# Multidimensional polynomial patterns over finite fields: bounds, counting
estimates and Gowers norm control

Borys Kuca  University of Crete, Department of mathematics and applied
mathematics, Voutes University Campus, Heraklion 71003, Greece
<EMAIL_ADDRESS>
###### Abstract.

We examine multidimensional polynomial progressions involving linearly
independent polynomials in finite fields, proving power saving bounds for sets
lacking such configurations. This jointly generalises earlier results of
Peluse (for the single dimensional case) and the author (for distinct degree
polynomials). In contrast to the cases studied in the aforementioned two
papers, a usual PET induction argument does not give Gowers norm control over
multidimensional progressions that involve polynomials of the same degrees.
The main challenge is therefore to obtain Gowers norm control, and we
accomplish this for all multidimensional polynomial progressions with pairwise
independent polynomials. The key inputs are: (1) a quantitative version of a
PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis,
Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box
norms in arbitrary finite abelian groups, motivated by earlier results of Tao,
Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box
norm smoothing technique, recently developed in the ergodic setting by the
author and Frantzikinakis; and (4) a new version of the multidimensional
degree lowering argument.

###### Key words and phrases:

polynomial Szemerédi theorem, polynomial progressions, Gowers norms.

###### 2020 Mathematics Subject Classification:

Primary: 11B30.

The author was supported by the Hellenic Foundation for Research and
Innovation, Project No: 1684.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Notation
  2. 2 Gowers norms for finite abelian groups
    1. 2.1 Basic definitions and properties
    2. 2.2 Concatenation of box norms for general groups
    3. 2.3 Weak inverse theorems for Gowers norms
  3. 3 The PET induction scheme
  4. 4 Control by a single box norm
  5. 5 Passing to a Gowers norm control
    1. 5.1 Norm smoothing for a linearly independent progression
    2. 5.2 Norm smoothing for a linearly dependent progression
    3. 5.3 The formalism for longer progressions
    4. 5.4 The induction scheme
  6. 6 Degree lowering
  7. A Standard technical lemmas

##  1\. Introduction

The last decade has witnessed considerable interest in quantifying the
polynomial Szemerédi theorem of Bergelson and Leibman, which asserts that all
dense subsets of integers contain polynomial patterns of a fairly general
form. In a series of papers, Peluse and Prendiville gave bounds for subsets of
natural numbers lacking polynomial progressions

(1) $\displaystyle x,\;x+p_{1}(n),\;\ldots,\;x+p_{\ell}(n)$

for fixed $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ which either have distinct
degrees [25, 26, 27] or are monomials of the same degree [28]. Even more
bounds have been obtained in the finite field setting, where one wants to
bound the size of subsets of ${\mathbb{F}}_{p}$ (for a large prime $p$)
lacking the patterns (1) [1, 3, 4, 16, 19, 20, 21, 23, 24]; some of these have
recently been extended to the setting of finite commutative rings [2]. Much
less is known in the multidimensional version of the aforementioned problems,
in which one looks at configurations such as

(2)
$\displaystyle(x_{1},\;x_{2}),\;(x_{1}+n^{2},\;x_{2}),\;(x_{1},\;x_{2}+n^{2}+n);$

the necessity of dealing with several different directions at once introduces
a number of technical issues that the existing methods found hard to deal
with. For the particular progression (2), good bounds in the finite field
setting have been obtained by Han, Lacey and Yang [16]; but when $n^{2}$ and
$n^{2}+n$ are replaced by higher degree polynomials of the same degree, no
bounds are currently known. Motivated by recent progress in ergodic theory [5,
6, 10, 11], we develop new techniques that enable us to deal with many of the
obstructions posed by multidimensional progressions. As a consequence, we give
power-saving bounds in the multidimensional polynomial Szemerédi theorem over
finite fields for progressions along linearly independent polynomials.

######  Theorem 1.1.

Let $d,D,\ell\in{\mathbb{N}}$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ be nonzero
vectors, and $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ be linearly independent
polynomials of degrees at most $d$ with zero constant terms. There exist
absolute constants $c=c(d,\ell)$, $C=C(d,\ell)>0$ such that all subsets of
${\mathbb{F}}_{p}^{D}$ with cardinality at least $Cp^{D-c}$ contain

(3)
$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}p_{1}(n),\;\ldots,\;{\textbf{x}}+{\textbf{v}}_{\ell}p_{\ell}(n)$

for some ${\textbf{x}}\in{\mathbb{F}}_{p}^{D}$,
$n\in{\mathbb{F}}_{p}\setminus{\\{0\\}}$.

For instance, Theorem 1.1 gives the first known bounds for subsets of
${\mathbb{F}}_{p}^{2}$ lacking the progression

$\displaystyle(x_{1},\;x_{2}),\;(x_{1}+n^{3},\;x_{2}),\;(x_{1},\;x_{2}+n^{3}+n)$

for some $x_{1},x_{2},n\in{\mathbb{F}}_{p}$ with $n\neq 0$.

Similarly to earlier works on the subject, Theorem 1.1 follows from a counting
estimate for the progression (3). In what follows, we let
$\operatorname*{\mathbb{E}}_{x\in X}=\frac{1}{X}\sum_{x\in X}$ for a finite
set $X$.

######  Theorem 1.2.

Let $d,D,\ell\in{\mathbb{N}}$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ be nonzero
vectors, and $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ be linearly independent
polynomials of degrees at most $d$ with zero constant terms. There exist
absolute constants $c=c(d,\ell)$, $C=C(d,\ell)>0$ such that for all 1-bounded
functions $f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have

(4)
$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))-\operatorname*{\mathbb{E}}_{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}}f_{0}({\textbf{x}})\prod_{j=1}^{\ell}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{j}({\textbf{x}}+{\textbf{v}}_{j}n)\mathclose{}\right|\leq
Cp^{-c}.$

Theorems 1.1 and 1.2 are joint generalisations of the results of Peluse in the
single dimensional case $D=1$ [24] and the results of the author for distinct
degree polynomials [19]. Yet Theorem 1.2, from which Theorem 1.1 follows in a
straightforward way, is significantly more difficult to prove than the
analogous results in [19, 24]. The arguments in [19, 24] essentially consist
of two steps. First, a PET induction argument establishes control of the
relevant counting operator

(5)
$\displaystyle\Lambda(f_{0},\ldots,f_{\ell})=\operatorname*{\mathbb{E}}_{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))$

by a Gowers norm of some degree $s$ (depending only on $\ell$ and the maximum
degree of the polynomials) of the function $f_{\ell}$ corresponding to the
polynomial $p_{\ell}$ of highest degree. Second, a degree lowering argument
allows us to pass from degree $s$ control to degree 1 control. Iterating this
argument for each function, we arrive at the identity (4).

The strategy outlined above breaks out quickly in the multidimensional case
$D>1$ whenever some of the polynomials have the same degree. The main issue in
this case is that the PET induction scheme that we use, developed in the
ergodic setting by Donoso, Ferré-Moragues, Koutsogiannis and Sun [5], only
allows us to control $\Lambda(f_{0},\ldots,f_{\ell})$ by a rather complicated
average of Gowers box norms of the functions involved. For instance, a PET
induction argument for the configuration (2) gives the bound

(6)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{x_{1},x_{2},n\in{\mathbb{F}}_{p}}f_{0}(x_{1},x_{2})f_{1}(x_{1}+n^{2},x_{2})f_{2}(x_{1},x_{2}+n^{2}+n)\mathclose{}\right|^{8}\\\
\leq\operatorname*{\mathbb{E}}_{h_{1},h_{2},h_{3}\in{\mathbb{F}}_{p}}\left\|f_{2}\right\|_{\begin{subarray}{c}2(h_{2}+h_{3}){\mathbf{e}}+2h_{1}{\mathbf{e}}_{2},\
2h_{2}{\mathbf{e}}+2h_{1}{\mathbf{e}}_{2},\
2h_{3}{\mathbf{e}}+2h_{1}{\mathbf{e}}_{2},\\\ 2h_{1}{\mathbf{e}}_{2},\
2(h_{2}+h_{3}){\mathbf{e}},\ 2h_{2}{\mathbf{e}},\
2h_{3}{\mathbf{e}}\end{subarray}}$

for all 1-bounded functions
$f_{0},f_{1},f_{2}:{\mathbb{F}}_{p}^{2}\to{\mathbb{C}}$, where
${\mathbf{e}}=(-1,1)$, ${\mathbf{e}}_{2}=(0,1)$, and the norm above is a
certain box norm whose direction vectors have coordinates polynomial in
$h_{1},h_{2},h_{3}$. In order to control $\Lambda(f_{0},\ldots,f_{\ell})$ by a
genuine Gowers norm, we therefore require two more ingredients compared to the
arguments in [19, 24].

First, we establish a quantitative concatenation result for averages of box
norms, which can be seen as a quantitative analogue of some of the results of
Tao and Ziegler from [31]. Its proof uses rather elementary facts about box
norms such as the Gowers-Cauchy-Schwarz inequality and inductive formula for
box norms. Inspired by quantitative concatenation arguments of Peluse and
Prendiville from [25, 26], the proof is somewhat simpler than the arguments in
these papers as it does not involve any inverse theorem for box norms. Despite
the simplicity of the tools involved, the result is rather general, and it
will likely find applications beyond the arguments in this paper. In our
setting, it allows us to pass from a control of
$\Lambda(f_{0},\ldots,f_{\ell})$ by an average of complicated box norms to a
control by a single box norm, and one of a relatively simple form. This has
previously been done in a fully qualitative way in the ergodic setting by
Donoso, Koutsogiannis, Ferré-Moragues and Sun [5]; by contrast, our finite-
field argument is quantitative. In the example above, it allows us to replace
the complicated average on the right hand side of (6) by a single box norm
$\left\|f_{2}\right\|_{{\mathbf{e}}^{\times s},{\mathbf{e}}_{2}^{\times s}}$
of degree $2s$ for some $s\in{\mathbb{N}}$, yielding a bound

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{x_{1},x_{2},n\in{\mathbb{F}}_{p}}f_{0}(x_{1},x_{2})f_{1}(x_{1}+n^{2},x_{2})f_{2}(x_{1},x_{2}+n^{2}+n)\mathclose{}\right|^{O(1)}\ll\left\|f_{2}\right\|_{{\mathbf{e}}^{\times
s},{\mathbf{e}}_{2}^{\times s}}+p^{-1}.$

Second, we adapt to the combinatorial setting a box norm smoothing technique
recently developed by Frantzikinakis and the author in ergodic theory [10,
11]. With its help, we replace the control of $\Lambda(f_{0},\ldots,f_{\ell})$
by a box norm with control by a Gowers norm. For instance, we show that

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{x_{1},x_{2},n\in{\mathbb{F}}_{p}}f_{0}(x_{1},x_{2})f_{1}(x_{1}+n^{2},x_{2})f_{2}(x_{1},x_{2}+n^{2}+n)\mathclose{}\right|^{O(1)}\ll\left\|f_{2}\right\|_{U^{s^{\prime}}({\mathbf{e}}_{2})}+p^{-1},$

where $\left\|f_{2}\right\|_{U^{s^{\prime}}({\mathbf{e}}_{2})}$ is the Gowers
norm of $f_{2}$ in the direction ${\mathbf{e}}_{2}=(0,1)$ of some degree
$s^{\prime}$.

The arguments above, combining the PET inductive scheme, quantitative
concatenation and box norm smoothing, yield Gowers norm control of
progressions involving not just linearly independent polynomials, but all
pairwise independent polynomials, giving the following result.

######  Theorem 1.3.

Let $d,D,\ell\in{\mathbb{N}}$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ be nonzero
vectors, and $p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ be pairwise independent
polynomials of degrees at most $d$ with zero constant terms. There exist
$c=c(d,\ell)$, $C=C(d,\ell)>0$ and $s=s(d,\ell)\in{\mathbb{N}}$ such that for
all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\mathclose{}\right|\leq
C\left\|f_{\ell}\right\|_{U^{s}({\textbf{v}}_{\ell})}^{c}+Cp^{-c}.$

We remark that in Theorems 1.1-1.3, it is not necessary for the polynomials to
have zero constant terms; if they do not, then the same results hold whenever
the polynomials $p_{j}-p_{j}(0)$ are linearly independent (resp. pairwise
independent). However, the assumption of zero constant terms makes the proofs
more convenient to write down, which is why we impose it.

Theorem 1.2 can be derived from Theorem 1.3 for linearly independent
polynomials by the same degree lowering argument that was used in [19] to
derive the special case of Theorem 1.2 for distinct degree polynomials; the
assumption of distinct degrees in [19] was only needed to obtain Gowers norm
control over the relevant counting operators while the degree lowering part
worked for any progression with linearly independent polynomials whose
counting operator was controlled by Gowers norms. We will however give an
alternative (and, arguably, cleaner) version of this degree lowering argument
which gives a better idea of what is going on.

###  1.1. Outline

We start the paper by giving an overview of the definitions and properties of
box and Gowers norms over general finite abelian groups in Section 2.1. We
then prove concatenation results for box norms (Section 2.2), followed by a
short discussion of weak inverse theorems for Gowers norms (Section 2.3). In
Section 3, we set up the PET induction scheme to control counting operators
for essentially distinct polynomials by an average of box norms. We upgrade
this in Section 4 to a control by a single box norm by combining the PET
results from Section 3 with the concatenation results from Section 2.2.
Section 5 is then fully dedicated to the box norm smoothing argument that
completes the proof of Theorem 1.3, giving control over the counts of
progressions with linearly independent polynomials by Gowers norms. The last
of the main sections of the paper, Section 6, contains the proof of Theorem
1.2, from which Theorem 1.1 follows easily. In Appendix A, we put together
various standard technical lemmas.

###  1.2. Notation

Throughout the paper, the letter $p$ always denotes a prime number. The labels
${\mathbb{N}},{\mathbb{N}}_{0},{\mathbb{Z}},{\mathbb{R}},{\mathbb{C}}$ denote
the sets of positive integers, nonnegative integers, integers, reals and
complex numbers. For integers $a<b$, we set $[a,b]=\\{a,a+1,\ldots,b\\}$,
abbreviating $[1,N]$ as $[N]$. We also let ${\mathbb{Z}}[n]$ denote the set of
single variable polynomials with coefficients in ${\mathbb{Z}}$.

We write elements of ${\mathbb{F}}_{p}^{D}$ (for some fixed
$D\in{\mathbb{N}}$) as ${\textbf{x}}=(x_{1},...,x_{D})$ and elements of
${\mathbb{F}}_{p}$ as $x$. We usually denote tuples of length $s$ as
${\underline{h}}=(h_{1},\ldots,h_{s})$. Given
${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s}$ and
${\underline{\epsilon}}\in\\{0,1\\}^{s}$, we also set
${\underline{h}}^{\underline{\epsilon}}=(h_{1}^{\varepsilon_{1}},\ldots,h_{s}^{\varepsilon_{s}})$,
where $h_{i}^{\varepsilon_{i}}=h_{i}$ if $\varepsilon_{i}=0$ and
$h_{i}^{\varepsilon_{i}}=h_{i}^{\prime}$ otherwise. Also, for
${\underline{u}}\in{\mathbb{N}}_{0}^{s}$, we define
$\textrm{supp}({\underline{u}})=\\{j\in[s]:\ u_{j}>0\\}$.

For a finite set $X$, we let $\operatorname*{\mathbb{E}}_{x\in
X}=\frac{1}{|X|}\sum_{x\in X}$ denote the average over $X$. If
$X={\mathbb{F}}_{p}^{D}$ or ${\mathbb{F}}_{p}$, then we suppress the set from
the notation and simply let
$\operatorname*{\mathbb{E}}_{\textbf{x}}=\operatorname*{\mathbb{E}}_{{\textbf{x}}\in{\mathbb{F}}_{p}^{D}}$
and
$\operatorname*{\mathbb{E}}_{n}=\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}$.

For a finite abelian group $G$ with additive notation and
$v_{1},\ldots,v_{s}\in G$, we let

$\langle v_{1},\ldots,v_{s}\rangle=\\{n_{1}v_{1}+\cdots+n_{s}v_{s}:\
n_{1},\ldots,n_{s}\in{\mathbb{Z}}\\}$

be the subgroup generated by $v_{1},\ldots,v_{s}$. Given a function
$f:G\to{\mathbb{C}}$ and a subgroup $H\subseteq G$, we denote
$\operatorname*{\mathbb{E}}(f|H)(x)=\operatorname*{\mathbb{E}}_{h\in H}f(x+h)$
to be any of the three equivalent things: the orthogonal projection on the
quotient group $G/H$, the conditional expectation with respect to the factor
$\\{x+H:\ x\in G\\}$, or the average of $f$ along the coset $x+H$. If
$H=\langle v\rangle$, then we also set
$\operatorname*{\mathbb{E}}(f|H)=\operatorname*{\mathbb{E}}(f|v)$.

We call a function $f:G\to{\mathbb{C}}$ 1-bounded if
$\left\|f\right\|_{\infty}:=\max_{x\in G}|f(x)|\leq 1$. We similarly define
$\|f\|_{s}=\left(\operatorname*{\mathbb{E}}_{x\in
G}|f({\textbf{x}})|^{s}\right)^{\frac{1}{s}}$ for $1\leq s<\infty$.

For $z\in{\mathbb{C}}$, we let ${\mathcal{C}}z=\overline{z}$ be the
conjugation operator.

We use the asymptotic notation in the standard way. If
$f,g:{\mathbb{N}}\to{\mathbb{C}}$, with $g$ taking positive real values, we
denote $f=O(g)$, $f\ll g$, $g\gg f$ or $g=\Omega(f)$ if there exists $C>0$
such that $|f(n)|\leq Cg(n)$ for sufficiently large $n$. If the constant $C$
depends on a parameter, we record this dependence with a subscript.

We let ${\textbf{v}}_{0}={\mathbf{e}}_{0}=\mathbf{0}$ be the zero vector in
${\mathbb{F}}_{p}^{D}$ and $p_{0}(n)=0$ be the zero polynomial.

