{6}$
$\displaystyle\lesssim
N^{-4-4\beta}\int_{NB}|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N})|^{6}\int_{NB}|{\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})|^{6}=N^{4-4\beta}\int_{B}|{\mathcal{E}}_{J_{1}}|^{6}\int_{B}|{\mathcal{E}}_{J_{2}}|^{6}.$

Then

$\sum_{B\subset\Delta}\int_{B}|{\mathcal{E}}_{J_{1}}|^{6}\int_{B}|{\mathcal{E}}_{J_{2}}|^{6}\lesssim
N^{4-4\beta}\int_{\Delta}dx\int_{(y,z)\in[0,N^{\beta-1}]^{4}\times[0,N^{\beta-1}]^{4}}|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)|^{6}dydz.$

Combining these two and using periodicity in the $y_{1},z_{1}$ variables we
get

$\int_{\Delta}|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}|^{6}\lesssim
N^{6-6\beta-2\delta}\times$
$\int_{(x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4})\in S}dx_{1}\ldots
dz_{4}\int_{y_{1},z_{1}\in[0,N^{\delta}]\atop{x_{2}\in[pN^{\delta},(p+1)N^{\delta}]\atop{x_{3}\in[lN^{\delta},(l+1)N^{\delta}]}}}|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)|^{6}dy_{1}dz_{1}dx_{2}dx_{3},$

where $S$ is characterized by

$0\leq
x_{1}\in[0,N^{\delta}],\;x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}\in[0,N^{\beta-1}].$

We seek to estimate the second integral uniformly over
$x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$. With these variables fixed,
we make the affine change of variables
$(y_{1},z_{1},x_{2},x_{3})\mapsto(u_{1},u_{2},w_{1},w_{2})$

$\begin{cases}u_{1}=(y_{1}+x_{1})+\frac{2h_{1}}{N}(x_{2}+y_{2})+\phi_{3}^{\prime}(\frac{h_{1}}{N})(x_{3}+y_{3})+\phi_{4}^{\prime}(\frac{h_{1}}{N})(x_{4}+y_{4})\\\
u_{2}=(z_{1}+x_{1})+\frac{2h_{2}}{N}(x_{2}+z_{2})+\phi_{3}^{\prime}(\frac{h_{2}}{N})(x_{3}+z_{3})+\phi_{4}^{\prime}(\frac{h_{2}}{N})(x_{4}+y_{4})\\\
w_{1}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}}{2N}\\\
w_{2}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}}{2N}\end{cases}.$

The Jacobian is $\sim\frac{1}{N^{2}}$, due to (3). The second integral is
comparable to

$N^{2}\int_{(u_{i},w_{i})\in[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{*}^{2}},\frac{a_{i}+O(1)}{M_{*}^{2}}]}\prod_{i=1}^{2}|\sum_{m_{i}\leq
M}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})|^{6}du_{1}dw_{1}du_{2}dw_{2}.$
(31)

Here $M_{*}=N^{\frac{1}{2}-\frac{\delta}{2}}$,
$a_{i}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{i}}{N})}{2}$ and
$\eta_{i}(m)={m^{3}}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h_{i}}{N})}{3!N^{2}}+{m^{4}}\frac{\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{i}}{N})}{4!N^{3}}+\ldots$.
Note that since $\frac{v}{N}=O(M^{-2})$ for $v$ equal to any of the variables
$x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$, we have dismissed the
contribution of these variables associated with quadratic (as well as the
higher order) terms. See Lemma 9.2.

We may apply again Theorem 2.4, using that $x_{3}=A(w_{1}-w_{2})$ with
$A=O(N)$. Note however that this time we cannot decouple into point masses (as
in (10)), since $M_{*}$ is significantly larger than 1. Instead, applying (11)
with $N=(\frac{M}{M_{*}})^{2}$ we dominate (31) by

$N^{2+\epsilon}\times$ (32)
$(\sum_{J_{1}^{\prime},J_{2}^{\prime}}[\int_{(u_{i},w_{i})\in[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{*}^{2}},\frac{a_{i}+O(1)}{M_{*}^{2}}]}\prod_{i=1}^{2}|\sum_{m_{i}\in
J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})|^{6}du_{1}dw_{1}du_{2}dw_{2}]^{\frac{1}{3}})^{3}.$

The intervals $J_{i}^{\prime}$ partitioning $[1,M]$ have length $M_{*}$. What
we have gained by doing this decoupling is that, when $m_{i}$ is confined to a
small interval $J_{i}^{\prime}=[h_{i}^{\prime},h_{i}^{\prime}+M_{*}]$, the
contribution of the term

