Approximate quadratic varieties
Luka Milićević†

00footnotetext: † Mathematical Institute of the Serbian Academy of Sciences
and Arts
† Email<EMAIL_ADDRESS>
  * Abstract

a A classical result in additive combinatorics, which is a combination of
Balog-Szemerédi-Gowers theorem and a variant of Freiman’s theorem due to
Ruzsa, says that if a subset $A$ of $\mathbb{F}_{p}^{n}$ contains at least
$c|A|^{3}$ additive quadruples, then there exists a subspace $V$, comparable
in size to $A$, such that $|A\cap V|\geq\Omega_{c}(|A|)$. Motivated by the
fact that higher order approximate algebraic structures play an important role
in the theory of uniformity norms, it would be of interest to find higher
order analogues of the mentioned result.
a In this paper, we study a quadratic version of the approximate property in
question, namely what it means for a set to be an approximate quadratic
variety. It turns out that information on the number of additive cubes, which
are 8-tuples of the form $(x,x+a,x+b,x+c,x+a+b,x+a+c,x+b+c,x+a+b+c)$, in a set
is insufficient on its own to guarantee quadratic structure, and it is
necessary to restrict linear structure in a given set, which is a natural
assumption in this context. With this in mind, we say that a subset $V$ of a
finite vector space $G$ is a _$(c_{0},\delta,\varepsilon)$ -approximate
quadratic variety_ if $|V|=\delta|G|$,
$\|\mathbbm{1}_{V}-\delta\|_{\mathsf{U}^{2}}\leq\varepsilon$ and $V$ contains
at least $c_{0}\delta^{7}|G|^{4}$ additive cubes. Our main result is the
structure theorem for approximate quadratic varieties, stating that such a set
has a large intersection with an exact quadratic variety of comparable size.

## §1 Introduction

Approximate algebraic structures are a major topic of study in additive
combinatorics. First result concerning such structures, and a cornerstone of
the area, is Freiman’s theorem [7]. As we shall work in vector spaces over a
fixed prime field $\mathbb{F}_{p}$ in this paper, we begin by recalling an
analogue of Freiman’s theorem in that setting, due to Ruzsa [22].

######  Theorem 1.1 (Ruzsa [22]).

Let $G$ be a finite-dimensional vector space over a prime field
$\mathbb{F}_{p}$. Suppose that $A\subseteq G$ satisfies $|A+A|\leq
K|A|$.111This condition is called _small doubling_ and set obeying this
condition are traditionally called _approximate groups_. Then there exists a
subspace $U\leq G$ of size $|U|\leq O_{K}(|A|)$ such that $A\subseteq U$.

Let us record here a variant of that theorem which has a condition involving
additive quadruples, which are $(x,y,z,w)$ such that $x+y=z+w$, instead of the
small doubling assumption. The variant is obtained after an application of
Balog-Szemerédi-Gowers theorem [3, 8], and appears more naturally in the
context of higher order Fourier analysis.

######  Theorem 1.2.

Let $G$ be a finite-dimensional vector space over a prime field
$\mathbb{F}_{p}$. Suppose that $A\subseteq G$ and that $A$ has at least
$c|A|^{3}$ additive quadruples. Then there exists a subspace $U\leq G$ of size
$|U|\leq O_{c}(|A|)$ such that $|U\cap A|\geq\Omega_{c}(|A|)$.

We may think of the sets satisfying the assumptions of Theorem 1.2 as
_approximate cosets_ (we choose this wording as the term _approximate
subgroup_ has a standard meaning of having a small doubling). The way such a
result is relevant in higher order Fourier analysis is via following
consequence, which was proved by Gowers and plays an important role in his
proof of an inverse theorem for $\mathsf{U}^{3}$ uniformity norm [8] (we shall
recall the definition of this norm slightly later).

######  Theorem 1.3 (Gowers [8]).

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_{p}$. Let
$A\subseteq G$ be a subset and let $\phi\colon A\to H$ be a map which
_respects_ at least $c|G|^{3}$ additive quadruples in $A$, meaning that
$\phi(x)+\phi(y)=\phi(z)+\phi(w)$ holds for at least $c|G|^{3}$ quadruples
$(x,y,z,w)\in A^{4}$ such that $x+y=z+w$. Then there exists an affine map
$\Phi\colon G\to H$ such that $\phi=\Phi$ holds for at least $\Omega_{c}(|G|)$
points in $A$.

Strictly speaking, Gowers obtained Theorem 1.3 in the case of cyclic groups,
but the same proof works in $\mathbb{F}_{p}^{n}$. We may think of this theorem
as a structure theorem for approximate affine homomorphisms. This is an
approximate structure in the context of maps between vector spaces.