##  2\. Gowers norms for finite abelian groups

###  2.1. Basic definitions and properties

Let $G$ be a finite abelian group. For a function $f:G\to{\mathbb{C}}$ and
$h\in G$, we define $\Delta_{h}f(x):=f(x)\overline{f(x+h)}$, and for
$h_{1},\ldots,h_{s}\in G$, we set

$\displaystyle\Delta_{h_{1},\ldots,h_{s}}f=\Delta_{h_{1}}\cdots\Delta_{h_{s}}f(x)=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f(x+{\underline{\epsilon}}\cdot{\underline{h}}).$

Given subgroups $H_{1},\ldots,H_{s}$, we define the _(Gowers) box norm of $f$
along $H_{1},\ldots,H_{s}$_ to be

$\displaystyle\left\|f\right\|_{H_{1},\ldots,H_{s}}$
$\displaystyle=\mathopen{}\left(\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{h_{1}\in
H_{1}}\cdots\operatorname*{\mathbb{E}}_{h_{s}\in
H_{s}}\Delta_{h_{1},\ldots,h_{s}}f(x)\mathclose{}\right)^{1/2^{s}}$
$\displaystyle=\mathopen{}\left(\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{h_{1}\in
H_{1}}\cdots\operatorname*{\mathbb{E}}_{h_{s}\in
H_{s}}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f(x+{\underline{\epsilon}}\cdot{\underline{h}})\mathclose{}\right)^{1/2^{s}},$

letting $s$ be its _degree_. For instance, if $G={\mathbb{F}}_{p}^{2}$,
$H_{1}=\langle(1,0)\rangle$ and $H_{2}=\langle(0,1)\rangle$, then

$\displaystyle\left\|f\right\|=\mathopen{}\left(\operatorname*{\mathbb{E}}_{\begin{subarray}{c}x_{1},x_{2},h_{1},h_{2}\in{\mathbb{F}}_{p}\end{subarray}}f(x_{1},x_{2})\overline{f(x_{1}+h_{1},x_{2})f(x_{1},x_{2}+h_{2})}f(x_{1}+h_{1},x_{2}+h_{2})\mathclose{}\right)^{1/4}.$

If some groups repeat, we also denote $H^{\times s}$ to indicate that $H$
appears $s$ times, e.g.

$\displaystyle\left\|f\right\|_{H_{1}^{\times 2},H_{2}^{\times
3},H_{3}}=\left\|f\right\|_{H_{1},H_{1},H_{2},H_{2},H_{2},H_{3}}.$

Whenever $H_{1}=\langle v_{1}\rangle,\ldots,H_{s}=\langle v_{s}\rangle$, we
also set

$\displaystyle\left\|f\right\|_{H_{1},\ldots,H_{s}}=\left\|f\right\|_{v_{1},\ldots,v_{s}}$

Lastly, we denote

(7) $\displaystyle\left\|f\right\|_{H^{\times
s}}=\left\|f\right\|_{U^{s}(H)},$

or $\left\|f\right\|_{U^{s}(v)}$ if $H=\langle v\rangle$, calling (7) the
_Gowers norm of $f$ along $H$ of degree $s$_.

Box norms are seminorms, and norms for $s\geq 2$. They satisfy a number of
other well-known properties, some of whose proofs can be found e.g. in [12,
30]. These include the following:

  1. (i)

(Monotonicity)

$\left\|f\right\|_{H_{1}}\leq\left\|f\right\|_{H_{1},H_{2}}\leq\left\|f\right\|_{H_{1},H_{2},H_{3}}\leq\cdots;$

  2. (ii)

(Box norms along subgroups) for subgroups $H_{1}^{\prime}\subseteq
H_{1},\ldots,H_{s}^{\prime}\subseteq H_{s}$, we have

(8)
$\displaystyle\left\|f\right\|_{H_{1},\ldots,H_{s}}\leq\left\|f\right\|_{H^{\prime}_{1},\ldots,H^{\prime}_{s}};$

  3. (iii)

(Inductive formula)

$\left\|f\right\|_{H_{1},\ldots,H_{s}}^{s}=\operatorname*{\mathbb{E}}_{h_{1}\in
H_{1}}\cdots\operatorname*{\mathbb{E}}_{h_{s^{\prime}}\in
H_{s^{\prime}}}\left\|f\right\|_{H_{s^{\prime}+1},\ldots,H_{s}}^{2^{s-s^{\prime}}}$

for $1\leq s^{\prime}\leq s$;

  4. (iv)

(Gowers-Cauchy-Schwarz inequality)

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{h_{1}\in
H_{1}}\cdots\operatorname*{\mathbb{E}}_{h_{s}\in
H_{s}}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f_{\underline{\epsilon}}(x+{\underline{\epsilon}}\cdot{\underline{h}})\mathclose{}\right|\leq\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}\left\|f_{\underline{\epsilon}}\right\|_{H_{1},\ldots,H_{s}}.$

We will frequently cite the aforementioned properties throughout the paper.

###  2.2. Concatenation of box norms for general groups

One of the key components of our argument is a quantitative concatenation
result for box norms. On various occasions in additive combinatorics, we have
to deal with expressions of the form

(9) $\displaystyle\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i1},\ldots,H_{is}},$

i.e. averages of box norms of $f$ along some subgroups $H_{i1},\ldots,H_{is}$
indexed by a finite set $I$. The main idea behind concatenation results is to
bound (9) from above by an average of box norms along larger subgroups
$H_{ij}+H_{i^{\prime}j}$, or even $H_{i_{1}j}+\cdots+H_{i_{k}j}$. The
rationale behind this move is that the larger subgroups may admit a more
explicit form that makes them easier to work with. The first concatenation
results have been developed by Tao and Ziegler [31] with the aim of proving
the existence of polynomial progressions in primes [32]. These results are
purely qualitative, though, hence not applicable to our context. In [5],
Donoso, Ferré-Moragues, Koutsogiannis and Sun used them to obtain a
(qualitative) box seminorm control on certain averages along polynomials; the
arguments in Section 4, in which we obtain quantitative box norm control over
our counting operators, can be perceived as a quantitative, finite field
version of the arguments from [5].

Certain quantitative concatenation results have been developed by Peluse and
Prendiville [25, 26, 28] in their works on polynomial progressions in natural
numbers. Our proofs are partly inspired by their techniques; specifically,
while proving Lemma 2.2, we use a trick observed in [25, Lemma 5.1]. Our
arguments are however somewhat more straightforward than those of Peluse and
Prendiville in that we do not use at any point an inverse theorem for degree 2
box norms, which is a starting point in their arguments. Instead, we rely
entirely on Lemma 2.2 and simple manipulations of the formulas for box norms
based on the inductive formula for box norms.

In order to illustrate better how concatenation works, we start with a simple
concatenation result for degree 1 norms.

######  Lemma 2.1 (Concatenation of degree 1 norms).

Let $G$ be a finite abelian group, $I$ be a finite indexing set and
$(H_{i})_{i\in I}$ be subgroups of $G$. Then for every 1-bounded function
$f:G\to{\mathbb{C}}$, we have

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i}}^{2}\mathclose{}\right)^{2}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\left\|f\right\|_{H_{i}+H_{i^{\prime}}}^{2}.$

###### Proof.

Expanding the definition of Gowers norms, we get

$\displaystyle\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i}}^{2}=\operatorname*{\mathbb{E}}_{i\in
I}\operatorname*{\mathbb{E}}_{x\in G}\operatorname*{\mathbb{E}}_{h_{i}\in
H_{i}}f(x)\overline{f(x+h_{i})}=\operatorname*{\mathbb{E}}_{x\in
G}f(x)\operatorname*{\mathbb{E}}_{i\in I}\operatorname*{\mathbb{E}}_{h_{i}\in
H_{i}}\overline{f(x+h_{i})}.$

Applying the Cauchy-Schwarz inequality in $x$ and using the 1-boundedness of
$f$, we infer that

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i}}^{2}\mathclose{}\right)^{2}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h_{i}\in H_{i},\\\
h^{\prime}_{i^{\prime}}\in
H_{i^{\prime}}\end{subarray}}f(x+h_{i})\overline{f(x+h^{\prime}_{i^{\prime}})}.$

A change of variables gives

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i}}^{2}\mathclose{}\right)^{2}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h\in
H_{i}+H_{i^{\prime}}\end{subarray}}f(x)\overline{f(x+h)},$

and the result follows from the definition of Gowers norms. ∎

The example below constitutes a very simple model case showing how
concatenation results will be applied in the later parts of the paper.

######  Example 1.

Let $G={\mathbb{F}}_{p}^{D}$ for some $D\in{\mathbb{N}}$ and
$I={\mathbb{F}}_{p}^{2}$. For linearly independent vectors
${\textbf{v}}_{1},{\textbf{v}}_{2}\in{\mathbb{F}}_{p}^{D}$, Lemma 2.1 gives

(10)
$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{h_{1},h_{2}\in{\mathbb{F}}_{p}}\left\|f\right\|_{h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2}}^{2}\mathclose{}\right)^{2}\leq\operatorname*{\mathbb{E}}_{h_{1},h_{1}^{\prime},h_{2},h_{2}^{\prime}\in{\mathbb{F}}_{p}}\left\|f\right\|_{\langle
h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2},h_{1}^{\prime}{\textbf{v}}_{1}+h_{2}^{\prime}{\textbf{v}}_{2}\rangle}^{2}$

when applied to the groups $H_{{\underline{h}}}=\langle
h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2}\rangle$ indexed by
${\underline{h}}\in{\mathbb{F}}_{p}^{2}$. Unless
$h_{1}h_{2}^{\prime}=h_{1}^{\prime}h_{2}$ (which happens for at most $p^{3}$
“bad” tuples $(h_{1},h_{1}^{\prime},h_{2},h_{2}^{\prime})$), we have

$\displaystyle\langle
h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2},h_{1}^{\prime}{\textbf{v}}_{1}+h_{2}^{\prime}{\textbf{v}}_{2}\rangle=\langle{\textbf{v}}_{1},{\textbf{v}}_{2}\rangle.$

Excluding the “bad” values $(h_{1},h_{1}^{\prime},h_{2},h_{2}^{\prime})$, we
have

(11)
$\displaystyle\operatorname*{\mathbb{E}}_{h_{1},h_{1}^{\prime},h_{2},h_{2}^{\prime}\in{\mathbb{F}}_{p}}\left\|f\right\|_{\langle
h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2},h_{1}^{\prime}{\textbf{v}}_{1}+h_{2}^{\prime}{\textbf{v}}_{2}\rangle}^{2}\leq\left\|f\right\|_{\langle{\textbf{v}}_{1},{\textbf{v}}_{2}\rangle}^{2}+p^{-1}.$

Combining (10) and (11), we obtain the inequality

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{h_{1},h_{2}\in{\mathbb{F}}_{p}}\left\|f\right\|_{h_{1}{\textbf{v}}_{1}+h_{2}{\textbf{v}}_{2}}^{2}\mathclose{}\right)^{2}\leq\left\|f\right\|_{\langle{\textbf{v}}_{1},{\textbf{v}}_{2}\rangle}^{2}+p^{-1}.$

Thus, we have successfully bounded an average of complicated box norms by a
single norm that involves only the “principal” directions
${\textbf{v}}_{1},{\textbf{v}}_{2}$ rather than linear combinations of them.

The argument becomes more complicated when we deal with averages of box norms
of degree greater than 1, i.e. when $s>1$ in (9). In handling this general
case, we will iteratively use the lemma below. Its proof is based on a trick
from the proof of [25, Lemma 5.1] and relies on rather elementary maneuvers
that involve the Gowers-Cauchy-Schwarz inequality, multiple applications of
the inductive formula for box norms and a simple change of variables.

######  Lemma 2.2.

Let $s\in{\mathbb{N}}$, $G$ be a finite abelian group, $I$ be a finite
indexing set and $H_{i},K_{i1},\ldots,K_{is}$ be subgroups of $G$ for each
$i\in I$. For each 1-bounded function $f:G\to{\mathbb{C}}$, we have

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i},K_{i1},\ldots,K_{is}}^{2^{s+1}}\mathclose{}\right)^{2^{2s+1}}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\left\|f\right\|_{K_{i1},\ldots,K_{is},K_{i^{\prime}1},\ldots,K_{i^{\prime}s},H_{i}+H_{i^{\prime}}}^{2^{2s+1}}$

Lemma 2.1 is thus a special case of Lemma 2.2 for $s=0$.

###### Proof.

Let $\delta=\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i},K_{i1},\ldots,K_{is}}^{2^{s+1}}$ and
$K_{i}=K_{i1}\times\cdots\times K_{is}$. The inductive formula for box norms
yields

$\displaystyle\delta=\operatorname*{\mathbb{E}}_{i\in
I}\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{k}}\in
K_{i}\end{subarray}}\operatorname*{\mathbb{E}}_{h\in
H_{i}}\Delta_{{\underline{k}},h}f(x).$

We expand

$\displaystyle\Delta_{{\underline{h}},k}f(x)=f(x)\overline{f(x+h)}\Delta_{{\underline{k}}}^{*}\Delta_{h}f(x),$

where
$\Delta_{{\underline{k}}}^{*}f(x)=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{0\\}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f(x+{\underline{\epsilon}}\cdot{\underline{k}})$,
and change the order of summation, so that

$\displaystyle\delta=\operatorname*{\mathbb{E}}_{x\in
G}f(x)\operatorname*{\mathbb{E}}_{i\in
I}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{k}}\in
K_{i}\end{subarray}}\operatorname*{\mathbb{E}}_{h\in
H_{i}}\overline{f(x+h)}\Delta^{*}_{{\underline{k}}}\Delta_{h}f(x).$

An application of the Cauchy-Schwarz inequality in $x$ gives

$\displaystyle\delta^{2}\leq\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{k}}\in K_{i},\\\
{\underline{k}}^{\prime}\in
K_{i^{\prime}}\end{subarray}}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h\in
H_{i},\\\ h^{\prime}\in
H_{i^{\prime}}\end{subarray}}\overline{f(x+h)}f(x+h^{\prime})\Delta^{*}_{{\underline{k}}}\Delta_{h}f(x)\overline{\Delta^{*}_{{\underline{k}}^{\prime}}\Delta_{h^{\prime}}f(x)}.$

We crucially observe that for each fixed $i,i^{\prime}\in I$ and $h\in
H_{i},h^{\prime}\in H_{i^{\prime}}$, the average

$\displaystyle\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{k}}\in K_{i},\\\
{\underline{k}}^{\prime}\in
K_{i^{\prime}}\end{subarray}}\overline{f(x+h)}f(x+h^{\prime})\Delta^{*}_{{\underline{k}}}\Delta_{h}f(x)\overline{\Delta^{*}_{{\underline{k}}^{\prime}}\Delta_{h^{\prime}}f(x)}$

is a box inner product along
$K_{i1},\ldots,K_{is},K_{i^{\prime}1},\ldots,K_{i^{\prime}s}$, and an
application of the Gowers-Cauchy-Schwarz inequality gives

$\displaystyle\delta^{2^{2s+1}}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\operatorname*{\mathbb{E}}_{x\in
G}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{k}}\in K_{i},\\\
{\underline{k}}^{\prime}\in
K_{i^{\prime}}\end{subarray}}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h\in
H_{i},\\\ h^{\prime}\in
H_{i^{\prime}}\end{subarray}}\Delta_{{\underline{k}},{\underline{k}}^{\prime}}\mathopen{}\left(f(x+h)\overline{f(x+h^{\prime})}\mathclose{}\right),$

Lastly, the inductive formula for box norms combined with a simple change of
variables implies that

$\displaystyle\delta^{2^{2s+1}}\leq\operatorname*{\mathbb{E}}_{i,i^{\prime}\in
I}\left\|f\right\|_{K_{i1},\ldots,K_{is},K_{i^{\prime}1},\ldots,K_{i^{\prime}s},H_{i}+H_{i^{\prime}}}^{2^{2s+1}},$

as claimed. ∎

By repeatedly applying Lemma 2.2 to (9) for $s>1$, we derive the following
concatenation result.

######  Proposition 2.3 (Concatenation of box norms, version I).

Let $s\in{\mathbb{N}}$, $G$ be a finite abelian group, $I$ be a finite
indexing set and $H_{i1},\ldots,H_{is}$ be subgroups of $G$ for each $i\in I$.
For all 1-bounded functions $f:G\to{\mathbb{C}}$, we have

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i1},\ldots,H_{is}}^{2^{s}}\mathclose{}\right)^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{s}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{H_{i_{\underline{\epsilon}}j}+H_{i_{{\underline{\epsilon}}^{\prime}}j}:\
j=1,\ldots,s,\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{s}\\\
\mathrm{with}\
(\epsilon_{1},\ldots,\epsilon_{s-j})=(\epsilon^{\prime}_{1},\ldots,\epsilon^{\prime}_{s-j}),\
\epsilon_{s+1-j}\neq\epsilon^{\prime}_{s+1-j}\\}\end{subarray}}.$

Importantly, the indices
${\underline{\epsilon}},{\underline{\epsilon}}^{\prime}$ are always distinct,
so that
$H_{i_{\underline{\epsilon}}j}+H_{i_{{\underline{\epsilon}}^{\prime}}j}$ is
strictly larger than
$H_{i_{\underline{\epsilon}}j},H_{i_{{\underline{\epsilon}}^{\prime}}j}$
unless one of the two subgroups is contained in the other.

###### Proof.

The proof of Proposition 2.3 relies on a gradual concatenation of the
“unconcatenated” subgroups $H_{i1},\ldots,H_{is}$ in the average using Lemma
2.2. We repeatedly use the inductive formula for box norms in order to
reinterpret the average in such a way that successive applications of Lemma
2.2 concatenate the subgroups $H_{i1},\ldots,H_{is}$ one by one.

Throughout, we shall assume that $h_{j}^{{\underline{\epsilon}}}$ is always an
element of $H_{i_{{\underline{\epsilon}}}j}$, and an average
$\operatorname*{\mathbb{E}}_{h_{j}^{{\underline{\epsilon}}}}$ always runs over
the subgroup $H_{i_{{\underline{\epsilon}}}j}$.