$\eta_{i}(m_{i})=\eta_{i}(h_{i}^{\prime}+m_{i}^{\prime})=\eta_{i}(h_{i}^{\prime})+\eta_{i}^{\prime}(h_{i}^{\prime})m_{i}^{\prime}+\eta_{i}^{\prime\prime}(h_{i}^{\prime})\frac{(m_{i}^{\prime})^{2}}{2}+O(\frac{(m_{i}^{\prime})^{3}}{N^{2}})$
(33)

can be neglected. To see this, note first that

$\eta_{i}^{\prime\prime}(h_{i}^{\prime})=\sum_{n\geq
3}\phi_{3}^{(n)}(\frac{h_{i}}{N})\frac{(h_{i}^{\prime})^{n-2}}{N^{n-1}(n-2)!}.$

Making another linear change of variables such that

$w_{i}^{\prime}=w_{i}+\frac{\eta_{i}^{\prime\prime}(h_{i}^{\prime})}{2}A(w_{1}-w_{2}),$

we write, using that $|x_{3}|\leq N^{\alpha-1}$

$\prod_{i=1}^{2}|\sum_{m_{i}\in
J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})|=\prod_{i=1}^{2}|\sum_{m_{i}^{\prime}\in[1,M_{*}]}e(m_{i}^{\prime}u_{i}^{\prime}+(m_{i}^{\prime})^{2}w_{i}^{\prime}+O((m_{i}^{\prime})^{3}\frac{N^{\alpha-1}}{N^{2}}))|.$

The range of $w_{i}^{\prime}$ is (a subset of)
$[\frac{a_{i}^{\prime}-O(1)}{M_{*}^{2}},\frac{a_{i}^{\prime}+O(1)}{M_{*}^{2}}]$,
where

$a_{i}^{\prime}=p+\frac{l}{2}\sum_{n\geq
2}\phi_{3}^{(n)}(\frac{h_{i}}{N})\frac{(h_{i}^{\prime})^{n-2}}{N^{n-2}(n-2)!}=p+l\frac{\phi_{3}^{{}^{\prime\prime}}(\frac{h_{i}+h_{i}^{\prime}}{N})}{2}.$

Since $\alpha-1-\frac{\beta}{2}\leq\delta$ by (29), we have that
$a_{i}-a_{i}^{\prime}=O(1)$. Thus, the quadratic term in (33) will not affect
the domain of integration. Moreover, the contribution of the higher order
terms in (33) is negligible (cf. Lemma 9.2), as long as we can guarantee that
we have $\frac{N^{\alpha-1}}{N^{2}}=O(M_{*}^{-3})$. This is equivalent to
$\delta\geq 1-\frac{2\beta}{3}$, and follows from (29) and the fact that
$\beta\leq\frac{6}{5}$. Under this assumption, we dominate (32) by

$N^{2+\epsilon}\left(\sum_{J_{1}^{\prime},J_{2}^{\prime}}\left[\prod_{i=1}^{2}\int_{[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{*}^{2}},\frac{a_{i}+O(1)}{M_{*}^{2}}]}|\sum_{m_{i}\in
J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i})|^{6}du_{i}dw_{i}\right]^{\frac{1}{3}}\right)^{3}.$

This is $\sim N^{2+2\delta+\epsilon}(\frac{M}{M_{*}})^{6}I_{a_{1}}I_{a_{2}}$,
where

$I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M_{*}^{2}},\frac{a+O(1)}{M_{*}^{2}}]}|\sum_{m\leq
M_{*}}e(mu+m^{2}w)|^{6}dudw$

is independent of $J_{1}^{\prime},J_{2}^{\prime}$. Recall that

$\begin{cases}a_{1}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})}{2}\\\
a_{2}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})}{2}\end{cases}.$ (34)

Assume now that $l\sim 2^{j}$, with

$N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}.$

For fixed $p,l$, and fixed $(a_{1},a_{2})$ (within a factor of $O(1)$), the
system (34) has $\lesssim(\frac{N}{2^{j}M})^{2}$ solutions $(h_{1},h_{2})$.
Getting back to (30), summing over $\Delta\subset\Omega$ we find that

$\int_{\Omega}|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}|^{6}\lesssim_{\epsilon}$
$N^{6-6\beta-2\delta}|S|N^{2+2\delta+\epsilon}N^{3+3\delta}\sum_{p\leq
M_{*}^{2}}\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim
N^{\alpha-1-\delta}}2^{-6j}\sum_{l\sim 2^{j}}(\sum_{|a-p|\lesssim
2^{j}}I_{a}^{1/3})^{6}.$

We use Cauchy–Schwarz to dominate this by

$N^{4+4\delta+\beta+\epsilon}\sum_{N^{\frac{4}{5}-\delta}\lesssim
2^{j}\lesssim
N^{\alpha-1-\delta}}\sum_{H\subset[1,M_{*}^{2}]\atop{|H|=2^{j}}}2^{-j}(\sum_{a\in
H}I_{a}^{2/3})^{3}.$

Using Lemma 4.2, it remains to check that

$\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim
N^{\alpha-1-\delta}}M_{*}^{4}2^{j}+\sum_{N^{\frac{4}{5}-\delta}\lesssim
2^{j}\lesssim N^{\alpha-1-\delta}}M_{*}^{6}2^{-j}\lesssim
N^{4-\beta-3\delta}.$

The first sum is in order, since $\alpha+\beta=3$. So is the second sum, as
long as $\delta\leq\frac{9}{5}-\beta$, which is guaranteed by (29).