Finally, let us mention another approximate algebraic structure. We have
already mentioned in passing the uniformity norms, whose definition we now
recall. Firstly, we recall that the _discrete multiplicative derivative
operator_ ${\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a}$ for shift $a\in G$ is defined by
${\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a}f(x)=f(x+a)\overline{f(x)}$ for functions $f\colon
G\to\mathbb{C}$.
Let $f\colon G\to\mathbb{C}$ be a function. The _Gowers uniformity norm_
$\|f\|_{\mathsf{U}^{k}}$, defined by Gowers in [9], is given by the formula

$\|f\|_{\mathsf{U}^{k}}=\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,a_{1},\dots,a_{k}}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a_{1}}\dots\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}\,_{a_{k}}f(x)\Big{)}^{1/2^{k}}.$

These norms are defined by a combinatorial average, which in the case of
$\mathsf{U}^{3}$ norm is taken over additive cubes, which will be crucial for
the rest of the paper. To be precise, an _additive cube_ is an 8-tuple of the
form

$\Big{(}x,x+a,x+b,x+c,x+a+b,x+a+c,x+b+c,x+a+b+c\Big{)}$

where $x,a,b,c\in G$ are any 4 elements of $G$.

In his inverse theorem for $\mathsf{U}^{3}$ norm, Gowers obtained a partial
description of functions with large value of the norm, which was sufficient
for the purposes of proving Szemerédi’s theorem for arithmetic progressions of
length 4. A qualitatively optimal inverse theorem was later obtained by Green
and Tao [14] for groups of odd order. In the vector space case their result is
the following.

######  Theorem 1.4 (Green and Tao [14]).

Suppose that $f\colon G\to\mathbb{D}$ satisfies $\|f\|_{\mathsf{U}^{3}}\geq
c_{0}$ and assume $p\geq 3$. Then there exists a quadratic polynomial $q\colon
G\to\mathbb{F}_{p}$ such that
$\Big{|}\mathop{\mathbb{E}}_{x}f(x)\exp\Big{(}\frac{2\pi
iq(x)}{p}\Big{)}\Big{|}\geq\Omega_{c_{0}}(1)$.

We may thus think of functions with large $\mathsf{U}^{3}$ norm as approximate
quadratic phases.

When it comes to higher order analogues of the results above, we have inverse
theorems for $\mathsf{U}^{k}$ norms, for higher $k$. These were proved by
Green, Tao and Ziegler [17] for cyclic groups, by Bergelson, Tao and Ziegler
[4] for finite vector spaces over prime fields of sufficiently large
characteristic and Tao and Ziegler [25] for all finite vector spaces. In
particular, in vector spaces in the case of high characteristic where $p\geq
k$, Theorem 1.4 of Green and Tao generalizes to the following result.

######  Theorem 1.5 (Bergelson, Tao, Ziegler [4]).

Suppose that $f\colon G\to\mathbb{D}$ satisfies $\|f\|_{\mathsf{U}^{k}}\geq
c_{0}$ and assume $p\geq k$. Then there exists a degree $k-1$ polynomial
$q\colon G\to\mathbb{F}_{p}$ such that
$\Big{|}\mathop{\mathbb{E}}_{x}f(x)\exp\Big{(}\frac{2\pi
iq(x)}{p}\Big{)}\Big{|}\geq\Omega_{c_{0}}(1)$.

Similarly to Theorem 1.4, we may think of this theorem as a structure theorem
for approximate polynomial forms.

Returning now to Theorem 1.3, we remark that there are satisfactory higher
order generalizations of approximate affine homomorphisms, namely approximate
polynomials and their multilinear variant, Freiman multihomorphisms, whose
definitions we now recall.
Let $G$ and $H$ be two abelian groups, let $A\subseteq G$ be a subset and let
$F\colon A\to H$ be a map. Similarly to
${\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a}$, let us write $\Delta_{a}F$ for the map from $A\cap A-a$
to $H$ given by $x\mapsto F(x+a)-F(x)$. We say that $F$ is an _$\varepsilon$
-approximate polynomial of degree at most $d$_ if
$\Delta_{a_{1}}\dots\Delta_{a_{d+1}}F(x)=0$ holds for at least
$\varepsilon|G|^{d+2}$ choices of $(d+2)$-tuples $(a_{1},\dots,a_{d+1},x)\in
G^{d+2}$ (and all $2^{k+1}$ resulting arguments of $F$ lie in its domain).
On the other hand, a _Freiman multihomomorphism of order $d$_ is a map
$\Phi\colon A\to H$, where now $A\subseteq G^{d}$, such that $\Phi$ is a
Freiman homomorphism in each principal direction, namely for each direction
$i\in[d]$, and each element $(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{d})$ of
$G_{1}\times\cdots\times G_{i-1}\times G_{i+1}\times\cdots\times G_{d}$, the
map that sends each $y_{i}$ such that $(x_{1},\dots,$ $x_{i-1},$ $y_{i},$
$x_{i+1},\dots,$ $x_{d})\in A$ to $\phi(x_{1},\dots,$ $x_{i-1},$ $y_{i},$
$x_{i+1},\dots,$ $x_{d})$ respects all additive quadruples.
Manners [20] proved a structure theorem for approximate polynomials over
cyclic groups and Gowers and the author [11] proved a structure theorem for
Freiman multihomomorphisms in finite vector spaces. Furthermore, in the same
paper, Gowers and the author obtained a structure theorem for approximate
polynomials in finite vector spaces in the case of high characteristic.
Our goal in this paper is to make a step towards completing the picture and
study the quadratic analogue of Theorem 1.2. Table 1 summarizes the discussion
of various approximate algebraic structures of linear and higher order.