The case $s=2$:

For illustrative purposes, we first prove Proposition 2.3 for degree 2 box
norms. Let $\delta=\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|^{4}_{H_{i1},H_{i2}}$. We first want to concatenate the
group $H_{i2}$. By Lemma 2.2, we have

(12) $\displaystyle\delta^{8}\leq\operatorname*{\mathbb{E}}_{i_{0},i_{1}\in
I}\left\|f\right\|_{H_{i_{0}1},H_{i_{1}1},H_{i_{0}2}+H_{i_{1}2}}^{8},$

and so the group $H_{i2}$ indeed got concatenated. The price we paid for this
is that while applying Lemma 2.2, the unconcatenated group $H_{i1}$ doubled
into $H_{i_{0}1}$ and $H_{i_{1}1}$. The next goal is therefore to concatenate
these groups one by one. Using the inductive formula for box norms, we can
rephrase (12) as

$\displaystyle\delta^{8}\leq\operatorname*{\mathbb{E}}_{i_{1}\in
I}\operatorname*{\mathbb{E}}_{h^{1}_{1}}\operatorname*{\mathbb{E}}_{i_{0}\in
I}\left\|\Delta_{h^{1}_{1}}f\right\|_{H_{i_{0}1},H_{i_{0}2}+H_{i_{1}2}}^{4}.$

Applying Lemma 2.2 to each $\operatorname*{\mathbb{E}}_{i_{0}\in
I}\left\|\Delta_{h^{1}_{1}}f\right\|_{H_{i_{0}1},H_{i_{0}2}+H_{i_{1}2}}^{4}$,
we deduce that

$\displaystyle\delta^{64}\leq\operatorname*{\mathbb{E}}_{i_{00},i_{01},i_{1}\in
I}\operatorname*{\mathbb{E}}_{h^{1}_{1}}\left\|\Delta_{h^{1}_{1}}f\right\|_{H_{i_{00}1}+H_{i_{01}1},H_{i_{00}2}+H_{i_{1}2},H_{i_{01}2}+H_{i_{1}2}}^{8}.$

Rephrasing the inequality above once more using the inductive formula for box
norms, we get

$\displaystyle\delta^{64}\leq\operatorname*{\mathbb{E}}_{i_{00},i_{01}\in
I}\operatorname*{\mathbb{E}}_{h^{00}_{1},h^{01}_{1}}\operatorname*{\mathbb{E}}_{i_{1}\in
I}\left\|\Delta_{h^{00}_{1}+h^{01}_{1}}f\right\|_{H_{i_{1}1},H_{i_{00}2}+H_{i_{1}2},H_{i_{01}2}+H_{i_{1}2}}^{8}.$

We then apply Lemma 2.2 for the last time, this time to each

$\operatorname*{\mathbb{E}}_{i_{1}\in
I}\left\|\Delta_{h^{00}_{1}+h^{01}_{1}}f\right\|_{H_{i_{1}1},H_{i_{00}2}+H_{i_{1}2},H_{i_{01}2}+H_{i_{1}2}}^{8},$

obtaining

$\displaystyle\delta^{2048}\leq\operatorname*{\mathbb{E}}_{i_{00},i_{01},i_{10},i_{11}\in
I}\operatorname*{\mathbb{E}}_{h^{00}_{1},h^{01}_{1}}\left\|\Delta_{h^{00}_{1}+h^{01}_{1}}f\right\|_{\begin{subarray}{c}H_{i_{10}1}+H_{i_{11}1},H_{i_{00}2}+H_{i_{10}2},H_{i_{00}2}+H_{i_{11}2},\\\
H_{i_{01}2}+H_{i_{10}2},H_{i_{01}2}+H_{i_{11}2}\end{subarray}}^{32}.$

The inductive formula for box norms then implies that

$\displaystyle\delta^{2048}\leq\operatorname*{\mathbb{E}}_{i_{00},i_{01},i_{10},i_{11}\in
I}\left\|f\right\|_{\begin{subarray}{c}H_{i_{00},1}+H_{i_{01},1},H_{i_{10}1}+H_{i_{11}1},H_{i_{00}2}+H_{i_{10}2},\\\
H_{i_{00}2}+H_{i_{11}2},H_{i_{01}2}+H_{i_{10}2},H_{i_{01}2}+H_{i_{11}2}\end{subarray}}^{64},$

and the exponent 64 can be dropped since $f$ is 1-bounded.

We note that the proof of Proposition 2.3 for $s=2$ relies on 3 applications
of Lemma 2.2. More generally, the proof for an arbitrary $s\geq 2$ will
require $1+2+\cdots+2^{s-1}=2^{s}-1$ applications of Lemma 2.2.

The general case:

We move on to prove the general case. Starting with
$\delta=\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i1},\ldots,H_{is}}^{2^{s}}$, we apply Lemma 2.2 to
bound

(13)
$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{i_{0},i_{1}\in
I}\left\|f\right\|_{H_{i_{0}1},\ldots,H_{i_{0}(s-1)},H_{i_{1}1},\ldots,H_{i_{1}(s-1)},H_{i_{0}s}+H_{i_{1}s}}^{2^{2s-1}}.$

We note that we passed from having $s$ unconcatenated groups indexed by $i$ to
$s-1$ unconcatenated groups indexed by $i_{0}$ and another $s-1$
unconcatenated groups indexed by $i_{1}$ (in addition to the concatenated
group $H_{i_{0}s}+H_{i_{1}s}$). Thus, the total number of groups almost
doubled, but what matters is that for each index $i_{0},i_{1}$, the number of
unconcatenated groups with this index went down by 1. At the next stage of the
argument, we will apply Lemma 2.2 twice to concatenate $H_{i_{0}(s-1)}$ first
and then $H_{i_{1}(s-1)}$. As a consequence, the two indices $i_{0},i_{1}$
will be replaced by four indices $i_{00},i_{01},i_{10},i_{11}$, and for each
of them we will have exactly $s-2$ unconcatenated groups. We will continue in
this manner: at each stage, the number of indices $i_{\underline{\epsilon}}$
will double, but the number of unconcatenated groups with each index
$i_{\underline{\epsilon}}$ will decrease by 1. Eventually, on the $s$-th step,
we will be left with $2^{s-1}$ unconcatenated groups
$H_{i_{\underline{\epsilon}}1}$, and $2^{s-1}$ applications of Lemma 2.2 will
allow us to concatenate them all without producing any new unconcatenated
groups. This will finish the argument.

This is the general strategy; let us see in detail what happens at the second
stage, i.e. after obtaining the bound (13). Using the induction formula for
box norms, we can rephrase (13) as

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{i_{1}\in
I}\operatorname*{\mathbb{E}}_{h^{1}_{1},\ldots,h^{1}_{s-1}}\operatorname*{\mathbb{E}}_{i_{0}\in
I}\left\|\Delta_{h^{1}_{1},\ldots,h^{1}_{s-1}}f\right\|_{H_{i_{0}1},\ldots,H_{i_{0}(s-1)},H_{i_{0}s}+H_{i_{1}s}}^{2^{s}}.$

For each fixed $i_{1},h^{1}_{1},\ldots,h^{1}_{s-1}$, we apply Lemma 2.2
separately to each average over $i_{0}$, obtaining

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{i_{1}\in
I}\operatorname*{\mathbb{E}}_{h^{1}_{1},\ldots,h^{1}_{s-1}}\operatorname*{\mathbb{E}}_{i_{00},i_{01}\in
I}\left\|\Delta_{h^{1}_{1},\ldots,h^{1}_{s-1}}f\right\|_{\begin{subarray}{c}H_{i_{00}1},\ldots,H_{i_{00}(s-2)},H_{i_{01}1},\ldots,H_{i_{01}(s-2)},\\\
H_{i_{00}(s-1)}+H_{i_{01}(s-1)},H_{i_{00}s}+H_{i_{1}s},H_{i_{01}s}+H_{i_{1}s}\end{subarray}}^{2^{2s-1}}.$

We rearrange the inequality above using the inductive formula for box norms as

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{i_{00},i_{01}\in
I}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h^{00}_{1},\ldots,h^{00}_{s-2},\\\
h^{01}_{1},\ldots,h^{01}_{s-2},\\\
h^{00}_{s-1},h^{01}_{s-1}\end{subarray}}\operatorname*{\mathbb{E}}_{i_{1}\in
I}\left\|\Delta_{\begin{subarray}{c}h^{00}_{1},\ldots,h^{00}_{s-2},\\\
h^{01}_{1},\ldots,h^{01}_{s-2},\\\
h^{00}_{s-1}+h^{01}_{s-1}\end{subarray}}f\right\|_{\begin{subarray}{c}H_{i_{1}1},\ldots,H_{i_{1}(s-1)},\\\
H_{i_{00}s}+H_{i_{1}s},H_{i_{01}s}+H_{i_{1}s}\end{subarray}}^{2^{s+1}}.$

in order to concatenate $H_{i_{1}(s-1)}$. By Lemma 2.2 applied separately to
each average over $i_{1}\in I$, we have

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{00},i_{01},\\\
i_{10},i_{11}\in
I\end{subarray}}\operatorname*{\mathbb{E}}_{\begin{subarray}{c}h^{00}_{1},\ldots,h^{00}_{s-2},\\\
h^{01}_{1},\ldots,h^{01}_{s-2},\\\
h^{00}_{s-1},h^{01}_{s-1}\end{subarray}}\left\|\Delta_{\begin{subarray}{c}h^{00}_{1},\ldots,h^{00}_{s-2},\\\
h^{01}_{1},\ldots,h^{01}_{s-2},\\\
h^{00}_{s-1}+h^{01}_{s-1}\end{subarray}}f\right\|_{\begin{subarray}{c}H_{i_{10}1},\ldots,H_{i_{10}(s-2)},H_{i_{11}1},\ldots,H_{i_{11}(s-2)},\\\
H_{i_{10}(s-1)}+H_{i_{11}(s-1)},H_{i_{00}s}+H_{i_{10}s},H_{i_{00}s}+H_{i_{11}s},\\\
H_{i_{01}s}+H_{i_{10}s},H_{i_{01}s}+H_{i_{11}s}\end{subarray}}^{2^{2s+1}}.$

An application of the inductive formula for box norms then gives

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{00},i_{01},\\\
i_{10},i_{11}\in
I\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}H_{i_{00}1},\ldots,H_{i_{00}(s-2)},H_{i_{01}1},\ldots,H_{i_{01}(s-2)},\\\
H_{i_{10}1},\ldots,H_{i_{10}(s-2)},H_{i_{11}1},\ldots,H_{i_{11}(s-2)},\\\
H_{i_{00}(s-1)}+H_{i_{01}(s-1)},H_{i_{10}(s-1)}+H_{i_{11}(s-1)},\\\
H_{i_{00}s}+H_{i_{10}s},H_{i_{00}s}+H_{i_{11}s},\\\
H_{i_{01}s}+H_{i_{10}s},H_{i_{01}s}+H_{i_{11}s}\end{subarray}}^{2^{4s-2}},$

which can be written more compactly as

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{00},i_{01},\\\
i_{10},i_{11}\in
I\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{H_{i_{\underline{\epsilon}}j}:\
{\underline{\epsilon}}\in\\{0,1\\}^{2},\ j=1,\ldots,s-2\\},\\\
\\{H_{i_{\underline{\epsilon}}(s-1)}+H_{i_{{\underline{\epsilon}}^{\prime}}(s-1)}:\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{2}\
\mathrm{with}\ \epsilon_{1}=\epsilon^{\prime}_{1},\
\epsilon_{2}\neq\epsilon^{\prime}_{2}\\},\\\
\\{H_{i_{\underline{\epsilon}}s}+H_{i_{{\underline{\epsilon}}^{\prime}}s}:\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{2}\
\mathrm{with}\
\epsilon_{1}\neq\epsilon^{\prime}_{1}\\}\end{subarray}}^{2^{4s-2}}.$

We have thus successfully concatenated all the groups
$H_{i_{\underline{\epsilon}}(s-1)}$ and $H_{i_{\underline{\epsilon}}s}$.

At this point, we stop keeping track of the ever more complicated powers of
the box norm on the right-hand, and instead use the 1-boundedness of $f$ to
replace the exponent by 1, so that

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{00},i_{01},\\\
i_{10},i_{11}\in
I\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{H_{i_{\underline{\epsilon}}j}:\
{\underline{\epsilon}}\in\\{0,1\\}^{2},\ j=1,\ldots,s-2\\},\\\
\\{H_{i_{\underline{\epsilon}}(s-1)}+H_{i_{{\underline{\epsilon}}^{\prime}}(s-1)}:\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{2}\
\mathrm{with}\ \epsilon_{1}=\epsilon^{\prime}_{1},\
\epsilon_{2}\neq\epsilon^{\prime}_{2}\\},\\\
\\{H_{i_{\underline{\epsilon}}s}+H_{i_{{\underline{\epsilon}}^{\prime}}s}:\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{2}\
\mathrm{with}\ \epsilon_{1}\neq\epsilon^{\prime}_{1}\\}\end{subarray}}.$

At the next stage, we concatenate the groups
$H_{i_{\underline{\epsilon}}(s-2)}$. Applying Lemma 2.2 and the induction
formula for box norms four times like before, each time to an average over
$i_{00},i_{01},i_{10},i_{11}$ respectively, we arrive at the inequality

$\displaystyle\delta^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{{\underline{\epsilon}}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{3}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{H_{i_{\underline{\epsilon}}j}:\
{\underline{\epsilon}}\in\\{0,1\\}^{3},\ j=1,\ldots,s-3\\},\\\
\\{H_{i_{\underline{\epsilon}}j}+H_{i_{{\underline{\epsilon}}^{\prime}}j}:\
j=s-2,s-1,s,\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{3}\\\
\mathrm{with}\
(\epsilon_{1},\ldots,\epsilon_{s-j})=(\epsilon^{\prime}1,\ldots,\epsilon^{\prime}_{s-j}),\
\epsilon_{s+1-j}\neq\epsilon^{\prime}_{s+1-j}\\}\end{subarray}}$

This time, we have successfully concatenated the groups
$H_{i_{\underline{\epsilon}}(s-2)}$. At the next step, 8 applications of Lemma
2.2 and the induction formula for box norms allow us to concatenate groups
$H_{i_{\underline{\epsilon}}(s-3)}$. Continuing the argument like this, we
arrive, after a total of

$\displaystyle 1+2+2^{2}+\cdots+2^{s-1}=2^{s}-1$

applications of Lemma 2.2 and the induction formula for box norms, at the
claimed inequality. ∎

For applications, the following weaker but notationally lighter corollary of
Proposition 2.3 is sufficient. It can be obtained by a straightforward
application of the monotonicity property of box norms to Proposition 2.3.

######  Corollary 2.4 (Concatenation of box norms, version II).

Let $s\in{\mathbb{N}}$, $G$ be a finite abelian group, $I$ be a finite
indexing set and $H_{i1},\ldots,H_{is}$ be subgroups of $G$ for each $i\in I$.
For all 1-bounded functions $f:G\to{\mathbb{C}}$, we have

(14) $\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i1},\ldots,H_{is}}^{2^{s}}\mathclose{}\right)^{O_{s}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{s}\end{subarray}}\left\|f\right\|_{\\{H_{i_{\underline{\epsilon}}j}+H_{i_{{\underline{\epsilon}}^{\prime}}j}:\
j=1,\ldots,s,\
{\underline{\epsilon}},{\underline{\epsilon}}^{\prime}\in\\{0,1\\}^{s},\
{\underline{\epsilon}}\neq{\underline{\epsilon}}^{\prime}\\}}.$

It is instructive to compare Corollary 2.4 with the relevant results from
[31]. One difference is in the setup itself: our argument is presented for
finite groups while the argument from [31] concerns ergodic seminorms for
countable group actions. But with a bit of extra work, our argument could
likely be extended to ergodic seminorms for at least ${\mathbb{Z}}$ actions
since it only relies on elementary maneuvers such as the Gowers-Cauchy-Schwarz
inequality; we however do not need this extension for the purposes of this
article. Apart from this, the most important difference is that we
quantitatively compare two averages of box norms while an analogous comparison
in [31] is fully qualitative. Moreover, in the norms on the right hand side of
(14), we only sum up subgroups with the same index $j$ whereas results from
[31] involve subgroups of the form $H_{ij}+H_{i^{\prime}j^{\prime}}$ for
various indices $j,j^{\prime}$. Finally, our argument necessitates the
introduction of indices $i_{\underline{\epsilon}}$ for $2^{s}$ values
${\underline{\epsilon}}$ on the right hand side of (14) while the arguments in
[31] allow to average on the right hand side over only two indices
$i,i^{\prime}$.

We shall use the following iterative consequence of Corollary 2.4, obtained
from an iterated application of Corollary 2.4. Its advantage is that it allows
us to take the directions in the concatenated box norms to be arbitrarily long
sums of the original directions rather than just double sums, as is the case
in Corollary 2.4.

######  Corollary 2.5 (Iterated concatenation of box norms).

Let $s,k\in{\mathbb{N}}$, $G$ be a finite abelian group, $I$ be a finite
indexing set and $H_{i1},\ldots,H_{is}$ be subgroups of $G$ for each $i\in I$.
There exist natural numbers $w_{1},\ldots,w_{k}=O_{s,k}(1)$ such that for all
1-bounded functions $f:G\to{\mathbb{C}}$, we have

(15) $\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i1},\ldots,H_{is}}^{2^{s}}\mathclose{}\right)^{O_{s,k}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{w_{1}+\cdots+w_{k}}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{\sum_{b_{1},\ldots,b_{k}\in\\{0,1\\}}H_{i_{{\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}}j}:\
j=1,\ldots,s,\\\
{\underline{\epsilon}}_{l0},{\underline{\epsilon}}_{l1}\in\\{0,1\\}^{w_{l}},\
{\underline{\epsilon}}_{l0}\neq{\underline{\epsilon}}_{l1},\
l=1,\ldots,k\\}\end{subarray}}.$

In (15) and other similar expressions, we interpret
${\underline{\epsilon}}:={\underline{\epsilon}}_{1}\cdots{\underline{\epsilon}}_{k}$
as the concatenation of the tuples
${\underline{\epsilon}}_{1},\ldots,{\underline{\epsilon}}_{k}$, so that the
length of the tuple ${\underline{\epsilon}}$ is the sum of lengths of
${\underline{\epsilon}}_{1},\ldots,{\underline{\epsilon}}_{k}$.

It will be important for our applications later on that in the sum

$\displaystyle\sum_{b_{1},\ldots,b_{k}\in\\{0,1\\}}H_{i_{{\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}}j}$

inside (15), the tuples
${\underline{\epsilon}}={\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}$
are all distinct for distinct values $(b_{1},\ldots,b_{k})$ due to the fact
that ${\underline{\epsilon}}_{l0}\neq{\underline{\epsilon}}_{l1}$ for all
$l=1,\ldots,k$.