##  7\. Proof of Theorem 1.2

This section shows that Theorem 1.3 implies Theorem 1.2. The argument is
inspired by [2].

The parameter $K$ will be very large and universal, independent of $N$,
$\phi_{k}$. The larger the $K$ we choose to work with, the smaller the
$\epsilon$ from the $N^{\epsilon}$ loss will be at the end of the section.

######  Proposition 7.1.

Assume $\alpha+\beta=3$ and $\frac{3}{2}\leq\alpha\leq 2.$ Assume
$\phi_{3},\phi_{4}:(0,3)\to{\mathbb{R}}$ are real analytic and satisfy (1),
(2) and (3). Let as before $\omega_{3}=[0,N^{\alpha}]$,
$\omega_{4}=[0,N^{\beta}]$ and

${\mathcal{E}}_{I,N}(x)=\sum_{n\in
I}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}+\phi_{4}(\frac{n}{N})x_{4}).$

We consider arbitrary integers $N_{0},M$ satisfying $1\leq
M\leq\frac{N_{0}}{K}$ and $N_{0}+[M,2M]\subset[\frac{N}{2},N]$.

Let $H_{1},H_{2}$ be intervals of length $\frac{M}{K}$ inside $N_{0}+[M,2M]$
such that ${\operatorname{dist}\,}(H_{1},H_{2})\geq\frac{M}{K}$. Then

$\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{6}\lesssim_{\epsilon}N^{7}M^{2+\epsilon}.$

###### Proof.

Write $H_{1}=N_{0}+I_{1}$, $H_{2}=N_{0}+I_{2}$ with $I_{1},I_{2}$ intervals of
length $\frac{M}{K}$ inside $[M,2M]$ and with separation $\geq\frac{M}{K}$. We
use the following expansion, certainly valid for all $m$ in $I_{i}$.

$\displaystyle\phi_{3}(\frac{N_{0}+m}{N})$ $\displaystyle=Q_{3}(m)+\sum_{n\geq
3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})}{n!}(\frac{m}{N})^{n}$
$\displaystyle=Q_{3}(m)+(\frac{M}{N})^{3}\sum_{n\geq
3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-3}}{n!}(\frac{m}{M})^{n}.$

Here $Q_{3}(m)=A+Bm+Cm^{2}$ with $B=O(\frac{1}{N})$, $C=O(\frac{1}{N^{2}})$.
We introduce the analogue $\tilde{\phi_{3}}$ of $\phi_{3}$ at scale $M$

$\tilde{\phi_{3}}(t)=\sum_{n\geq
3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-3}}{n!}t^{n}.$

This series is convergent as long as $\frac{N_{0}}{N}+t\in(0,3)$, so the new
function is certainly real analytic on $(0,2)$, since $N_{0}\leq N$.

Let $\delta>0$ be conveniently small. By choosing $K$ large enough we can make
$\frac{M}{N}$ as small as we wish, so we may guarantee that for each
$t\in[\frac{1}{2},1]$ we have

$|\tilde{\phi}_{3}^{(3)}(t)-{\phi}_{3}^{(3)}(\frac{N_{0}}{N})|\leq\delta.$
(35)

Thus, we can guarantee (3) for $\tilde{\phi}_{3}$, with a slightly smaller,
but uniform value of $A_{4}$. The same will work with (1) and (2), as it will
soon become clear. To this end, we may also enforce

$|\tilde{\phi}_{3}^{(4)}(t)|\leq\delta.$ (36)

We also define, with $Q_{4}(m)=D+Em+Fm^{2}$ satisfying $E=O(\frac{1}{N})$,
$F=O(\frac{1}{N}^{2})$

$\displaystyle\phi_{4}(\frac{N_{0}+m}{N})$ $\displaystyle=Q_{4}(m)+\sum_{n\geq
3}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})}{n!}(\frac{m}{N})^{n}$
$\displaystyle=Q_{4}(m)+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{3!}(\frac{m}{N})^{3}+(\frac{M}{N})^{4}\sum_{n\geq
4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-4}}{n!}(\frac{m}{M})^{n}.$

The last two terms are equal to

$\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}(\frac{M}{N})^{3}\tilde{\phi}_{3}(\frac{m}{M})+(\frac{M}{N})^{4}\sum_{n\geq
4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(n)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})n!}(\frac{M}{N})^{n-4}(\frac{m}{M})^{n}.$

Let $\tilde{\phi}_{4}$ be the analogue of $\phi_{4}$ at scale $M$ defined by

$\tilde{\phi}_{4}(t)=\sum_{n\geq
4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(n)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})n!}(\frac{M}{N})^{n-4}t^{n}.$

As before, by choosing $K$ large enough, we can arrange that for all
$t\in[\frac{1}{2},1]$

$|\tilde{\phi}_{4}^{(4)}(t)-\frac{\phi_{4}^{(4)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(4)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}|\leq\delta.$

Combining this with (35) and (36) we may arrange that

$\det\begin{bmatrix}\tilde{\phi}_{3}^{(3)}(t)&\tilde{\phi}_{3}^{(4)}(t)\\\
\tilde{\phi}_{4}^{(3)}(s)&\tilde{\phi}_{4}^{(4)}(s)\end{bmatrix}-\det\begin{bmatrix}\phi_{3}^{(3)}(\frac{N_{0}}{N})&\phi_{3}^{(4)}(\frac{N_{0}}{N})\\\
\phi_{4}^{(3)}(\frac{N_{0}}{N})&\phi_{4}^{(4)}(\frac{N_{0}}{N})\end{bmatrix}$

is as small in absolute value as we wish, uniformly over
$t,s\in[\frac{1}{2},1]$. In particular, we can guarantee (2) for the pair
$(\tilde{\phi}_{3},\tilde{\phi}_{4})$, with slightly modified, but uniform
values of $A_{2},A_{3}$. Similar comments apply regarding (1).

Now

$\displaystyle|{\mathcal{E}}_{H_{k},N}(x)|$ $\displaystyle=|\sum_{m\in
I_{k}}e(mx_{1}+(m^{2}+2N_{0}m)x_{2}+\phi_{3}(\frac{N_{0}+m}{N})x_{3}+\phi_{4}(\frac{N_{0}+m}{N})x_{4})|$
$\displaystyle=|\sum_{m\in
I_{i}}e(m(x_{1}+2N_{0}x_{2}+Bx_{3}+Ex_{4})+m^{2}(x_{2}+Cx_{3}+Fx_{4})$
$\displaystyle+(\frac{M}{N})^{3}\tilde{\phi}_{3}(\frac{m}{M})(x_{3}+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}x_{4})$
$\displaystyle+(\frac{M}{N})^{4}\tilde{\phi}_{4}(\frac{m}{M})x_{4})|.$

Recall $N_{0}\sim N$, $B,E=O(1/N)$, $C,F=O(1/N^{2})$. We make the change of
variables

$\begin{cases}y_{1}=x_{1}+2N_{0}x_{2}+Bx_{3}+Ex_{4}\\\
y_{2}=x_{2}+Cx_{3}+Fx_{4}\\\
y_{3}=(\frac{M}{N})^{3}(x_{3}+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}x_{4})\\\
y_{4}=(\frac{M}{N})^{4}x_{4}\end{cases}.$

Due to periodicity, we may extend the range of $x_{1}$ to $[0,N_{0}]$. This
linear transformation maps
$[0,N_{0}]\times[0,1]\times\omega_{3}\times\omega_{4}$ to a subset of a box
$\tilde{\omega}_{1}\times\tilde{\omega}_{2}\times\tilde{\omega}_{3}\times\tilde{\omega_{4}}$
centered at the origin, with dimensions roughly
$N_{0},1,M^{3}N^{\alpha-3},M^{4}N^{\beta-4}$.

Thus

$|{\mathcal{E}}_{H_{k},N}(x)|=|{\mathcal{E}}_{I_{k},M}(y)|$

where

${\mathcal{E}}_{I_{k},M}(y)=\sum_{m\in
I_{k}}e(my_{1}+m^{2}y_{2}+\tilde{\phi}_{3}(\frac{m}{M})y_{3}+\tilde{\phi}_{4}(\frac{m}{M})y_{4}).$

We may write, using again periodicity in $y_{1}$ and $y_{2}$

$\displaystyle\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{6}$
$\displaystyle=\frac{1}{N_{0}}\int_{[0,N_{0}]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{6}$
$\displaystyle\leq(\frac{N}{M})^{7}\int_{[0,1]\times[0,1]\times\tilde{\omega}_{3}\times\tilde{\omega_{4}}}|{\mathcal{E}}_{I_{1},M}(y){\mathcal{E}}_{I_{2},M}(y)|^{6}.$

Finally, we use Theorem 1.3 with $N=M$, noting that
$\tilde{\omega}_{3}\subset[-M^{\alpha},M^{\alpha}]$ and
$\tilde{\omega}_{4}\subset[-M^{\beta},M^{\beta}]$, to estimate the last
expression by

$(\frac{N}{M})^{7}M^{9+\epsilon}=N^{7}M^{2+\epsilon}.$

∎

We can now prove Theorem 1.2.