###### Proof.

We induct on $k$. For $k=1$, we set
$w_{1}=s,{\underline{\epsilon}}_{10}={\underline{\epsilon}},{\underline{\epsilon}}_{11}={\underline{\epsilon}}^{\prime}$,
and the result is just Corollary 2.4. For $k\geq 2$, we let $w_{2}$ be the
degree of the box norms on the right hand side of (14). Rephrasing the right
hand side of (14) as

(16)
$\displaystyle\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{w_{1}}\end{subarray}}\left\|f\right\|_{\\{H_{i_{{\underline{\epsilon}}_{10}}j}+H_{i_{{\underline{\epsilon}}_{11}}j}:\
j=1,\ldots,s,\
{\underline{\epsilon}}_{10},{\underline{\epsilon}}_{11}\in\\{0,1\\}^{w_{1}},\
{\underline{\epsilon}}_{10}\neq{\underline{\epsilon}}_{11}\\}},$

we apply Corollary 2.4 again, this time taking (16) as the left hand side.
Thus, the indexing set $I$ is replaced by $I^{\\{0,1\\}^{w_{1}}}$ and the role
of indices $j=1,\ldots,s$ is played by the tuples
$(j,{\underline{\epsilon}}_{10},{\underline{\epsilon}}_{11})$. By Corollary
2.4, a $O_{s,k}(1)$ power of (16) is bounded from above by

(17)
$\displaystyle\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{w_{1}+w_{2}}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{H_{i_{{\underline{\epsilon}}_{10}{\underline{\epsilon}}_{20}}j}+H_{i_{{\underline{\epsilon}}_{11}{\underline{\epsilon}}_{20}}j}+H_{i_{{\underline{\epsilon}}_{10}{\underline{\epsilon}}_{21}}j}+H_{i_{{\underline{\epsilon}}_{11}{\underline{\epsilon}}_{21}}j}:\
j=1,\ldots,s,\\\
{\underline{\epsilon}}_{10},{\underline{\epsilon}}_{11}\in\\{0,1\\}^{w_{1}},\
{\underline{\epsilon}}_{10}\neq{\underline{\epsilon}}_{11},\
{\underline{\epsilon}}_{20},{\underline{\epsilon}}_{21}\in\\{0,1\\}^{w_{1}},\
{\underline{\epsilon}}_{20}\neq{\underline{\epsilon}}_{21}\\}\end{subarray}}.$

Arguing in the same way for $k\geq 3$ (e.g. applying Corollary 2.4 to (17) for
$k=3$, with $w_{3}$ being the degree of the norm on the right hand side of
(17)), we deduce Corollary 2.5 in full. ∎

It is instructive to see what Corollary 2.5 gives when $s=1$. In this case, an
iterated application of Lemma 2.1 in place of Corollary 2.4 gives simply

(18) $\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{i\in
I}\left\|f\right\|_{H_{i}}^{2}\mathclose{}\right)^{2^{k}}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}i_{\underline{\epsilon}}\in
I,\\\
{\underline{\epsilon}}\in\\{0,1\\}^{k}\end{subarray}}\left\|f\right\|_{\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{k}}H_{i_{\underline{\epsilon}}}}^{2},$

corresponding to taking $w_{1}=\cdots=w_{k}=1$ in Corollary 2.5 and
${\underline{\epsilon}}_{lb_{l}}=b_{l}$ for $1\leq l\leq k$,
$b_{l}\in\\{0,1\\}$.

###  2.3. Weak inverse theorems for Gowers norms

We now specialise to the case $G={\mathbb{F}}_{p}^{D}$ in order to discuss
various notational conventions and inverse theorems for Gowers norms that we
shall use throughout the paper. For multiplicative derivatives, we set

$\displaystyle\Delta_{{\textbf{v}}_{1},\ldots,{\textbf{v}}_{s};{\underline{h}}}f=\Delta_{{\textbf{v}}_{1}h_{1},\ldots,{\textbf{v}}_{s}h_{s}}f\quad\textrm{and}\quad\Delta_{s,{\textbf{v}};{\underline{h}}}f=\Delta_{{\textbf{v}}h_{1},\ldots,{\textbf{v}}h_{s}}f,$

so that

$\displaystyle\left\|f\right\|_{{\textbf{v}}_{1},\ldots,{\textbf{v}}_{s}}^{2^{s}}=\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\Delta_{{\textbf{v}}_{1},\ldots,{\textbf{v}}_{s};{\underline{h}}}f({\textbf{x}})\quad\textrm{and}\quad\left\|f\right\|^{2^{s}}_{U^{s}({\textbf{v}})}=\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\Delta_{s,{\textbf{v}};{\underline{h}}}f({\textbf{x}}).$

We move on to discuss inverse theorems. For $s=1$, we can rephrase the Gowers
norm as

$\displaystyle\left\|f\right\|_{U^{1}({\textbf{v}})}^{2}=\operatorname*{\mathbb{E}}_{{\textbf{x}}}f({\textbf{x}})\operatorname*{\mathbb{E}}(\overline{f}|{\textbf{v}})({\textbf{x}}),$

and so $\left\|f\right\|_{U^{1}({\textbf{v}})}^{2}$ is a correlation of $f$
with a v-invariant function. For $s>1$, a similar statement can be obtained by
considering degree $s$ dual functions in place of the conditional expectation.
Letting
$\Delta_{s,{\textbf{v}};{\underline{h}}}^{*}f(x)=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{\underline{0}\\}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f({\textbf{x}}+({\underline{\epsilon}}\cdot{\underline{h}}){\textbf{v}})$,
we define the  degree $s$ dual function of $f$ along v to be

$\displaystyle{\mathcal{D}}_{s,{\textbf{v}}}f({\textbf{x}})=\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\Delta_{s,{\textbf{v}};{\underline{h}}}^{*}f(x)=\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{\underline{0}\\}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f({\textbf{x}}+({\underline{\epsilon}}\cdot{\underline{h}}){\textbf{v}}).$

This gives us a weak inverse theorem for the norm $U^{s}({\textbf{v}})$:

$\displaystyle\left\|f\right\|_{U^{s}({\textbf{v}})}^{2^{s}}=\operatorname*{\mathbb{E}}_{\textbf{x}}f({\textbf{x}}){\mathcal{D}}_{s,{\textbf{v}}}f({\textbf{x}}).$

These weak inverse theorems can be easily deduced from the definitions of
Gowers norms in contrast to strong inverse theorems developed by Green, Tao,
Ziegler and Manners [13, 14, 15, 22], which establish deep connections between
Gowers norms and nilsequences. Luckily for the quantitative content, we will
only need the former; any use of strong inverse theorems for Gowers norms of
degree at least 3 would necessarily lead to much worse bounds in our main
results. An exception is a strong inverse theorem for $U^{2}({\textbf{v}})$,
which gives good bounds, and which we will prove and use in later parts of the
paper.

To simplify notation later on, we define ${\mathfrak{D}}_{s}({\textbf{v}})$ to
be the collection of dual functions along v of degree at most $s$, and
${\mathfrak{D}}({\textbf{v}})$ to be the collection of dual functions along v
of all degrees.

##  3\. The PET induction scheme

We move on to describe a PET induction scheme that allows us to control the
counting operator (5) by an average of box norms. We start with an elementary
lemma that addresses the case of linear averages, originally proved in the
ergodic setting by Host.

######  Lemma 3.1 (Box norm control for linear averages, [17, Proposition 1]).

Let $D,\ell\in{\mathbb{N}}$,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}\in{\mathbb{Z}}^{D}$ and
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ be 1-bounded. Then

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{x}\operatorname*{\mathbb{E}}_{n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)\cdots
f_{\ell}({\textbf{x}}+{\textbf{v}}_{\ell}n)\mathclose{}\right|\leq\left\|f_{\ell}\right\|_{{\textbf{v}}_{\ell},{\textbf{v}}_{\ell}-{\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}-{\textbf{v}}_{\ell-1}}.$

The key PET bound in this direction is the following. In what follows, we call
two polynomials essentially distinct if their difference is nonconstant, and
we let $\textrm{supp}({\underline{u}})=\\{j:\ u_{j}>0\\}$.

######  Proposition 3.2 (PET bound).

Let $d,D,\ell\in{\mathbb{N}}$. There exist natural numbers
$s,s^{\prime}=O_{d,D,\ell}(1)$ with the following property: for all
essentially distinct polynomials
${\textbf{p}}_{1},\ldots,{\textbf{p}}_{\ell}\in{\mathbb{Z}}[n]^{D}$ with
degrees $\deg{\textbf{p}}_{\ell}=\max_{j\in[\ell]}\deg{\textbf{p}}_{j}=d$ and
coefficients ${\textbf{p}}_{j}(n)=\sum_{i=0}^{d}{\textbf{a}}_{ji}n^{i}$, and
for all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have the bound

(19)
$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{2^{s^{\prime}}}\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f_{\ell}\right\|_{{{\textbf{c}}_{1}({\underline{h}})},\ldots,{{\textbf{c}}_{s}({\underline{h}})}}^{2^{s}}$

for nonzero polynomials
${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}:{\mathbb{Z}}^{s^{\prime}}\to{\mathbb{Z}}^{D}$.
Moreover, the polynomials ${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}$ are
independent of the functions, and they take the form

(20)
$\displaystyle{\textbf{c}}_{j}({\underline{h}})=\sum_{\begin{subarray}{c}{\underline{u}}\in{\mathbb{N}}_{0}^{s^{\prime}},\\\
|{\underline{u}}|\leq
d-1\end{subarray}}c_{{\underline{u}}}({\textbf{a}}_{\ell(|{\underline{u}}|+1)}-{\textbf{a}}_{w_{j{\underline{u}}}(|{\underline{u}}|+1)}){\underline{h}}^{\underline{u}},$

where:

  1. (i)

for each $j\in[s]$, the indices $w_{j{\underline{u}}}$ take value in the set
$\\{0,\ldots,\ell\\}$ (with
${\textbf{a}}_{0(|{\underline{u}}|+1)}:=\mathbf{0}$) and have the property
$w_{j{\underline{u}}}=w_{j{\underline{u}}^{\prime}}$ whenever
$\textrm{supp}({\underline{u}})=\textrm{supp}({\underline{u}}^{\prime})$;

  2. (ii)

$c_{{\underline{u}}}$ is the coefficient of $n{\underline{h}}^{\underline{u}}$
in the multinomial expansion of
$(n+h_{1}+\cdots+h_{s^{\prime}})^{|{\underline{u}}|+1}$.

Proposition 3.2 is essentially a quantitative restatement of Propositions 5.3
and 5.6 from [5] restricted to finite fields. In its proof, we therefore
copiously cite relevant results from [5, 6].

###### Proof.

Let $s\in{\mathbb{N}}$ and
${\textbf{q}}_{1},\ldots,{\textbf{q}}_{\ell}\in{\mathbb{Z}}[n,h_{1},\ldots,h_{s}]^{D}$.
Given the ordered polynomial family
${\mathcal{Q}}=({\textbf{q}}_{1},\ldots,{\textbf{q}}_{\ell})$ and
$m=1,\ldots,\ell$, we define the new polynomial family

$\displaystyle\partial_{m}{\mathcal{Q}}=$
$\displaystyle(\tilde{{\textbf{q}}}_{1}-\tilde{{\textbf{q}}}_{m},\ldots,\tilde{{\textbf{q}}}_{\ell}-\tilde{{\textbf{q}}}_{m},T_{h_{s+1}}\tilde{{\textbf{q}}}_{1}-\tilde{{\textbf{q}}}_{m},\ldots,T_{h_{s+1}}\tilde{{\textbf{q}}}_{\ell}-\tilde{{\textbf{q}}}_{m})^{*},$

where
$T_{h_{s+1}}{\textbf{q}}(n,{\underline{h}})={\textbf{q}}(n+h_{s+1},{\underline{h}})$,
$\tilde{{\textbf{q}}}(n,{\underline{h}})={\textbf{q}}(n,{\underline{h}})-{\textbf{q}}(0,{\underline{h}})$
and the ∗ operation removes all zero polynomials and all subsequent copies of
the same polynomial whenever it repeats several times.

If
$g_{0,{\underline{h}}},\ldots,g_{\ell,{\underline{h}}}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$
are 1-bounded functions for each ${\underline{h}}\in{\mathbb{F}}_{p}^{s}$,
then an application of the Cauchy-Schwarz inequality in
${\underline{h}},{\textbf{x}}$ gives

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}g_{0,{\underline{h}}}({\textbf{x}})g_{1,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{1}(n,{\underline{h}}))\cdots
g_{\ell,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{\ell}(n,{\underline{h}}))\mathclose{}\right|^{2}\\\
\leq\operatorname*{\mathbb{E}}_{({\underline{h}},h_{s+1})\in{\mathbb{F}}_{p}^{s+1}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}g_{1,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{1}(n,{\underline{h}}))\cdots
g_{\ell,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{\ell}(n,{\underline{h}}))\\\
\overline{g_{1,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{1}(n+h_{s+1},{\underline{h}}))\cdots
g_{\ell,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{\ell}(n+h_{s+1},{\underline{h}}))}.$

Making the change of variables
${\textbf{x}}\mapsto{\textbf{x}}-{\textbf{q}}_{m}(n,{\underline{h}})$, we can
rewrite the inequality above as

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}g_{0,{\underline{h}}}({\textbf{x}})g_{1,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{1}(n,{\underline{h}}))\cdots
g_{\ell,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{\ell}(n,{\underline{h}}))\mathclose{}\right|^{2}\\\
\leq\operatorname*{\mathbb{E}}_{({\underline{h}},h_{s+1})\in{\mathbb{F}}_{p}^{s+1}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}{g}_{\mathbf{0},{\underline{h}},h_{s+1}}({\textbf{x}})\prod_{{\textbf{q}}\in\partial_{m}{\mathcal{Q}}}{g}_{{\textbf{q}},{\underline{h}},h_{s+1}}({\textbf{x}}+{\textbf{q}}(n,{\underline{h}},h_{s+1})).$

The function $g_{\mathbf{0},{\underline{h}},h_{s+1}}$ is given by the formula

$g_{\mathbf{0},{\underline{h}},h_{s+1}}({\textbf{x}})=\prod_{j:\tilde{{\textbf{q}}}_{j}=\tilde{{\textbf{q}}}_{m}}g_{j,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{j}(0,{\underline{h}})-{\textbf{q}}_{m}(0,{\underline{h}})),$

and for ${\textbf{q}}\in\partial_{m}{\mathcal{Q}}$, we have

$\displaystyle{g}_{{\textbf{q}},{\underline{h}},h_{s+1}}=$
$\displaystyle\prod_{j:\tilde{{\textbf{q}}}_{j}-\tilde{{\textbf{q}}}_{m}={\textbf{q}}}g_{j,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{j}(0,{\underline{h}})-{\textbf{q}}_{m}(0,{\underline{h}}))$
$\displaystyle\prod_{j:T_{h_{s+1}}\tilde{{\textbf{q}}}_{j}-\tilde{{\textbf{q}}}_{m}={\textbf{q}}}\overline{g_{j,{\underline{h}}}({\textbf{x}}+{\textbf{q}}_{j}(h_{s+1},{\underline{h}})-{\textbf{q}}_{m}(0,{\underline{h}}))}$

We call the family ${\mathcal{Q}}$ _nice_ 111In the language of [5, 6], the
family is nondegenerate and $\ell$-standard. if
${\textbf{q}}_{1},\ldots,{\textbf{q}}_{\ell}$ are essentially distinct as
polynomials in $n$ (meaning that ${\textbf{q}}_{j}-{\textbf{q}}_{j}$ is
nonconstant in $n$ for $i\neq j$) and the polynomial ${\textbf{q}}_{\ell}$ has
maximum degree in $n$. By [6, Theorem 4.2], there exist
$s^{\prime}\in{\mathbb{N}}_{0}$ (which can be chosen to be $O_{d,D,\ell}(1)$,
see footnote 29 in [5]) and $m_{1},\ldots,m_{s^{\prime}}\in{\mathbb{N}}$ such
that the family
${\mathcal{P}}^{\prime}:=\partial_{m_{s^{\prime}}}\cdots\partial_{m_{1}}{\mathcal{P}}$
is nice and its members are linear in $n$. The bound on $s^{\prime}$ and the
fact that $|\partial_{m}{\mathcal{Q}}|\leq 2|{\mathcal{Q}}|$ for any nice
family ${\mathcal{Q}}$ implies that $s:=|{\mathcal{P}}^{\prime}|\leq
2^{s^{\prime}}|{\mathcal{P}}|=O_{d,D,\ell}(1)$.

Let
${\textbf{b}}_{11}({\underline{h}})n+{\textbf{b}}_{10}({\underline{h}}),\ldots,{\textbf{b}}_{s1}({\underline{h}})n+{\textbf{b}}_{s0}({\underline{h}})$
be the elements of ${\mathcal{P}}^{\prime}$. By a repeated application of the
Cauchy-Schwarz inequality and a change of variables as illustrated above, we
get the bound

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{2^{s^{\prime}}}\\\
\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}{f}_{0,{\underline{h}}}({\textbf{x}})\prod_{j=1}^{s}{f}_{j,{\underline{h}}}({\textbf{x}}+{\textbf{b}}_{j1}({\underline{h}})n+{\textbf{b}}_{j0}({\underline{h}}))$

for some 1-bounded functions
${f}_{0,{\underline{h}}},\ldots,{f}_{s,{\underline{h}}}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$.
Importantly, we have
${f}_{s,{\underline{h}}}({\textbf{x}})={\mathcal{C}}^{s^{\prime}}f_{\ell}({\textbf{x}}+{\textbf{q}}({\underline{h}}))$
for some ${\textbf{q}}\in{\mathbb{Z}}[{\underline{h}}]^{D}$; this crucial fact
is (essentially) the property from [6, Theorem 4.2] that the tuple
${\mathcal{P}}^{\prime}$ is standard for $f_{\ell}$.

Since the family ${\mathcal{P}}^{\prime}$ is nice, the coefficients
${\textbf{b}}_{11},\ldots,{\textbf{b}}_{s1}$ are all distinct. By Lemma 3.1,
the result follows upon letting ${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}$ be
the polynomials ${\textbf{b}}_{\ell 1},{\textbf{b}}_{\ell
1}-{\textbf{b}}_{11},\ldots,{\textbf{b}}_{\ell 1}-{\textbf{b}}_{(\ell-1)1}$.
It remains to show that the coefficients of the polynomials
${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}$ have the claimed properties. The
fact that they are nonzero is a consequence of the distinctness of
${\textbf{b}}_{11},\ldots,{\textbf{b}}_{s1}$. The other properties follow from
the complicated coefficient tracking scheme developed in [5, Section 5], which
we briefly outline.