Choose $K$ large enough, depending on $\epsilon$. Write ${\mathcal{H}}_{n}(I)$
for the collection of dyadic intervals in $I$ with length $\frac{N}{2K^{n}}$.
We write $H_{1}\not\sim H_{2}$ to imply that $H_{1},H_{2}$ are not neighbors.
Then

$|{\mathcal{E}}_{I,N}(x)|\leq
3\max_{H\in{\mathcal{H}}_{1}(I)}|{\mathcal{E}}_{H,N}(x)|+K^{10}\max_{H_{1}\not\sim
H_{2}\in{\mathcal{H}}_{1}(I)}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{1/2}.$

We repeat this inequality until we reach intervals in ${\mathcal{H}}_{l}$ of
length $\sim 1$, that is $K^{l}\sim N$. We have

$\displaystyle|{\mathcal{E}}_{I,N}(x)|$ $\displaystyle\lesssim
3^{l}+l3^{l}K^{10}\max_{1\leq n\leq
l}\max_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim
H_{2}\in{\mathcal{H}}_{n+1}(H)}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{1/2}$
$\displaystyle\lesssim(\log N)N^{\log_{K}3}\max_{1\leq n\leq
l}\max_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim
H_{2}\in{\mathcal{H}}_{n+1}(H)}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{1/2}.$

Using Corollary 7.1 we finish the proof

$\displaystyle\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}$
$\displaystyle|{\mathcal{E}}_{I,N}|^{12}$
$\displaystyle\lesssim_{K}N^{\log_{K}3}\sum_{n}\sum_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim
H_{2}\in{\mathcal{H}}_{n+1}(H)}\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)|^{6}$
$\displaystyle\lesssim_{K,\epsilon}N^{\epsilon+\log_{K}3}\sum_{n}{K^{n}}N^{7}(\frac{N}{K^{n}})^{2}$
$\displaystyle\lesssim_{K,\epsilon}N^{\epsilon+\log_{K}3}N^{9}.$

Choosing $K$ large enough, we may make $\log_{K}3$ as small as we wish.

##  8\. Other values of $p$

The reason Theorem 1.2 was accessible via the bilinear result in Theorem 1.3
has to do with the fact that $6$ is the critical exponent for the decoupling
for the parabola (at the canonical scale). Thus, our arguments rely
fundamentally on this dimensional reduction. In [6], the small cap decoupling
for the parabola is settled, and the associated critical exponents lie between
4 and 6. In principle, this new tool can be used to determine $L^{p}$ moments
for curves in ${\mathbb{R}}^{4}$, in the range $8\leq p\leq 12$.

There are many possible things to consider in this direction. One is the
following extension of Conjecture 1.1, that we use to illustrate a different
type of obstruction that appears in this regime. This was observed in [1], in
a related context.

######  Conjecture 8.1 (Square root cancellation in $L^{p}$).

Let $11\leq p\leq 12$. Assume $\alpha\geq\beta\geq 0$ satisfy
$\alpha+\beta=\frac{p}{2}-3$. Let $\phi_{3}$, $\phi_{4}$, $\omega_{3}$,
$\omega_{4}$ be as in Theorem 1.2. Then

$\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{[\frac{N}{2},N],N}|^{p}\lesssim_{\epsilon}N^{p-3+\epsilon}.$

The case $\beta=0$ was proved in [6] in the larger range $9\leq p<12$. As
mentioned earlier, when $\beta=0$, the curve collapses to a three dimensional
curve. However, the next result shows that the restriction $p\geq 11$ is
needed if $\beta>0.$ The new obstruction can be described as constructive
interference on spatially disjoint blocks.

######  Theorem 8.2.

Assume $p<11$. Let $\omega_{3}=[-N^{\alpha},N^{\alpha}]$,
$\omega_{4}=[-N^{\beta},N^{\beta}]$, $\alpha\geq\beta$ and
$\alpha+\beta=\frac{p}{2}-3$. Assume also that $\beta>0$.

Then, for some $\delta>0$ and $\phi_{3}(t)=t^{3}$, $\phi_{4}(t)=t^{4}$ we have

$\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{[\frac{N}{2},N],N}|^{p}\gtrsim
N^{p-3+\delta}.$

###### Proof.

Lemma 8.3 shows that the integral is greater than

$\sum_{J\subset
I}\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{J,N}|^{p},$

where the sum runs over intervals $J$ of length $M<N$, partitioning
$[\frac{N}{2},N]$. The parameter $M$ will be determined later. In some sense,
the components ${\mathcal{E}}_{J,N}$ behave as if they were spatially
supported on pairwise disjoint sets.

By periodicity

$\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{J,N}|^{p}=N^{-3}\int_{[-N^{2},N^{2}]\times[-N,N]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{J,N}|^{p}.$

Write $H=[h+1,h+M]$. Note that

$|{\mathcal{E}}_{J,N}(x)|=|\sum_{1\leq m\leq
M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})|$

where

$\begin{cases}y_{1}=x_{1}+2h{x_{2}}+\frac{3h^{2}}{N^{3}}x_{3}+\frac{4h^{3}}{N^{4}}x_{4}\\\
y_{2}=x_{2}+\frac{3h}{N^{3}}x_{3}+\frac{6h^{2}}{N^{4}}x_{4}\\\
y_{3}=\frac{x_{3}}{N^{3}}+\frac{4h}{N^{4}}x_{4}\\\
y_{4}=\frac{x_{4}}{N^{4}}\end{cases}.$

This change of variables maps
$[-N^{2},N^{2}]\times[-N,N]\times\omega_{3}\times\omega_{4}$ to a set
containing

$S=[-o(N^{2}),o(N^{2})]\times[-o(N),o(N)]\times[-o(N^{\alpha-3}),o(N^{\alpha-3})]\times[-o(N^{\beta-4}),o(N^{\beta-4})].$

We have used that $3\geq\alpha\geq\beta-1$. Thus

$\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{J,N}(x)|^{p}dx$
$\geq N^{4}\int_{S}|\sum_{1\leq m\leq
M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})|^{p}dy.$

Let

$M=\max\\{N^{1-\frac{\alpha}{3}},N^{1-\frac{\beta}{4}}\\}.$

Note that since $\beta>0$ we have that $M=N^{1-\epsilon}$ for some
$\epsilon>0$. Note also that

$[0,o(M^{-3})]\times[0,o(M^{-4})]\subset[0,o(N^{\alpha-3})]\times[0,o(N^{\beta-4})].$

Using constructive interference we get

$\int_{S}|\sum_{1\leq m\leq
M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})|^{p}dy\gtrsim M^{p-10}N^{3}.$

Putting things together we conclude that

$\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}|{\mathcal{E}}_{[\frac{N}{2},N],N}|^{p}\gtrsim\frac{N}{M}N^{4}M^{p-10}N^{3}.$

Note that this is $\geq N^{p-3+\delta}$, for some $\delta>0$.

∎

######  Lemma 8.3.

Let ${\mathcal{R}}$ be a collections of rectangular boxes $R$ in
${\mathbb{R}}^{n}$, with pairwise disjoint doubles $2R$. Let $F$ be a Schwartz
function in ${\mathbb{R}}^{n}$ that can be written as

$F=\sum_{R\in{\mathcal{R}}}F_{R},$

with the spectrum of $F_{R}$ inside $R$. Then for each $2\leq p\leq\infty$ we
have

$(\|F_{R}\|_{p})_{l^{p}({\mathcal{R}})}\lesssim\|F\|_{p}.$

The implicit constant is independent of $F$ and ${\mathcal{R}}$.

###### Proof.

Interpolate between $2$ and $\infty$. ∎

##  9\. Auxiliary results

This section records two auxiliary results that are used repeatedly throughout
the paper.

######  Lemma 9.1.

Assume $t,s\in[\frac{1}{2},1]$ satisfy $|t-s|\sim 1$. The Jacobian of the
transformation $y=\psi(x)$

$\begin{cases}y_{1}=x_{1}+2tx_{2}+\phi_{3}^{\prime}(t)x_{3}+\phi_{4}^{\prime}(t)x_{4}\\\
y_{2}=x_{1}+2sx_{2}+\phi_{3}^{\prime}(s)x_{3}+\phi_{4}^{\prime}(s)x_{4}\\\
y_{3}=\frac{2x_{2}}{N}+\phi_{3}^{\prime\prime}(t)\frac{x_{3}}{N}+\phi_{4}^{\prime\prime}(t)\frac{x_{4}}{N}\\\
y_{4}=\frac{2x_{2}}{N}+\phi_{3}^{\prime\prime}(s)\frac{x_{3}}{N}+\phi_{4}^{\prime\prime}(s)\frac{x_{4}}{N}\end{cases}$

is $\sim\frac{1}{N^{2}}.$

Moreover, $\psi$ maps cubes $Q$ with side length $L$ to subsets of rectangular
boxes of dimensions roughly $L\times L\times\frac{L}{N}\times\frac{L}{N}$.

If the cube $Q$ is centered at the origin, $\psi(Q)$ contains the rectangular
box $[-o(L),o(L)]^{2}\times[-o(\frac{L}{N}),o(\frac{L}{N})]^{2}$.