The coefficients
${\textbf{b}}_{11}({\underline{h}}),\ldots,{\textbf{b}}_{s1}({\underline{h}})$
are polynomials in ${\underline{h}}$ of the form

(21)
$\displaystyle{\textbf{b}}_{j1}({\underline{h}})=\sum_{\begin{subarray}{c}{\underline{u}}\in{\mathbb{N}}_{0}^{s^{\prime}},\\\
|{\underline{u}}|\leq
d-1\end{subarray}}c_{{\underline{u}}}({\textbf{a}}_{w_{j{\underline{u}}}(|{\underline{u}}|+1)}-{\textbf{a}}_{w_{{\underline{u}}}(|{\underline{u}}|+1)}){\underline{h}}^{\underline{u}},$

where:

  1. (i)

the indices $w_{{\underline{u}}},w_{j{\underline{u}}}$ take value in the set
$\\{0,\ldots,\ell\\}$ (with
${\textbf{a}}_{0(|{\underline{u}}|+1)}:=\mathbf{0}$) and have the property
$w_{j{\underline{u}}}=w_{j{\underline{u}}^{\prime}}$ and
$w_{\underline{u}}=w_{{\underline{u}}^{\prime}}$ whenever
$\textrm{supp}({\underline{u}})=\textrm{supp}({\underline{u}}^{\prime})$;

  2. (ii)

we have $w_{s{\underline{u}}}=\ell$ for all ${\underline{u}}$;

  3. (iii)

$c_{{\underline{u}}}$ is the coefficient of $n{\underline{h}}^{\underline{u}}$
in the multinomial expansion of
$(n+h_{1}+\cdots+h_{s^{\prime}})^{|{\underline{u}}|+1}$, i.e.

$\displaystyle
c_{{\underline{u}}}={{u_{s^{\prime}}+\cdots+u_{1}+1}\choose{u_{s^{\prime}},\ldots,u_{1},1}}.$

The properties (i)-(iii) of the polynomials
${\textbf{b}}_{11},\ldots,{\textbf{b}}_{s1}$ correspond to the properties
(P1)-(P4) from [5, Definition 5.5]. Specifically, in the language of [5], the
formula (21) corresponds to the coefficient ${\underline{h}}^{\underline{u}}$
having symbol $(w_{1{\underline{u}}},\ldots,w_{s{\underline{u}}})$ and type
$(c_{\underline{u}},w_{{\underline{u}}},|{\underline{u}}|+1)$. Then the
general formula (21) corresponds to the property (P1) from [5, Definition
5.5], and the property (i) above corresponds to the property (P3) from [5,
Definition 5.5]. Similarly, the property (ii) above is a restatement of the
property (P4) from [5, Definition 5.5] Lastly, the formula for
$c_{\underline{u}}$ in (iii) is a restatement of the property (P2) from [5,
Definition 5.5] applied to the polynomials
${\textbf{b}}_{j1}({\underline{h}})n+{\textbf{b}}_{j0}({\underline{h}})$. That
the polynomials ${\textbf{b}}_{11},\ldots,{\textbf{b}}_{s1}$ enjoy the
structural properties (i)-(iii) is a consequence of [5, Proposition 5.6] (and
more precisely, the remark right below its proof), which asserts that the
properties (P1)-(P4) from [5, Definition 5.5] are preserved while applying the
van der Corput operation ${\mathcal{Q}}\mapsto\partial_{m}{\mathcal{Q}}$;
hence, applying this procedure iteratively, the properties (P1)-(P4) are
enjoyed by the family ${\mathcal{P}}^{\prime}$. Finally, that the coefficients
of the polynomials ${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}$ enjoy the
properties stated in Proposition 3.2 is a consequence of the fact that
${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}$ equal
${\textbf{b}}_{s1}-{\textbf{b}}_{11},\ldots,{\textbf{b}}_{s1}-{\textbf{b}}_{(s-1)1},{\textbf{b}}_{s1}$.

∎

######  Example 2.

We illustrate the rather abstract content of Proposition 3.2 and its technical
proof for the family

${\mathcal{P}}=\\{n^{2}{\textbf{v}}_{1},(n^{2}+n){\textbf{v}}_{2}\\},$

where ${\textbf{v}}_{1},{\textbf{v}}_{2}\in{\mathbb{Z}}^{D}$ and
$D\in{\mathbb{N}}$. Let
$f_{0},f_{1},f_{2}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ and

$\delta=\mathopen{}\left|\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+n^{2}{\textbf{v}}_{1})f_{2}({\textbf{x}}+(n^{2}+n){\textbf{v}}_{2})\mathclose{}\right|.$

An application of the Cauchy Schwarz inequality in x gives

$\displaystyle\delta^{2}\leq\operatorname*{\mathbb{E}}_{h_{1}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{1}({\textbf{x}}+n^{2}{\textbf{v}}_{1})\overline{f_{1}}({\textbf{x}}+(n+h_{1})^{2}{\textbf{v}}_{1})$
$\displaystyle
f_{2}({\textbf{x}}+(n^{2}+n){\textbf{v}}_{2})\overline{f_{2}}({\textbf{x}}+((n+h_{1})^{2}+(n+h_{1})){\textbf{v}}_{2}).$

After the change of variables
${\textbf{x}}\mapsto{\textbf{x}}-n^{2}{\textbf{v}}_{1}$, we get

(22)
$\displaystyle\delta^{2}\leq\operatorname*{\mathbb{E}}_{h_{1}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{0,h_{1}}({\textbf{x}})f_{1,h_{1}}({\textbf{x}}+2h_{1}n{\textbf{v}}_{1})f_{2,h_{1}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2})$
$\displaystyle
f_{3,h_{1}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(2h_{1}n+n){\textbf{v}}_{2})$

for

$\displaystyle
f_{j,h_{1}}({\textbf{x}})=\begin{cases}f_{1}({\textbf{x}}),\;&j=0\\\
\overline{f_{1}}({\textbf{x}}+h_{1}^{2}{\textbf{v}}_{1}),\;&j=1\\\
f_{2}({\textbf{x}}),\;&j=2\\\
\overline{f_{2}}({\textbf{x}}+(h_{1}^{2}+h_{1}){\textbf{v}}_{2}),\;&j=3.\end{cases}$

We note that in (22), we are dealing with the polynomial family

$\displaystyle\partial_{1}{\mathcal{P}}=\\{2h_{1}n{\textbf{v}}_{1},\;n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2},\;n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(2h_{1}n+n){\textbf{v}}_{2}\\}$

Subsequently, we apply the Cauchy-Schwarz inequality in ${\textbf{x}},h_{1}$
to (22), obtaining

$\displaystyle\delta^{4}\leq\operatorname*{\mathbb{E}}_{h_{1},h_{2}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{1,h_{1}}({\textbf{x}}+2h_{1}n{\textbf{v}}_{1})\overline{f_{1,h_{1}}}({\textbf{x}}+2h_{1}(n+h_{2}){\textbf{v}}_{1})$
$\displaystyle
f_{2,h_{1}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2})\overline{f_{2,h_{1}}}({\textbf{x}}+(n+h_{2})^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(n+h_{2}){\textbf{v}}_{2})$
$\displaystyle
f_{3,h_{1}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(2h_{1}n+n){\textbf{v}}_{2})$
$\displaystyle\overline{f_{3,h_{1}}}({\textbf{x}}+(n+h_{2})^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(2h_{1}(n+h_{2})+(n+h_{2})){\textbf{v}}_{2}).$

Performing the change of variables
${\textbf{x}}\mapsto{\textbf{x}}-2h_{1}n{\textbf{v}}_{1}$, we have

(23)
$\displaystyle\delta^{4}\leq\operatorname*{\mathbb{E}}_{h_{1},h_{2}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{0,h_{1},h_{2}}({\textbf{x}})f_{1,h_{1},h_{2}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}-2h_{1}n{\textbf{v}}_{1})$
$\displaystyle
f_{2,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2h_{2}n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}-2h_{1}n{\textbf{v}}_{1})$
$\displaystyle
f_{3,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2h_{1}n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2})$
$\displaystyle
f_{4,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2(h_{1}+h_{2})n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}),$

where the functions $f_{j,h_{1},h_{2}}$ are given by

$\displaystyle
f_{j,h_{1},h_{2}}({\textbf{x}})=\begin{cases}f_{1,h_{1}}({\textbf{x}})\overline{f_{1,h_{1}}}({\textbf{x}}+2h_{1}h_{2}{\textbf{v}}_{1}),\;&j=0\\\
f_{2,h_{1}}({\textbf{x}}),\;&j=1\\\
\overline{f_{2,h_{1}}}({\textbf{x}}+h_{2}^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+h_{2}{\textbf{v}}_{2}),\;&j=2\\\
f_{3,h_{1}}({\textbf{x}}),\;&j=3\\\
\overline{f_{3,h_{1}}}({\textbf{x}}+h_{2}^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(2h_{1}h_{2}+h_{2}){\textbf{v}}_{2}),\;&j=4.\end{cases}$

The polynomial family that we see in (23) is
$\partial_{1}\partial_{1}{\mathcal{P}}$.

We apply the Cauchy-Schwarz inequality one more time, this time in
$h_{1},h_{2},{\textbf{x}}$ to (23). In doing so, we obtain an average over a
product of 8 functions:

$\displaystyle\delta^{8}\leq\operatorname*{\mathbb{E}}_{h_{1},h_{2},h_{3}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{1,h_{1},h_{2}}({\textbf{x}}+n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}-2h_{1}n{\textbf{v}}_{1})$
$\displaystyle\overline{f_{1,h_{1},h_{2}}}({\textbf{x}}+(n+h_{3})^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+(n+h_{3}){\textbf{v}}_{2}-2h_{1}(n+h_{3}){\textbf{v}}_{1})$
$\displaystyle
f_{2,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2h_{2}n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}-2h_{1}n{\textbf{v}}_{1})$
$\displaystyle\overline{f_{2,h_{1},h_{2}}}({\textbf{x}}+((n+h_{3})^{2}+2h_{2}(n+h_{3}))({\textbf{v}}_{2}-{\textbf{v}}_{1})+(n+h_{3}){\textbf{v}}_{2}-2h_{1}(n+h_{3}){\textbf{v}}_{1})$
$\displaystyle
f_{3,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2h_{1}n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2})$
$\displaystyle\overline{f_{3,h_{1},h_{2}}}({\textbf{x}}+((n+h_{3})^{2}+2h_{1}(n+h_{3}))({\textbf{v}}_{2}-{\textbf{v}}_{1})+(n+h_{3}){\textbf{v}}_{2})$
$\displaystyle
f_{4,h_{1},h_{2}}({\textbf{x}}+(n^{2}+2(h_{1}+h_{2})n)({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2})$
$\displaystyle\overline{f_{4,h_{1},h_{2}}}({\textbf{x}}+((n+h_{3})^{2}+2(h_{1}+h_{2})(n+h_{3}))({\textbf{v}}_{2}-{\textbf{v}}_{1})+(n+h_{3}){\textbf{v}}_{2}).$

Performing the change of variables
${\textbf{x}}\mapsto{\textbf{x}}-(n^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+n{\textbf{v}}_{2}-2h_{1}n{\textbf{v}}_{1})$,
we get

(24)
$\displaystyle\delta^{8}\leq\operatorname*{\mathbb{E}}_{h_{1},h_{2},h_{3}}\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}$
$\displaystyle
f_{0,{\underline{h}}}({\textbf{x}})f_{1,{\underline{h}}}({\textbf{x}}+2h_{3}n({\textbf{v}}_{2}-{\textbf{v}}_{1}))f_{2,{\underline{h}}}({\textbf{x}}+2h_{2}n({\textbf{v}}_{2}-{\textbf{v}}_{1}))$
$\displaystyle
f_{3,{\underline{h}}}({\textbf{x}}+2(h_{2}+h_{3})n({\textbf{v}}_{2}-{\textbf{v}}_{1}))f_{4,{\underline{h}}}({\textbf{x}}+2h_{1}n{\textbf{v}}_{2})$
$\displaystyle
f_{5,{\underline{h}}}({\textbf{x}}+2h_{3}n({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}n{\textbf{v}}_{2})f_{6,{\underline{h}}}({\textbf{x}}+2h_{2}n({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}n{\textbf{v}}_{2})$
$\displaystyle
f_{7,{\underline{h}}}({\textbf{x}}+2(h_{2}+h_{3})n({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}n{\textbf{v}}_{2}),$

where

$\displaystyle
f_{j,{\underline{h}}}({\textbf{x}})=\begin{cases}f_{1,{\underline{h}}}({\textbf{x}}),\;&j=0\\\
\overline{f_{1,h_{1},h_{2}}}({\textbf{x}}+h_{3}^{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+h_{3}{\textbf{v}}_{2}-2h_{1}h_{3}{\textbf{v}}_{1}),\;&j=1\\\
f_{2,h_{1},h_{2}}({\textbf{x}}),\;&j=2\\\
\overline{f_{2,h_{1},h_{2}}}({\textbf{x}}+(h_{3}^{2}+2h_{2}h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1})+h_{3}{\textbf{v}}_{2}-2h_{1}h_{3}{\textbf{v}}_{1}),\;&j=3\\\
f_{3,h_{1},h_{2}}({\textbf{x}}),\;&j=4\\\
\overline{f_{3,h_{1},h_{2}}}({\textbf{x}}+(h_{3}^{2}+2h_{1}h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1})+h_{3}{\textbf{v}}_{2}),\;&j=5\\\
f_{4,h_{1},h_{2}}({\textbf{x}}),\;&j=6\\\
\overline{f_{4,h_{1},h_{2}}}({\textbf{x}}+(h_{3}^{2}+2(h_{1}+h_{2})h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1})+h_{3}{\textbf{v}}_{2}),\;&j=7.\end{cases}$

The polynomials in the average (24) correspond to the family
$\partial_{1}\partial_{1}\partial_{1}{\mathcal{P}}$, and they are all linear
in $n$. Moreover, it is easy to check that
$f_{7,{\underline{h}}}({\textbf{x}})=\overline{f_{2}({\textbf{x}}+{\textbf{q}}({\underline{h}}))}$
for some polynomial ${\textbf{q}}\in{\mathbb{Z}}^{D}[{\underline{h}}]$. By
Lemma 3.1, we have

$\displaystyle\delta^{8}\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{3}}\left\|f_{2}\right\|_{\begin{subarray}{c}2(h_{2}+h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},\
2h_{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},\
2h_{3}({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},\\\
2h_{1}{\textbf{v}}_{2},\ 2(h_{2}+h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1}),\
2h_{2}({\textbf{v}}_{2}-{\textbf{v}}_{1}),\
2h_{3}({\textbf{v}}_{2}-{\textbf{v}}_{1})\end{subarray}},$

i.e. the polynomials ${\textbf{c}}_{1},\ldots,{\textbf{c}}_{7}$ take the form

$\displaystyle
2(h_{2}+h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},\
2h_{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},\
2h_{3}({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{1}{\textbf{v}}_{2},$
$\displaystyle 2h_{1}{\textbf{v}}_{2},\
2(h_{2}+h_{3})({\textbf{v}}_{2}-{\textbf{v}}_{1}),\
2h_{2}({\textbf{v}}_{2}-{\textbf{v}}_{1}),\
2h_{3}({\textbf{v}}_{2}-{\textbf{v}}_{1}).$

They can be rewritten in the form

$\displaystyle
2h_{1}\varepsilon_{1}{\textbf{v}}_{2}+2h_{2}\varepsilon_{2}({\textbf{v}}_{2}-{\textbf{v}}_{1})+2h_{3}\varepsilon_{3}({\textbf{v}}_{2}-{\textbf{v}}_{1})$

for ${\underline{\epsilon}}\in\\{0,1\\}^{3}\setminus{\\{\underline{0}\\}}$. We
note that they satisfy the formula (20), which in our case takes the form

$\displaystyle\sum_{i=1}^{3}2({\textbf{v}}_{2}-{\textbf{a}}_{w_{jh_{i}}2})h_{i},$

with
${\textbf{a}}_{02}=\mathbf{0},{\textbf{a}}_{12}={\textbf{v}}_{1},{\textbf{a}}_{22}={\textbf{v}}_{2}$
and $w_{jh_{1}}\in\\{0,2\\},w_{jh_{2}},w_{jh_{3}}\in\\{1,2\\}$.

##  4\. Control by a single box norm

The control over the counting operator by an average of box norms as provided
by Proposition 3.2 is insufficient for our purposes. In this section, we use
Corollary 2.5 in order to concatenate the complicated polynomial subgroups
appearing in the right hand side of (19) and obtain control of the counting
operator by a single box norm.

We start with a simple lemma which bounds the zero set of polynomials. Its
proof is inspired by and almost identical to that of [6, Lemma 2.11].

######  Lemma 4.1 (Zero sets of polynomials).

Suppose that $g\in{\mathbb{Z}}[h_{1},\ldots,h_{s}]$ has degree
$d\in{\mathbb{N}}$. Then
$\mathopen{}\left|\\{{\underline{h}}\in{\mathbb{F}}_{p}^{s}:\
g({\underline{h}})=0\\}\mathclose{}\right|\ll_{s,d}p^{s-1}$.

###### Proof.

We induct on $s$ and assume throughout that the characteristic $p$ is large
enough depending on the coefficient of $g$ so that $g$ is nonconstant modulo
$p$. For $s=1$, the polynomial $g$ has at most $d$ roots in
${\mathbb{F}}_{p}$, and so the statement follows easily. Suppose it holds for
some $s\in{\mathbb{N}}$. A polynomial $g$ in $s+1$ variables
$({\underline{h}},h_{s+1})$ can be rewritten as

$\displaystyle
g({\underline{h}},h_{s+1})=\sum_{j=0}^{d}g_{j}({\underline{h}})h_{s+1}^{j}.$

The identity $g({\underline{h}},h_{s+1})=0$ holds in two cases. Either
$g_{0}({\underline{h}})=\cdots=g_{d}({\underline{h}})=0$, or at least one
$g_{j}({\underline{h}})$ is nonzero. In the former case, we apply the
inductive hypothesis to conclude that there are $O_{d,s}(p^{s-1})$ values
${\underline{h}}$ at which all the polynomials $g_{0},\ldots,g_{d}$ vanish. In
the latter case, we fix ${\underline{h}}$; then the polynomial
$g_{\underline{h}}(h_{s+1})=g({\underline{h}},h_{s+1})$ is nonzero, and by the
$s=1$ case, it vanishes for at most $d$ values $h_{s+1}$. Combining these two
bounds, we get that

$\displaystyle\mathopen{}\left|\\{{\underline{h}}\in{\mathbb{F}}_{p}^{s+1}:\
g({\underline{h}},h_{s+1})=0\\}\mathclose{}\right|\ll_{s,d}p^{s-1}\cdot
p+dp^{s}\ll_{s,d}p^{s},$

as claimed. ∎

In what follows, we let
$A_{d,s^{\prime}}:=\left\\{{\underline{u}}\in{\mathbb{N}}_{0}^{s^{\prime}}:\
|{\underline{u}}|\leq d\right\\}$. The result below can be thought as the
quantitative, finite-field version of [5, Proposition 5.2].