###### Proof.

Let $\phi_{1}(u)=u$, $\phi_{2}(u)={u^{2}}$. Then the Jacobian is

$\frac{1}{N^{2}}\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\
\phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\
\phi_{1}^{\prime\prime}(t)&\phi_{2}^{\prime\prime}(t)&\phi_{3}^{\prime\prime}(t)&\phi_{4}^{\prime\prime}(t)\\\
\phi_{1}^{\prime\prime}(s)&\phi_{2}^{\prime\prime}(s)&\phi_{3}^{\prime\prime}(s)&\phi_{4}^{\prime\prime}(s)\end{array}\right].$
(37)

Note that

$\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\
\phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\
\phi_{1}^{\prime\prime}(t)&\phi_{2}^{\prime\prime}(t)&\phi_{3}^{\prime\prime}(t)&\phi_{4}^{\prime\prime}(t)\\\
\phi_{1}^{\prime\prime}(s)&\phi_{2}^{\prime\prime}(s)&\phi_{3}^{\prime\prime}(s)&\phi_{4}^{\prime\prime}(s)\end{array}\right]=\lim_{\epsilon\to
0}\frac{1}{\epsilon^{2}}\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\
\phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\
\phi_{1}^{\prime}(t+\epsilon)&\phi_{2}^{\prime}(t+\epsilon)&\phi_{3}^{\prime}(t+\epsilon)&\phi_{4}^{\prime}(t+\epsilon)\\\
\phi_{1}^{\prime}(s+\epsilon)&\phi_{2}^{\prime}(s+\epsilon)&\phi_{3}^{\prime}(s+\epsilon)&\phi_{4}^{\prime}(s+\epsilon)\end{array}\right].$

A generalization of the Mean-Value Theorem (see [9], Voll II, part V, Chap 1,
No. 95) guarantees that

$\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\
\phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\
\phi_{1}^{\prime}(t+\epsilon)&\phi_{2}^{\prime}(t+\epsilon)&\phi_{3}^{\prime}(t+\epsilon)&\phi_{4}^{\prime}(t+\epsilon)\\\
\phi_{1}^{\prime}(s+\epsilon)&\phi_{2}^{\prime}(s+\epsilon)&\phi_{3}^{\prime}(s+\epsilon)&\phi_{4}^{\prime}(s+\epsilon)\end{array}\right]=$
$\displaystyle=\epsilon^{2}(t-s)^{2}(t+\epsilon-s)(s+\epsilon-t)\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(\tau_{1})&\phi_{2}^{\prime}(\tau_{1})&\phi_{3}^{\prime}(\tau_{1})&\phi_{4}^{\prime}(\tau_{1})\\\
\phi_{1}^{\prime\prime}(\tau_{2})&\phi_{2}^{\prime\prime}(\tau_{2})&\phi_{3}^{\prime\prime}(\tau_{2})&\phi_{4}^{\prime\prime}(\tau_{2})\\\
\phi_{1}^{\prime\prime\prime}(\tau_{3})&\phi_{2}^{\prime\prime\prime}(\tau_{3})&\phi_{3}^{\prime\prime\prime}(\tau_{3})&\phi_{4}^{\prime\prime\prime}(\tau_{3})\\\
\phi_{1}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{2}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{3}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{4}^{\prime\prime\prime\prime}(\tau_{4})\end{array}\right]$
$\displaystyle=\epsilon^{2}(t-s)^{2}(t+\epsilon-s)(s+\epsilon-t)\operatorname{det}\left[\begin{array}[]{cccc}\phi_{3}^{\prime\prime\prime}(\tau_{3})&\phi_{4}^{\prime\prime\prime}(\tau_{3})\\\
\phi_{3}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{4}^{\prime\prime\prime\prime}(\tau_{4})\end{array}\right]$

for some $\tau_{i}\in[\frac{1}{2},1]$ depending on $t,s,\epsilon$. The
conclusion follows by letting $\epsilon\to 0$ and using (2).

The second statement is immediate. To prove the last one, assume

$y\in[-cL,cL]^{2}\times[-c\frac{L}{N},c\frac{L}{N}]^{2},$

for some small enough $c$, independent of $N$. We need to prove that
$y=\psi(x)$ for some $x\in Q$. This can be seen by solving for $x$. For
example,

$x_{1}\sim
N^{2}\operatorname{det}\left[\begin{array}[]{cccc}y_{1}&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\
y_{2}&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\
y_{3}&\frac{\phi_{2}^{\prime\prime}(t)}{N}&\frac{\phi_{3}^{\prime\prime}(t)}{N}&\frac{\phi_{4}^{\prime\prime}(t)}{N}\\\
y_{4}&\frac{\phi_{2}^{\prime\prime}(s)}{N}&\frac{\phi_{3}^{\prime\prime}(s)}{N}&\frac{\phi_{4}^{\prime\prime}(s)}{N}\end{array}\right].$

This and (1) show that

$|x_{1}|\lesssim
N^{2}(\frac{|y_{1}|+|y_{2}|}{N^{2}}+\frac{|y_{3}|+|y_{4}|}{N}).$

The same inequality holds for all $x_{i}$, which proves the desired statement.

∎

######  Lemma 9.2.

Let $\gamma$ be a Schwartz function supported on $[-2,2]$. Define the smooth
Weyl sums for $u,w,v\in{\mathbb{R}}$

$G(u,w,v)=\sum_{k\in{\mathbb{Z}}}\gamma(k/M)e(ku+k^{2}w+k^{3}v).$

Let $1\leq b\leq q\leq M$ with $(b,q)=1$. Assume that
${\operatorname{dist}\,}(w-\frac{b}{q},{\mathbb{Z}}):=\varphi\leq\frac{1}{qM}$
and that $|v|\lesssim\frac{1}{M^{3}}$. Then for each $\epsilon>0$ we have

$|G(u,w,v)|\lesssim_{\epsilon}\frac{M^{\epsilon}}{q^{1/2}}\min\\{M,\frac{1}{\varphi^{1/2}}\\}$

if

$u\in{\mathcal{M}}=\bigcup_{m\in{\mathbb{Z}}}[\frac{m}{q}-\varphi
M^{1+\epsilon},\frac{m}{q}+\varphi M^{1+\epsilon}]$

and

$|G(u,w,v)|\lesssim_{\epsilon}M^{-100}$

if

$u\not\in{\mathcal{M}}.$

###### Proof.

Invoking periodicity, we may assume that $w=\frac{b}{q}+\varphi$ with
$|\varphi|\leq\frac{1}{Mq}$. Using the representation $k=rq+k_{1}$, $0\leq
k_{1}\leq q-1$ and the Poisson summation formula we get

$G(u,w,v)=\sum_{k_{1}=0}^{q-1}e(k_{1}^{2}b/q)\sum_{r\in{\mathbb{Z}}}\gamma(\frac{k_{1}+rq}{M})e((rq+k_{1})u+(rq+k_{1})^{2}\varphi+(rq+k_{1})^{3}v)$
$=\sum_{m\in{\mathbb{Z}}}\left[\frac{1}{q}\sum_{k_{1}=0}^{q-1}e(k_{1}^{2}b/q-k_{1}m/q)\right]\left[\int_{\mathbb{R}}\gamma(y/M)e((u+\frac{m}{q})y+\varphi
y^{2}+vy^{3})dy\right]$ $=\sum_{m\in{\mathbb{Z}}}S(b,m,q)J(u,v,\varphi,m,q)$
(38)

where

$S(b,m,q)=\frac{1}{q}\sum_{k=0}^{q-1}e(k^{2}b/q-km/q)$ $\displaystyle
J(u,v,\varphi,m,q)$
$\displaystyle=\int_{{\mathbb{R}}}\gamma(y/M)e((u+\frac{m}{q})y+\varphi
y^{2}+vy^{3})dy$
$\displaystyle=M\int_{{\mathbb{R}}}\gamma(z)e(M(u+\frac{m}{q})z+\varphi
M^{2}z^{2}+vM^{3}z^{3})dz.$

If $|\varphi|\lesssim\frac{1}{M^{2}}$, we are content with the bound
$|J(u,v,\varphi,m,q)|\lesssim M$.

Assume now that $|\varphi|\gg\frac{1}{M^{2}}$. The classical van der Corput
estimate (second derivative test) reads

$|\int_{{\mathbb{R}}}\gamma(z)e(Az+Bz^{2}+Cz^{3})dz|\lesssim|B|^{-1/2},$

if $|B|\gg|C|$. In our case $|B|=|\varphi|M^{2}\gg 1\gtrsim|vM^{3}|=|C|$. In
either case we get

$|J(u,v,\varphi,m,q)|\lesssim\min\\{M,|\varphi|^{-1/2}\\}.$

On the other hand, repeated integration by parts (first derivative test) shows
that for each $\alpha>0$

$|J(u,v,\varphi,m,q)|\lesssim_{\alpha}\frac{1}{A^{\alpha}}$

when $|A|=M|u+\frac{m}{q}|\geq M^{\epsilon}\varphi M^{2}$. Thus, when
$u\in{\mathcal{M}}$, only $O(M^{\epsilon})$ values of $m$ will have a non-
negligible contribution to the sum, while if $u\not\in{\mathcal{M}}$ then the
contribution from all $m$ will be negligible.

Combining these with the classical estimate

$|S(b,m,q)|\lesssim\frac{1}{\sqrt{q}}$

finishes the argument.

∎

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