######  Proposition 4.2 (Concatenation of box norms along polynomials).

Let $d,D,s,s^{\prime}\in{\mathbb{N}}$. There exists
$s^{\prime\prime}=O_{d,s}(1)$ with the following property: for all 1-bounded
functions $f:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$ and polynomials
${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}\in{\mathbb{Z}}[h_{1},\ldots,h_{s^{\prime}}]$
of the form

$\displaystyle{\textbf{c}}_{j}({\underline{h}})=\sum_{\begin{subarray}{c}{\underline{u}}\in
A_{d,s^{\prime}}\end{subarray}}{\textbf{v}}_{j{\underline{u}}}{\underline{h}}^{{\underline{u}}},$

we have

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f\right\|_{{{\textbf{c}}_{1}({\underline{h}})},\ldots,{{\textbf{c}}_{s}({\underline{h}})}}^{2^{s}}\mathclose{}\right)^{O_{d,s,s^{\prime}}(1)}\leq\left\|f\right\|_{G_{1}^{\times
s^{\prime\prime}},\ldots,G_{s}^{\times
s^{\prime\prime}}}+O_{d,s,s^{\prime}}(p^{-1}),$

where $G_{j}=\langle{\textbf{v}}_{j{\underline{u}}}:\ {\underline{u}}\in
A_{d,s^{\prime}}\rangle$ for $j\in[s]$.

Before we prove Proposition 4.2 in full, we illustrate the proof with a simple
example.

######  Example 3.

Suppose first that $s=1$ and
${\textbf{c}}({\underline{h}})={\textbf{u}}+\sum_{i=1}^{s^{\prime}}{\textbf{v}}_{i}h_{i}$,
so that we are dealing with an expression

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f\right\|_{{\textbf{u}}+{\textbf{v}}_{1}h_{1}+\cdots+{\textbf{v}}_{s^{\prime}}h_{s^{\prime}}}^{2}.$

By the inequality (18), we have

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f\right\|_{{\textbf{u}}+{\textbf{v}}_{1}h_{1}+\cdots+{\textbf{v}}_{s^{\prime}}h_{s^{\prime}}}^{2}\mathclose{}\right)^{2^{k}}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{h}}_{\underline{\epsilon}}\in{\mathbb{F}}_{p}^{s^{\prime}},\\\
{\underline{\epsilon}}\in\\{0,1\\}^{k}\end{subarray}}\left\|f\right\|_{\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{k}}\langle{\textbf{u}}+h_{{\underline{\epsilon}}1}{\textbf{v}}_{1}+\cdots+h_{{\underline{\epsilon}}s^{\prime}}{\textbf{v}}_{s^{\prime}}\rangle}^{2},$

where we are setting
$h_{\underline{\epsilon}}=(h_{{\underline{\epsilon}}1},\ldots,h_{{\underline{\epsilon}}s^{\prime}})$.
We pick $k=\lceil\log_{2}s^{\prime}\rceil$, so that $k$ is the smallest
natural number satisfying $2^{k}\geq s^{\prime}$, and we claim that for almost
all tuples $({\underline{h}}_{\underline{\epsilon}})_{\underline{\epsilon}}$,
the group
$\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{k}}\langle{\textbf{u}}+h_{{\underline{\epsilon}}1}{\textbf{v}}_{1}+\cdots+h_{{\underline{\epsilon}}s^{\prime}}{\textbf{v}}_{s^{\prime}}\rangle$
equals all of
$\langle{\textbf{u}},{\textbf{v}}_{1},\ldots,{\textbf{v}}_{s^{\prime}}\rangle$.
Since

$\displaystyle\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{k}}\langle{\textbf{u}}+h_{{\underline{\epsilon}}1}{\textbf{v}}_{1}+\cdots+h_{{\underline{\epsilon}}s^{\prime}}{\textbf{v}}_{s^{\prime}}\rangle=\langle{\textbf{u}}+h_{{\underline{\epsilon}}1}{\textbf{v}}_{1}+\cdots+h_{{\underline{\epsilon}}s^{\prime}}{\textbf{v}}_{s^{\prime}}:\;{\underline{\epsilon}}\in\\{0,1\\}^{k}\rangle,$

the claim will follow whenever

(25)
$\displaystyle\langle(h_{{\underline{\epsilon}}1},\ldots,h_{{\underline{\epsilon}}s^{\prime}}):\;{\underline{\epsilon}}\in\\{0,1\\}^{k}\rangle={\mathbb{F}}_{p}^{s^{\prime}}.$

Ordering ${\underline{\epsilon}}\in\\{0,1\\}^{k}$ as
${\underline{\epsilon}}_{1},\ldots,{\underline{\epsilon}}_{k}$ and using
$2^{k}\geq{s^{\prime}}$, we observe that the matrix
$(h_{{\underline{\epsilon}}_{j}1},\ldots,h_{{\underline{\epsilon}}_{j}s^{\prime}})_{j\in[s^{\prime}]}$
is square. Its determinant is a nonconstant polynomial in
$(h_{{\underline{\epsilon}}_{j}i})_{j,i\in[s^{\prime}]}$, and Lemma 4.1
guarantees that it has full rank for all but a $O_{s^{\prime}}(p^{-1})$
proportion of $(h_{{\underline{\epsilon}}_{j}i})_{i,j\in[s^{\prime}]}$. It
follows that (25) holds for all except a $O_{s^{\prime}}(p^{-1})$ proportion
of ${\underline{h}}_{\underline{\epsilon}}$, and so

$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f\right\|_{{\textbf{u}}+{\textbf{v}}_{1}h_{1}+\cdots+{\textbf{v}}_{s^{\prime}}h_{s^{\prime}}}^{2}\mathclose{}\right)^{2^{k}}\leq\left\|f\right\|_{\langle{\textbf{u}},{\textbf{v}}_{1},\ldots,{\textbf{v}}_{s^{\prime}}\rangle}^{2}+O_{s^{\prime}}(p^{-1}).$

###### Proof of Proposition 4.2.

For $j=1,\ldots,s$ and ${\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}$, we
set $H_{{\underline{h}},j}=\langle{\textbf{c}}_{j}({\underline{h}})\rangle$.
For some $k\in{\mathbb{N}}$ to be chosen later, Corollary 2.5 gives us

(26)
$\displaystyle\mathopen{}\left(\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f\right\|_{{{\textbf{c}}_{1}({\underline{h}})},\ldots,{{\textbf{c}}_{s}({\underline{h}})}}^{2^{s}}\mathclose{}\right)^{O_{s,k}(1)}\leq\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{h}}_{\underline{\epsilon}}\in{\mathbb{F}}_{p}^{s^{\prime}},\\\
{\underline{\epsilon}}\in\\{0,1\\}^{w_{1}+\cdots+w_{k}}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{\sum_{b_{1},\ldots,b_{k}\in\\{0,1\\}}H_{{\underline{h}}_{{\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}}j}:\
j=1,\ldots,s,\\\
{\underline{\epsilon}}_{l0},{\underline{\epsilon}}_{l1}\in\\{0,1\\}^{w_{l}},\
{\underline{\epsilon}}_{l0}\neq{\underline{\epsilon}}_{l1},\
l=1,\ldots,k\\}\end{subarray}}$

for some $w_{1},\ldots,w_{k}=O_{s,k}(1)$. Unpacking the definitions of the
groups $H_{{\underline{h}},j}$ and polynomials ${\textbf{c}}_{j}$, we obtain
the identity

(27)
$\displaystyle\sum_{b_{1},\ldots,b_{k}\in\\{0,1\\}}H_{{\underline{h}}_{{\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}}j}=\left\langle\sum_{\begin{subarray}{c}{\underline{u}}\in
A_{d,s^{\prime}}\end{subarray}}{\textbf{v}}_{j{\underline{u}}}{\underline{h}}_{{\underline{\epsilon}}_{\underline{b}}}^{\underline{u}}:\
{\underline{b}}\in\\{0,1\\}^{k}\right\rangle$

where
${\underline{\epsilon}}_{\underline{b}}:={\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}$.
Our goal is to show that for all $j$ and almost all choices of
$({\underline{h}}_{{\underline{\epsilon}}_{\underline{b}}})_{{\underline{b}}\in\\{0,1\\}^{k}}$,
the group $G_{j}$ lies inside (27) as long as $k$ is sufficiently large. This
will follow whenever

$\displaystyle\left\langle\mathopen{}\left({\underline{h}}_{{\underline{\epsilon}}_{\underline{b}}}^{\underline{u}}\mathclose{}\right)_{\begin{subarray}{c}{\underline{u}}\in
A_{d,s^{\prime}}\end{subarray}}:\
{\underline{b}}\in\\{0,1\\}^{k}\right\rangle$

spans all of ${\mathbb{F}}_{p}^{|A_{d,s^{\prime}}|}$. To show the latter
statement, it suffices to show that the matrix

$M_{1}=\mathopen{}\left({\underline{h}}_{{\underline{\epsilon}}_{\underline{b}}}^{\underline{u}}\mathclose{}\right)_{\begin{subarray}{c}{\underline{u}}\in
A_{d,s^{\prime}},{\underline{b}}\in\\{0,1\\}^{k}\end{subarray}},$

indexed in one direction by $|A_{d,s^{\prime}}|$ possible exponents
${\underline{u}}$ and in the other direction by $2^{k}$ choices of
${\underline{b}}\in\\{0,1\\}^{k}$, has full rank. Picking the smallest
$k\in{\mathbb{N}}$ satisfying $2^{k}\geq|A_{d,s^{\prime}}|$ (hence
$s^{\prime\prime}=O_{k,s}(1)=O_{d,s,s^{\prime}}(1)$), we can find a subset
$B\subseteq\\{0,1\\}^{k}$ with $|B|=|A_{d,s^{\prime}}|$, so that the submatrix

(28) $\displaystyle
M_{2}=\mathopen{}\left({\underline{h}}_{{\underline{\epsilon}}_{\underline{b}}}^{\underline{u}}\mathclose{}\right)_{\begin{subarray}{c}{\underline{u}}\in
A_{d,s^{\prime}},\ {\underline{b}}\in B\end{subarray}}$

is a square matrix and retains the rank of $M_{1}$ as a matrix over
${\mathbb{Z}}[({\underline{h}}_{\underline{\epsilon}})_{\underline{\epsilon}}]$.
Let $w:=w_{1}+\cdots+w_{k}$. For each
$({\underline{h}}_{{\underline{\epsilon}}})_{{\underline{\epsilon}}\in\\{0,1\\}^{w}}\in{\mathbb{F}}_{p}^{s^{\prime}2^{w}}$,
we define
$g(({\underline{h}}_{{\underline{\epsilon}}})_{{\underline{\epsilon}}})$ to be
the determinant of (28). The function $g$ is a polynomial map from
${\mathbb{F}}_{p}^{s^{\prime}2^{w}}$ to ${\mathbb{F}}_{p}$; moreover, it is
nonconstant. One way to see it is to enumerate the elements of
$A_{d,s^{\prime}}$ and $B$ in some arbitrary fashion (as
${\underline{u}}_{1},\ldots,{\underline{u}}_{d^{\prime}}$ and
${\underline{b}}_{1},\ldots,{\underline{b}}_{d^{\prime}}$ for
$d^{\prime}=|B|=|A_{d,s^{\prime}}|$) and observe that the coefficient of the
monomial
$\prod_{j=1}^{d^{\prime}}{\underline{h}}_{{\underline{\epsilon}}_{{\underline{b}}_{j}}}^{{\underline{u}}_{j}}$
in $g$ is nonzero222This monomial does appear at least once with nonzero
coefficient when computing $g$ inductively, and since
${\underline{\epsilon}}_{{\underline{b}}}\neq{\underline{\epsilon}}_{{\underline{b}}^{\prime}}$
for ${\underline{b}}\neq{\underline{b}}^{\prime}$, it appears exactly once in
the computation. For instance, for $s^{\prime},d=2$, the matrix $M_{2}$ takes
the form
$M_{2}=\begin{pmatrix}h_{11}^{2}&h_{11}h_{12}&h_{12}^{2}&h_{11}&h_{12}&1\\\
h_{21}^{2}&h_{21}h_{22}&h_{22}^{2}&h_{21}&h_{22}&1\\\
h_{31}^{2}&h_{31}h_{32}&h_{32}^{2}&h_{31}&h_{32}&1\\\
h_{41}^{2}&h_{41}h_{42}&h_{42}^{2}&h_{41}&h_{42}&1\\\
h_{51}^{2}&h_{51}h_{52}&h_{52}^{2}&h_{51}&h_{52}&1\\\
h_{61}^{2}&h_{61}h_{62}&h_{62}^{2}&h_{61}&h_{62}&1\\\ \end{pmatrix}$ (upon
relabelling ${\underline{h}}_{{\underline{\epsilon}}_{{\underline{b}}_{j}}}$
as ${\underline{h}}_{j}$), and it is clear that
$h_{11}^{2}h_{21}h_{22}h_{32}^{2}h_{41}h_{52}$, the product of the diagonal
entries of $M_{2}$, has a nonzero coefficient in $\det(M_{2})$. .

By Lemma 4.1, for all but a $O_{d,s,s^{\prime}}(p^{-1})$ proportion of values
$({\underline{h}}_{{\underline{\epsilon}}})_{{\underline{\epsilon}}\in\\{0,1\\}^{w}}\in{\mathbb{F}}_{p}^{s^{\prime}2^{w}}$,
the polynomial
$g(({\underline{h}}_{{\underline{\epsilon}}})_{{\underline{\epsilon}}})$ is
nonzero. For all these “good” values, the matrix $M_{2}$ is invertible, hence
$M_{1}$ has full rank, and hence the group
$\sum_{{\underline{b}}\in\\{0,1\\}}H_{{\underline{h}}_{{\underline{\epsilon}}_{{\underline{b}}}}j}$
contains $G_{j}$. Taking the union bound, we conclude that for all but a
$O_{d,s,s^{\prime}}(p^{-1})$ proportion of “bad” values
$({\underline{h}}_{{\underline{\epsilon}}})_{{\underline{\epsilon}}\in\\{0,1\\}^{w}}\in{\mathbb{F}}_{p}^{s^{\prime}2^{w}}$,
all the groups
$\sum_{{\underline{b}}\in\\{0,1\\}}H_{{\underline{h}}_{{\underline{\epsilon}}_{{\underline{b}}}}j}$
in the right hand side of (26) contain $G_{j}$. Letting $s^{\prime\prime}\leq
4^{w}=O_{d,s,s^{\prime}}(1)$ be the number of

$({\underline{\epsilon}}_{10},{\underline{\epsilon}}_{11},\ldots,{\underline{\epsilon}}_{k0},{\underline{\epsilon}}_{k1})\in\\{0,1\\}^{w_{1}}\times\\{0,1\\}^{w_{1}}\times\cdots\times\\{0,1\\}^{w_{k}}\times\\{0,1\\}^{w_{k}}$

with ${\underline{\epsilon}}_{l0}\neq{\underline{\epsilon}}_{l1}$ for
$l=1,\ldots,k$, we then have

(29)
$\mathopen{}\left(\operatorname*{\mathbb{E}}_{\begin{subarray}{c}{\underline{h}}_{\underline{\epsilon}}\in{\mathbb{F}}_{p}^{s^{\prime}},\\\
{\underline{\epsilon}}\in\\{0,1\\}^{w}\end{subarray}}\left\|f\right\|_{\begin{subarray}{c}\\{\sum_{b_{1},\ldots,b_{k}\in\\{0,1\\}}H_{{\underline{h}}_{{\underline{\epsilon}}_{1b_{1}}\cdots{\underline{\epsilon}}_{kb_{k}}}j}:\
j=1,\ldots,s,\\\
{\underline{\epsilon}}_{l0},{\underline{\epsilon}}_{l1}\in\\{0,1\\}^{w_{l}},\
{\underline{\epsilon}}_{l0}\neq{\underline{\epsilon}}_{l1},\
l=1,\ldots,k\\}\end{subarray}}\mathclose{}\right)^{O_{d,s,s^{\prime}}(1)}\\\
\leq\left\|f\right\|_{G_{1}^{\times s^{\prime\prime}},\ldots,G_{s}^{\times
s^{\prime\prime}}}+O_{d,s,s^{\prime}}(p^{-1}),$

where the error term corresponds to the “bad” values
$({\underline{h}}_{\underline{\epsilon}})_{\underline{\epsilon}}$. The result
follows upon combining (26) with (29) and the bound on $k$. ∎

Propositions 3.2 and 4.2 together give the following control by a single box
norm. The proposition below should be compared with (and was inspired by) [5,
Theorem 2.5], which gives its qualitative, ergodic theoretic counterpart.

######  Proposition 4.3 (Control by a single box norm).

Let $d,D,\ell\in{\mathbb{N}}$. There exists a natural number
$s=O_{d,D,\ell}(1)$ with the following property: for all essentially distinct
polynomials
${\textbf{p}}_{1},\ldots,{\textbf{p}}_{\ell}\in{\mathbb{Z}}[n]^{D}$ with
degrees $\deg{\textbf{p}}_{\ell}=\max_{j\in[\ell]}\deg{\textbf{p}}_{j}=d$ and
coefficients ${\textbf{p}}_{j}(n)=\sum_{i=0}^{d}{\textbf{a}}_{ji}n^{i}$, and
for all 1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have the bound

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{O_{d,D,\ell}(1)}\\\
\leq\left\|f_{\ell}\right\|_{{\textbf{a}}_{\ell d_{\ell 0}}^{\times
s},({\textbf{a}}_{\ell d_{\ell 1}}-{\textbf{a}}_{1d_{\ell 1}})^{\times
s},\ldots,({\textbf{a}}_{\ell
d_{\ell(\ell-1)}}-{\textbf{a}}_{(\ell-1)d_{\ell(\ell-1)}})^{\times
s}}+O_{d,D,\ell}(p^{-1}),$

where $d_{j\ell}=\deg({\textbf{p}}_{\ell}-{\textbf{p}}_{j})$ and
${\textbf{p}}_{0}=\mathbf{0}$.

###### Proof.

By Proposition 3.2, we have

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{2^{s^{\prime}}}\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|f_{\ell}\right\|_{{{\textbf{c}}_{1}({\underline{h}})},\ldots,{{\textbf{c}}_{s}({\underline{h}})}}^{2^{s}}$

for some $s^{\prime}=O_{d,D,\ell}(1)$ and nonzero polynomials
${\textbf{c}}_{1},\ldots,{\textbf{c}}_{s}:{\mathbb{Z}}^{s^{\prime}}\to{\mathbb{Z}}^{\ell}$
that take the form

$\displaystyle{\textbf{c}}_{j}({\underline{h}})=\sum_{\begin{subarray}{c}{\underline{u}}\in
A_{d-1,s^{\prime}}\end{subarray}}c_{{\underline{u}}}({\textbf{a}}_{\ell(|{\underline{u}}|+1)}-{\textbf{a}}_{w_{j{\underline{u}}}(|{\underline{u}}|+1)}){\underline{h}}^{\underline{u}},$

where:

  1. (i)

the indices $w_{j{\underline{u}}}$ take value in the set $\\{0,\ldots,\ell\\}$
(with ${\textbf{a}}_{0(|{\underline{u}}|+1)}:=\mathbf{0}$) and have the
property $w_{j{\underline{u}}}=w_{j{\underline{u}}^{\prime}}$ whenever
$\textrm{supp}({\underline{u}})=\textrm{supp}({\underline{u}}^{\prime})$;

  2. (ii)

$c_{{\underline{u}}}$ is the coefficient of $n{\underline{h}}^{\underline{u}}$
in the multinomial expansion of
$(n+h_{1}+\cdots+h_{s^{\prime}})^{|{\underline{u}}|+1}$; in particular,
$c_{\underline{u}}$ is nonzero.

Let

$\displaystyle
G_{j}=\langle{\textbf{a}}_{\ell(|{\underline{u}}|+1)}-{\textbf{a}}_{w_{j{\underline{u}}}(|{\underline{u}}|+1)}:\
{\underline{u}}\in A_{d-1,s^{\prime}}\rangle.$

Proposition 4.2 then gives

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{O_{d,D,\ell}(1)}\leq\left\|f_{\ell}\right\|_{G_{1}^{\times
s^{\prime\prime}},\ldots,G_{s}^{\times
s^{\prime\prime}}}+O_{d,D,\ell}(p^{-1})$

for some $s^{\prime\prime}=O_{d,D,\ell}(1)$.

For each $j=1,\ldots,s$, the polynomials ${\textbf{c}}_{j}$ are nonzero, and
so there exists ${\underline{u}}\in{\mathbb{N}}_{0}^{s^{\prime}}$ for which
$0\leq w_{j{\underline{u}}}<\ell$. Since
$w_{j{\underline{u}}}=w_{j{\underline{u}}^{\prime}}$ for all
${\underline{u}}^{\prime}$ with the same support as ${\underline{u}}$, we can
assume that ${\underline{u}}$ satisfies $|{\underline{u}}|=d_{\ell
w_{j{\underline{u}}}}-1.$ Letting $w_{j}:=w_{j{\underline{u}}}$, we deduce
that ${\textbf{a}}_{\ell d_{\ell w_{j}}}-{\textbf{a}}_{w_{j}d_{\ell w_{j}}}\in
G_{j}$. Since ${\textbf{a}}_{\ell d_{\ell w_{j}}}-{\textbf{a}}_{w_{j}d_{\ell
w_{j}}}\in G_{j}$ for each $j=1,\ldots,s$, we deduce from the property (8)
that

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{O_{d,D,\ell}(1)}\\\
\leq\left\|f_{\ell}\right\|_{({\textbf{a}}_{\ell d_{\ell
w_{1}}}-{\textbf{a}}_{w_{1}d_{\ell w_{1}}})^{\times
s^{\prime\prime}},\ldots,({\textbf{a}}_{\ell d_{\ell
w_{s}}}-{\textbf{a}}_{w_{s}d_{\ell w_{s}}})^{\times
s^{\prime\prime}}}+O_{d,D,\ell}(p^{-1}).$

Given that $w_{1},\ldots,w_{s}\in\\{0,\ldots,\ell-1\\}$, each vector
${\textbf{a}}_{\ell d_{\ell 0}},{\textbf{a}}_{\ell d_{\ell
1}}-{\textbf{a}}_{1d_{\ell 1}},\ldots,{\textbf{a}}_{\ell
d_{\ell(\ell-1)}}-{\textbf{a}}_{(\ell-1)d_{\ell(\ell-1)}}$ appears at most
$s^{\prime\prime\prime}=ss^{\prime\prime}$ times in the norm above, and so we
can bound

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{p}}_{1}(n))\cdots
f_{\ell}({\textbf{x}}+{\textbf{p}}_{\ell}(n))\mathclose{}\right|^{O_{d,D,\ell}(1)}\\\
\leq\left\|f_{\ell}\right\|_{{\textbf{a}}_{\ell d_{\ell 0}}^{\times
s^{\prime\prime\prime}},({\textbf{a}}_{\ell d_{\ell 1}}-{\textbf{a}}_{1d_{\ell
1}})^{\times s^{\prime\prime\prime}},\ldots,({\textbf{a}}_{\ell
d_{\ell(\ell-1)}}-{\textbf{a}}_{(\ell-1)d_{\ell(\ell-1)}})^{\times
s^{\prime\prime\prime}}}+O_{d,D,\ell}(p^{-1}).$

The result follows upon relabelling $s^{\prime\prime\prime}$ as $s$. ∎

We now extend Proposition 4.3 to counting operators for polynomials
progressions twisted by products of dual functions, which will naturally
appear later. In order to do so, we need first the following lemma that allows
us to get rid of dual functions from the counting operator.

######  Proposition 4.4 (Removing dual functions, [8, Proposition 6.1]).

Let $d,D,L\in{\mathbb{N}}$ be integers,
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{L}\in{\mathbb{Z}}^{D}$ be direction
vectors and $q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials of degree at
most $d$. There exists a natural number $s=O_{d,L}(1)$ such that for all
1-bounded functions
$A:{\mathbb{F}}_{p}^{D}\times{\mathbb{F}}_{p}\to{\mathbb{C}}$ and
${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}$ with
${\mathcal{D}}_{j}\in{\mathfrak{D}}_{d}({\textbf{u}}_{j})$, we have

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}A({\textbf{x}},n)\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{j}q_{j}(n))\mathclose{}\right|^{2^{s}}\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{\textbf{x}}\operatorname*{\mathbb{E}}_{n}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}A({\textbf{x}},n+{\underline{\epsilon}}\cdot{\underline{h}})\mathclose{}\right|.$

We then combine Proposition 4.3 with Proposition 4.4, obtaining the following
generalisation of Proposition 4.3, which is a quantitative finite-field
version of [10, Proposition B.1].

######  Proposition 4.5 (Control by a single box norm II).

Let $d,D,\ell,L\in{\mathbb{N}}$. There exist $s\in{\mathbb{N}}$ with the
following property: for all

  * •

vectors ${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell}$,
${\textbf{u}}_{1},\ldots,{\textbf{u}}_{L}\in{\mathbb{Z}}^{D}$,

  * •

polynomials $p_{1},\ldots,p_{\ell}$, $q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ of
degree at most $d$ and coefficients $p_{j}(n)=\sum_{i=0}^{d}a_{ji}n^{i}$,

  * •

1-bounded functions
$f_{0},\ldots,f_{\ell}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$,

  * •

1-bounded functions ${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}$ satisfying
${\mathcal{D}}_{j}\in{\mathfrak{D}}_{d}({\textbf{u}}_{j})$,

we have the bound

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{j}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\leq\left\|f_{\ell}\right\|_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}}+O_{d,\ell,L}(p^{-1})$

for some vectors

(30) $\displaystyle{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}\in\\{a_{\ell
d_{j\ell}}{\textbf{v}}_{\ell}-a_{jd_{j\ell}}{\textbf{v}}_{j}:\
j=0,\ldots,\ell-1\\},$

where $d_{j\ell}=\deg({\textbf{v}}_{\ell}p_{\ell}-{\textbf{v}}_{j}p_{j})$ and
${\textbf{v}}_{0}=\mathbf{0},p_{0}=0$. Moreover, the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}$ are independent of
$f_{0},\ldots,f_{\ell}$.

Proposition (4.5) tells us that whenever our counting operator is twisted by
dual functions, we can safely ignore them and obtain a box norm control
involving the same directions as in Proposition 4.3 except that the degree of
such norm will be much higher.

We note that while Proposition 4.5 follows from Proposition 4.3, the values of
$s$ and the constants obtained in Proposition 4.5 do not depend on $D$. This
is because we could split ${\mathbb{F}}_{p}^{D}$ into cosets of the subspace
generated by
${\textbf{v}}_{1},\ldots,{\textbf{v}}_{\ell},{\textbf{u}}_{1},\ldots,{\textbf{u}}_{L}$,
which has dimension at most $\ell+L$, carry out the entire analysis there, and
then lift the result back to ${\mathbb{F}}_{p}^{D}$.

###### Proof.

Applying Proposition 4.4, we can find a natural number $s^{\prime}=O_{d,L}(1)$
such that

(31)
$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{j}q_{j}(n))\mathclose{}\right|^{2^{s^{\prime}}}\\\
\leq\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f_{j}({\textbf{x}}+{\textbf{p}}_{j}(n+{\underline{\epsilon}}\cdot{\underline{h}}))\mathclose{}\right|.$

If $d=1$ and $p_{j}(n):=a_{j1}n+a_{j0}$ for all $j=1,\ldots,\ell$, then the
right hand side of (31) equals

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f_{j}({\textbf{x}}+{\textbf{v}}_{j}(a_{j1}n+a_{j0}+a_{j1}{\underline{\epsilon}}\cdot{\underline{h}}))\mathclose{}\right|$
(32)
$\displaystyle=\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}\Delta_{{\textbf{v}}_{j}a_{j1}h_{1},\ldots,{\textbf{v}}_{j}a_{j1}h_{s^{\prime}}}f_{j}({\textbf{x}}+{\textbf{v}}_{j}(a_{j1}n+a_{j0}))\mathclose{}\right|$

Lemma 3.1 implies that (32) is bounded from above by

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}}\left\|\Delta_{{\textbf{v}}_{\ell}a_{\ell
1}h_{1},\ldots,{\textbf{v}}_{\ell}a_{\ell
1}h_{s^{\prime}}}f_{\ell}\right\|_{{\textbf{v}}_{\ell}a_{\ell
1},{\textbf{v}}_{\ell}a_{\ell
1}-{\textbf{v}}_{\ell}a_{11},\ldots,{\textbf{v}}_{\ell}a_{\ell
1}-{\textbf{v}}_{\ell}a_{(\ell-1)1}}.$

The inductive formula for box norms then gives

$\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}f_{j}({\textbf{x}}+{\textbf{v}}_{j}p_{j}(n))\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}({\textbf{x}}+{\textbf{u}}_{j}q_{j}(n))\mathclose{}\right|^{O_{d,\ell,L}(1)}\\\
\leq\left\|f_{\ell}\right\|_{{\textbf{v}}_{\ell}a_{\ell 1}^{\times
s^{\prime}+1},{\textbf{v}}_{\ell}a_{\ell
1}-{\textbf{v}}_{\ell}a_{11},\ldots,{\textbf{v}}_{\ell}a_{\ell
1}-{\textbf{v}}_{\ell}a_{(\ell-1)1}},$

completing the proof in the case $d=1$.

Suppose now that $d>1$. We first show that for almost all
${\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}$, the leading coefficients of
the polynomials

(33)
$\displaystyle\\{{\textbf{v}}_{\ell}p_{\ell}(n+\underline{1}\cdot{\underline{h}})-{\textbf{v}}_{j}p_{j}(n+{\underline{\epsilon}}\cdot{\underline{h}})\colon\;{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}},\
j\in\\{0,\ldots,\ell\\},\
(j,{\underline{\epsilon}})\neq(\ell,\underline{1})\\}$

are nonzero rational multiples of the leading coefficients of the polynomials

${\textbf{v}}_{\ell}p_{\ell},{\textbf{v}}_{\ell}p_{\ell}-{\textbf{v}}_{1}p_{1},\ldots,{\textbf{v}}_{\ell}p_{\ell}-{\textbf{v}}_{\ell-1}p_{\ell-1}.$

Indeed, to analyse what the leading coefficients of the polynomials in (33)
are, we need to consider four cases:

  1. (i)

${\textbf{v}}_{\ell}\neq{\textbf{v}}_{j}$;

  2. (ii)

${\textbf{v}}_{\ell}={\textbf{v}}_{j}$ and $a_{\ell d}\neq a_{jd}$;

  3. (iii)

${\textbf{v}}_{\ell}={\textbf{v}}_{j}$, $a_{\ell d}=a_{jd}$ and
${\underline{\epsilon}}=\underline{1}$;

  4. (iv)

${\textbf{v}}_{\ell}={\textbf{v}}_{j}$, $a_{\ell d}=a_{jd}$ and
${\underline{\epsilon}}\neq\underline{1}$.

In the first three cases, the leading coefficient is
${\textbf{v}}_{\ell}a_{\ell d_{\ell j}}-{\textbf{v}}_{j}a_{jd_{\ell j}}$,
where $d_{\ell j}:=\deg({\textbf{v}}_{\ell}p_{\ell}-{\textbf{v}}_{j}p_{j})$
equals $d$ in the first two cases and $0<d_{\ell j}<d$ in the third one. Thus,
the leading coefficient of
${\textbf{v}}_{\ell}p_{\ell}(n+\underline{1}\cdot{\underline{h}})-{\textbf{v}}_{j}p_{j}(n+{\underline{\epsilon}}\cdot{\underline{h}})$
in these cases is the same as the leading coefficient of
${\textbf{v}}_{\ell}p_{\ell}-{\textbf{v}}_{j}p_{j}$. In the last case, the
leading coefficient takes the form
$c_{j{\underline{\epsilon}}}({\underline{h}}){\textbf{v}}_{\ell}$ for

$c_{j{\underline{\epsilon}}}=a_{\ell
d}d(\underline{1}-{\underline{\epsilon}})\cdot{\underline{h}}+a_{\ell(d-1)}-a_{j(d-1)},$

which is nonzero for all ${\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}$
except a subset
$A_{j{\underline{\epsilon}}}\subset{\mathbb{F}}_{p}^{s^{\prime}}$ of size
$|A_{j{\underline{\epsilon}}}|\leq p^{s^{\prime}-1}$.

By Proposition 4.3, there exist $s\in{\mathbb{N}}$ and vectors

(34)
$\displaystyle{\textbf{b}}_{1}({\underline{h}}),\ldots,{\textbf{b}}_{s}({\underline{h}})\in$
$\displaystyle\\{a_{\ell d_{\ell j}}{\textbf{v}}_{\ell}-a_{jd_{\ell
j}}{\textbf{v}}_{j}\colon\ j\in\\{0,\ldots,\ell-1\\}\\}$
$\displaystyle\cup\\{c_{j{\underline{\epsilon}}}({\underline{h}}){\textbf{v}}_{\ell}\colon\
j\in\\{0,\ldots,\ell\\},\
{\underline{\epsilon}}\neq\underline{1},\;{\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}\\}$

such that for every ${\underline{h}}\in{\mathbb{F}}_{p}^{s^{\prime}}$, we have

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}}}\operatorname*{\mathbb{E}}_{n\in{\mathbb{F}}_{p}}f_{0}({\textbf{x}})\cdot\prod_{j=1}^{\ell}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f_{j}({\textbf{x}}+{\textbf{p}}_{j}(n+{\underline{\epsilon}}\cdot{\underline{h}}))\mathclose{}\right|^{O_{d,\ell,L}(1)}\leq\left\|f_{\ell}\right\|_{{\textbf{b}}_{1}({\underline{h}}),\ldots,{\textbf{b}}_{s}({\underline{h}})}+O_{d,\ell,L}(p^{-1}).$

As long as $c_{j{\underline{\epsilon}}}({\underline{h}})\neq 0$, the vector
$c_{j{\underline{\epsilon}}}({\underline{h}}){\textbf{v}}_{\ell}$ generates
the same subgroup inside ${\mathbb{F}}_{p}^{D}$ as $a_{\ell d_{\ell
0}}{\textbf{v}}_{\ell}$ (here $d_{\ell 0}=\deg p_{\ell}$), and so for every
${\underline{h}}\in{\mathbb{N}}^{s^{\prime}}\setminus{A}$, where
$A=\bigcup_{(j,{\underline{\epsilon}})\neq(\ell,\underline{1})}A_{j{\underline{\epsilon}}}$,
we can replace all the vectors ${\textbf{b}}_{i}$ of the form
$c_{j{\underline{\epsilon}}}({\underline{h}}){\textbf{v}}_{\ell}$ by vectors
$a_{\ell d_{\ell 0}}{\textbf{v}}_{\ell}$. The result follows from this fact,
the bound $|A|\ll_{d,\ell}p^{s^{\prime}-1}$ on the exceptional set, the
identity (31), and the last inequality above. ∎

##  5\. Passing to a Gowers norm control

So far, we have shown in Proposition 3.2 via a PET argument that a counting
operator for a multidimensional progression is controlled by an average of box
norms, and then we used concatenation results from Section 2.2 to show in
Propositions 4.3 and 4.5 that we can in fact control the counting operator by
a single box norm. While aesthetically pleasing, this result alone does not
give us any immediate advantage for proving Theorem 1.2 due to the lack of a
usable inverse theorem for general box norms. Its utility, however, lies in
the fact that it can be used as an intermediate step in establishing control
by a proper Gowers norm in Theorem 1.3 as long as the polynomials are pairwise
independent. Passing from a box norm control to a Gowers norm control can be
accomplished via a “box norm smoothing” argument, developed by Frantzikinakis
and the author in the ergodic setting [10, 11]. This is the first appearance
of this argument in the combinatorial setting, and before we present it in the
full generality, we illustrate the underlying idea with two examples.

###  5.1. Norm smoothing for a linearly independent progression

In this section, let

$\displaystyle\Lambda(f_{0},f_{1},f_{2})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n^{2})f_{2}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))$

be the counting operator for the progression

(35)
$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n^{2},\;{\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n).$

Our goal is to sketch a proof, modulo some simplifying assumptions, that for
some absolute $c>0$, we obtain the degree 1 control

(36)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2})|\ll\left\|f_{2}\right\|_{{\textbf{v}}_{2}}^{c}+p^{-c}.$

This is a special case of Theorem 1.3. The argument for this progression is
strictly simpler than the argument in the general case, as we use the fact
that it is a 3-point pattern involving linearly independent polynomials.
However, the argument showcases the key idea that allows us to prove Theorem
1.3 for arbitrary progressions along pairwise independent polynomials.

First, Proposition 4.3 gives $s\in{\mathbb{N}}$ such that for any 1-bounded
$f_{0},f_{1},f_{2}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have the bound

(37)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2})|^{O(1)}\leq\left\|f_{2}\right\|_{{\textbf{v}}_{2}^{\times
s},({\textbf{v}}_{2}-{\textbf{v}}_{1})^{\times s}}+O(p^{-1}).$

For the purpose of this example only, suppose that we can replace the
complicated box norm $\left\|f_{2}\right\|_{{\textbf{v}}_{2}^{\times
s},({\textbf{v}}_{2}-{\textbf{v}}_{1})^{\times s}}$ by a simpler box norm
$\left\|f_{2}\right\|_{{\textbf{v}}_{2}-{\textbf{v}}_{1}}$, so that the
following bound

(38)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2})|^{O(1)}\leq\left\|f_{2}\right\|_{{\textbf{v}}_{2}-{\textbf{v}}_{1}}+O(p^{-1})$

holds instead. Fix $f_{0},f_{2},f_{2}$. We aim to illustrate how we can pass
from controlling the counting operator by the norm
$\left\|f_{2}\right\|_{{\textbf{v}}_{2}-{\textbf{v}}_{1}}$ as in (38) to a
control by the norm $\left\|f_{2}\right\|_{{\textbf{v}}_{2}}$ as in (36). Our
argument follows a two-step ping-pong strategy. First, we show that the
counting operator is controlled by the norm
$\left\|f_{1}\right\|_{{\textbf{v}}_{1}}$. Then, we use this auxiliary control
to deduce that the operator is controlled by the norm
$\left\|f_{2}\right\|_{{\textbf{v}}_{2}}$. Thus, we pass information first
from $f_{2}$ to $f_{1}$ and then back from $f_{1}$ to $f_{2}$, like a tennis
ball from one end of a table to another; therefore we call these steps ping
and pong respectively.

Let

$\displaystyle\tilde{f}_{2}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{2}(n^{2}+n))\overline{f_{1}}({\textbf{x}}+{\textbf{v}}_{1}n^{2}-{\textbf{v}}_{2}(n^{2}+n)),$

so that
$\Lambda(f_{0},f_{1},f_{2})=\operatorname*{\mathbb{E}}_{\textbf{x}}f_{2}({\textbf{x}})\overline{\tilde{f_{2}}({\textbf{x}})}$.
Letting $\delta=|\Lambda(f_{0},f_{1},f_{2})|$, we deduce from Lemma A.1 that

$\displaystyle\Lambda(f_{0},f_{1},\tilde{f}_{2})\geq\delta^{2}.$

We then use the control (38) to deduce that, upon assuming that $\delta\gg
p^{-c}$ for a sufficiently small $c>0$, we have

$\displaystyle\left\|\tilde{f}_{2}\right\|_{{\textbf{v}}_{2}-{\textbf{v}}_{1}}\gg\delta^{O(1)}.$

From Lemma A.1 we obtain

$\Lambda(f_{0},f_{1},\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1})))\\\
=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n^{2})\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1}))({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))\gg\delta^{O(1)}.$

At this point, we crucially observe that the function
$\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1}))$
is invariant under shifts by ${\textbf{v}}_{2}-{\textbf{v}}_{1}$, implying
that

$\displaystyle\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1}))({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))=\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1}))({\textbf{x}}+{\textbf{v}}_{1}(n^{2}+n)).$

Using this identity, we rewrite the inequality above as

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n^{2})\operatorname*{\mathbb{E}}(\tilde{f}_{2}|({\textbf{v}}_{2}-{\textbf{v}}_{1}))({\textbf{x}}+{\textbf{v}}_{1}(n^{2}+n))\gg\delta^{O(1)}.$

Importantly, this inequality involves the counting operator for the
progression

(39)
$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n^{2},\;{\textbf{x}}+{\textbf{v}}_{1}(n^{2}+n),$

which is essentially single dimensional because both polynomials run along the
same direction ${\textbf{v}}_{1}$. It follows from [24, Theorem 2.1] that the
counting operator for (39) is controlled by
$\left\|f_{1}\right\|_{{\textbf{v}}_{1}}$ with power-saving bounds. We
therefore deduce, upon assuming once more that the constant $c>0$ is
sufficiently small, that
$\left\|f_{1}\right\|_{{\textbf{v}}_{1}}\gg\delta^{O(1)}$. As promised, we
have passed from a control of the original counting operator by the norm
$\left\|\tilde{f}_{2}\right\|_{{\textbf{v}}_{2}-{\textbf{v}}_{1}}$ to the
control by a norm $\left\|f_{1}\right\|_{{\textbf{v}}_{1}}$ of the form

(40)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2})|\ll\left\|f_{1}\right\|_{{\textbf{v}}_{1}}^{c^{\prime}}+p^{-c^{\prime}}$

for some absolute $c^{\prime}>0$. This completes the ping step.

At this point, we could decompose
$f_{1}=\operatorname*{\mathbb{E}}(f_{1}|{\textbf{v}}_{1})+(f_{1}-\operatorname*{\mathbb{E}}(f_{1}|{\textbf{v}}_{1}))$,
deduce from the newly established norm control (40) that the second term
contributes at most $O(p^{-c^{\prime}})$ to $\Lambda(f_{0},f_{1},f_{2})$, and
use the ${\textbf{v}}_{1}$-invariance of
$\operatorname*{\mathbb{E}}(f_{1}|{\textbf{v}}_{1})$ to conclude that

$\displaystyle\Lambda(f_{0},f_{1},f_{2})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}(f_{0}\operatorname*{\mathbb{E}}(f_{1}|{\textbf{v}}_{1}))({\textbf{x}})f_{2}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))+O(p^{-c^{\prime}}).$

The claim (36) would then follow directly from [24, Theorem 2.1]. However,
this argument does not work for more general progressions, so we now present
an alternative method that is more readily generalisable.

We argue similarly as in the ping step, but this time we will pass from a
control by $\left\|f_{1}\right\|_{{\textbf{v}}_{1}}$ to a control by
$\left\|f_{2}\right\|_{{\textbf{v}}_{2}}$; this is the essence of the pong
step. Letting

$\displaystyle\tilde{f}_{1}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{1}n^{2})\overline{f_{2}}({\textbf{x}}+{\textbf{v}}_{2}(n+n^{2})-{\textbf{v}}_{1}n^{2})$

we deduce from Lemma A.1 that

$\displaystyle\Lambda(f_{0},\tilde{f}_{1},{f}_{2})\geq\delta^{2}.$

Using this fact and the auxiliary control (40) for $\tilde{f}_{1}$ in place of
$f_{1}$, we infer that
$\left\|\tilde{f}_{1}\right\|_{{\textbf{v}}_{1}}\gg\delta^{O(1)}$ (again,
under the assumption that $\delta\gg p^{-c}$ for sufficiently small $c>0$). An
application of Lemma A.1 gives

$\displaystyle\Lambda(f_{0},\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1}),f_{2})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1})({\textbf{x}}+{\textbf{v}}_{1}n^{2})f_{2}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))\gg\delta^{O(1)}.$

Crucially, the function
$\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1})$ is
${\textbf{v}}_{1}$-invariant, implying that

$\displaystyle\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1})({\textbf{x}}+{\textbf{v}}_{1}n^{2})=\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1})({\textbf{x}}).$

Letting
$g=f_{0}\cdot\operatorname*{\mathbb{E}}(\tilde{f}_{1}|{\textbf{v}}_{1})$, we
have thus showed that

$\displaystyle\operatorname*{\mathbb{E}}_{{\textbf{x}},n}g({\textbf{x}})f_{2}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))\gg\delta^{O(1)}.$

From this and [24, Theorem 2.1], we deduce that
$\left\|f_{2}\right\|_{{\textbf{v}}_{2}}\gg\delta^{O(1)}$. The claim follows.

The argument becomes more complicated when the progression has length greater
than 3, the polynomials are pairwise independent rather than linearly
independent, and we do not start with the simplifying assumption (38) instead
of the much weaker but more accurate control (37). The next example will
address the aforementioned technicalities and give a sense of changes that
have to be made to tackle the more general case.

###  5.2. Norm smoothing for a linearly dependent progression

In this section, let

$\displaystyle\Lambda(f_{0},f_{1},f_{2},f_{3})=\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0}({\textbf{x}})f_{1}({\textbf{x}}+{\textbf{v}}_{1}n)f_{2}({\textbf{x}}+{\textbf{v}}_{2}n^{2})f_{3}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))$

be the counting operator for the progression

(41)
$\displaystyle{\textbf{x}},\;{\textbf{x}}+{\textbf{v}}_{1}n,\;{\textbf{x}}+{\textbf{v}}_{2}n^{2},\;{\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n).$

Our goal is to sketch the proof of the following special case of Theorem 1.3.

######  Proposition 5.1.

There exists $s\in{\mathbb{N}}$ and $c>0$ with the following property: for
every $D\in{\mathbb{N}}$ and 1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have the
bound

(42)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|\ll\left\|f_{3}\right\|_{U^{s}({\textbf{v}}_{3})}^{c}+p^{-c}.$

In Proposition 5.3, we upgrade Proposition 5.1 to one in which $\Lambda$ is
controlled by a Gowers norm of any of $f_{1},f_{2},f_{3}$.

The progression (41) is no longer linearly independent, and so while proving
Proposition 5.1, we cannot for instance invoke the results of Peluse [24]
after completing the ping step. For that reason, the 4-point pattern (41)
captures better the difficulties present in the general case than the 3-point
progression (35) while still offering the benefits of concreteness.

Our starting point is Proposition 4.3 which gives $s_{0},s_{1}\in{\mathbb{N}}$
and $c>0$ with the property that for any 1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$, we have

(43)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|\leq\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times s_{1}}}^{c}+O(p^{-c}).$

In fact, in the statement of Proposition 4.3, we have $s_{0}=s_{1}$ (which we
can always assume by taking $s=\max(s_{0},s_{1})$ and using the monotonicity
of box norms), but it will be more illustrative to describe these two numbers
using separate labels. Proposition 5.1 will follow by an iterated application
of the following result, which enables us to replace the vectors
${\textbf{v}}_{3}-{\textbf{v}}_{2}$ one by one with (possibly many copies of)
${\textbf{v}}_{3}$. In doing so, we “smooth out” or “uniformise” the original
norm until we reach one that only involves a large number of
${\textbf{v}}_{3}$.

######  Proposition 5.2 (Norm smoothing for (41)).

Let $D,s_{0},s_{1}\in{\mathbb{N}}$ and $c>0$ be such that (43) holds for all
1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$. Then there
exist $s^{\prime}=O_{s_{0},s_{1}}(1)$ and $c^{\prime}\gg_{c,s_{0},s_{1}}1$
such that the bound

(44)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|\ll_{c,s_{0},s_{1}}\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s^{\prime}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1}}^{c^{\prime}}+p^{-c^{\prime}}$

holds for all 1-bounded functions
$f_{0},f_{1},f_{2},f_{3}:{\mathbb{F}}_{p}^{D}\to{\mathbb{C}}$.

It is crucial that the norm in (43) involves the direction
${\textbf{v}}_{3}-{\textbf{v}}_{2}$ rather than ${\textbf{v}}_{2}$, otherwise
the argument would not work.

###### Proof of Proposition 5.2.

Let $\delta=|\Lambda(f_{0},f_{1},f_{2},f_{3})|$, and assume without loss of
generality that $\delta\gg_{c,s_{0},s_{1}}p^{-c^{\prime}}$ for a sufficiently
small $0<c^{\prime}\leq c/2$. Like in Section 5.1, our argument follows a two-
step ping-pong strategy. Starting with the assumption that we can control the
counting operator by the norm $\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times s_{1}}}$ of $f_{3}$, we
first show that the counting operator is also controlled by the norm
$\left\|f_{2}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1},{\textbf{v}}_{2}^{\times s_{2}}}$ of $f_{2}$ for some
$s_{2}=O_{s_{0},s_{1}}(1)$. Then, we use this auxiliary control to deduce that
the operator is controlled by the norm
$\left\|f_{3}\right\|_{{\textbf{v}}_{3}^{\times
s^{\prime}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times s_{1}-1}}$ for some
$s^{\prime}=O_{s_{0},s_{1}}(1)$. As before, we call these two steps ping and
pong respectively. In what follows, we allow all the quantities to depend on
$c,s_{0},s_{1}$.

Step 1 (ping): Obtaining control by a norm of $f_{2}$.

Set

$\tilde{f}_{3}({\textbf{x}})=\operatorname*{\mathbb{E}}_{n}\overline{f_{0}}({\textbf{x}}-{\textbf{v}}_{3}(n^{2}+n))\overline{f_{1}}({\textbf{x}}+{\textbf{v}}_{1}n-{\textbf{v}}_{3}(n^{2}+n))\overline{f_{2}}({\textbf{x}}+{\textbf{v}}_{2}n^{2}-{\textbf{v}}_{3}(n^{2}+n)),$

so that

$\displaystyle\Lambda(f_{0},f_{1},f_{2},\tilde{f}_{3})\geq\delta^{2}.$

by Lemma A.1. Thus, at the expense of losing an exponent, we have replaced an
arbitrary function $f_{3}$ in the counting operator by the structured term
$\tilde{f}_{3}$. We deduce from the inequality (43) and the assumption
$\delta\gg p^{-c/2}$ that
$\left\|\tilde{f}_{3}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times s_{1}}}\gg\delta^{O(1)}$.

Our goal is to get to the point where we can apply the inverse theorem for
degree 1 seminorm so as to correlate $\tilde{f}_{3}$, or some derivate of it,
with a $({\textbf{v}}_{3}-{\textbf{v}}_{2})$-invariant function. This is
accomplished by Corollary A.3(i), which gives that

$\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})\cdot
f_{1,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{1}n)\cdot
f_{2,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))\mathclose{}\right|$

is bounded from below by $\Omega(\delta^{O(1)})$ for

$\displaystyle
f_{j,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})=\Delta_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}({\textbf{x}})\quad\mathrm{for}\quad
j=0,1,2$

and 1-bounded, $({\textbf{v}}_{3}-{\textbf{v}}_{2})$-invariant functions
$u_{{\underline{h}},{\underline{h}}^{\prime}}$. This invariance property
yields the identity

(45)
$u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{3}(n^{2}+n))=u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))$

which allows us to rewrite the average over
${\underline{h}},{\underline{h}}^{\prime}$ as

$\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})\cdot
f_{1,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{1}n)\cdot
f_{2,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))\mathclose{}\right|.$

Thus, we have replaced ${\textbf{v}}_{3}$ in the counting operator by
${\textbf{v}}_{2}$. Importantly, in the new counting operator, both quadratic
polynomials $n^{2}$ and $n^{2}+n$ lie along the same direction
${\textbf{v}}_{2}$. By the pigeonhole principle, we can find a set
$B\subset{\mathbb{F}}_{p}^{2(s_{0}+s_{1}-1)}$ of size
$|B|\gg\delta^{O(1)}p^{2(s_{0}+s_{1}-1)}$ such that

$\displaystyle\mathopen{}\left|\operatorname*{\mathbb{E}}_{{\textbf{x}},n}f_{0,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}})\cdot
f_{1,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{1}n)\cdot
f_{2,{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}n^{2})\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}({\textbf{x}}+{\textbf{v}}_{2}(n^{2}+n))\mathclose{}\right|\gg\delta^{O(1)}$

for each $({\underline{h}},{\underline{h}}^{\prime})\in B$. It follows from
Proposition 4.3 that there exists $s_{2}\in{\mathbb{N}}$ such that

$\displaystyle\operatorname*{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{F}}_{p}^{s_{0}+s_{1}-1}}1_{B}({\underline{h}},{\underline{h}}^{\prime})\cdot\left\|f_{2,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{U^{s_{2}}({\textbf{v}}_{2})}\gg\delta^{O(1)}.$

Extending by nonnegativity to all
$({\underline{h}},{\underline{h}}^{\prime})$, invoking the definition of
$f_{2,{\underline{h}},{\underline{h}}^{\prime}}$, changing variables to
replace ${\underline{h}}-{\underline{h}}^{\prime}$ by ${\underline{h}}$ and
using the induction formula for box norms together with the Hölder inequality,
we deduce that

$\displaystyle\left\|f_{2}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1},{\textbf{v}}_{2}^{\times s_{2}}}\gg\delta^{O(1)}.$

Hence, the norm $\left\|f_{2}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1},{\textbf{v}}_{2}^{\times s_{2}}}$ controls the counting operator
$\Lambda(f_{0},f_{1},f_{2},f_{3})$ in the sense that

(46)
$\displaystyle|\Lambda(f_{0},f_{1},f_{2},f_{3})|^{O(1)}\ll\left\|f_{2}\right\|_{{\textbf{v}}_{3}^{\times
s_{0}},({\textbf{v}}_{3}-{\textbf{v}}_{2})^{\times
s_{1}-1},{\textbf{v}}_{2}^{\times s_{2}}}+p^{-1}$

for all 1-bounded functions $f_{0},f_{1},f_{2},f_{3}$.

Starting with a control of $\Lambda(f_{0},f_{1},f_{2},f_{3})$ by a norm of
$f_{3}$, we have arrived at a control by a norm of $f_{2}$; in this sense we
passed information from $f_{3}$ to $f_{2}$. The bound (46) is not particularly
useful as an independent result because of the $s_{0}$ vectors