Degree | Sets | Maps $G\to H$ | Forms $G\to\mathbb{F}_{p}$
---|---|---|---
Linear | Theorem 2 | Theorem 3 | Inverse theorem for $\|\cdot\|_{\mathsf{U}^{2}}$ norm
Higher | ?? |  | Structure theorem for
---
approximate polynomials and
Freiman multihomorphisms
Inverse theorem for $\|\cdot\|_{\mathsf{U}^{k}}$ norm
Table 1: Various approximate algebraic structures.

Returning to Theorem 1.2, we may phrase the assumption on the number of
additive quadruples as follows. Let $A$ be a subset of $G$ and let $x,a,b\in
G$ be chosen uniformly and independently at random. Then

$\mathbb{P}(x+a+b\in A|x+a,x+b,x\in A)=\frac{\mathbb{P}(x+a+b,x+a,x+b,x\in
A)}{\mathbb{P}(x+a,x+b,x\in A)}=\frac{Q}{|A|^{3}},$

where $Q$ is the number of additive quadruples in $A$. Thus, $A$ has at least
$c_{0}|A|^{3}$ additive quadruples if and only the probability of ’completing
an additive quadruple’ $\mathbb{P}(x+a+b\in A|x+a,x+b,x\in A)$ is at least
$c_{0}$. In other words, $A$ is approximately closed under the relevant
operation.

In order to find a quadratic generalization, it is natural to count the
additive cubes. Observe that if $V=\\{x\in G\colon q(x)=0\\}$ is a quadratic
variety defined by a quadratic map $q\colon G\to\mathbb{F}_{p}^{r}$, then $V$
is closed under ’completing additive cubes’ due to the following identity

$\phi(x+a+b+c)=\phi(x+a+b)+\phi(x+a+c)+\phi(x+b+c)-\phi(x+a)-\phi(x+b)-\phi(x+c)+\phi(x).$

However, this time being approximately closed under completing additive cubes
is no longer sufficient on its own to guarantee that $V$ has any quadratic
structure at all. Consider the following example. Let $G=U\oplus T$ be a
direct sum and let $\pi\colon G\to T$ be the projection onto $T$ associated
with this direct sum. Let $S$ be a Sidon subset (a set without non-trivial
additive quadruples) of $T$ and set $V=U+S$. Observe that if
$x+a+b,x+a,x+b,x\in V$ then we have $\pi(x+a+b)+\pi(x)=\pi(x+a)+\pi(x+b)$,
and, since $S$ is Sidon, this means that $\pi(x)\in\\{\pi(x+a),\pi(x+b)\\}$ so
$a\in U$ or $b\in U$. Suppose now that 7 points
$x,x+a,x+b,x+c,x+a+b,x+a+c,x+b+c$ all belong to $V$. If $a,b,c\in U$, then
$x+a+b+c\in V$. Otherwise, suppose without loss of generality that $a\notin
U$. Observation about additive quadruples in $V$ implies that $b,c\in U$, so
$x+a+b+c\in x+a+U\subseteq V$, so actually $V$ is closed under completing
additive cubes.
The set $V$ above has no quadratic structure, and has some, but not
significant linear structure. Since our aim is to obtain a purely quadratic
structure theorem, we impose further condition on $V$ and assume that it has
negligible linear structure. This leads us to the following definition,
formulated by Gowers (personal communication), which also forbids the example
above.

######  Definition 1.6.

Let $V\subseteq G$. We say that $V$ is a _$(c_{0},\delta,\varepsilon)$
-approximate quadratic variety_ if $|V|=\delta|G|$,
$\|\mathbbm{1}_{V}-\delta\|_{\mathsf{U}^{2}}\leq\varepsilon$ and
$\mathop{\mathbb{E}}_{x,a,b,c}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a,b,c}\mathbbm{1}_{V}(x)=c_{0}\delta^{7}$.

We think of parameters $c_{0},\delta$ and $\varepsilon$ as satisfying the
relationship $c_{0}\gg\delta\gg\varepsilon$. As a further motivation for
imposing the $\mathsf{U}^{2}$ condition, we note the following connection with
approximate quadratic polynomials.

######  Proposition 1.7.

Let $G$ and $H$ be two finite-dimensional vector spaces over $\mathbb{F}_{p}$
and let $A\subseteq G$. Let $F\colon A\to H$ satisfy
$\Delta_{a}\Delta_{b}\Delta_{c}F(x)=0$ for at least $c_{0}|G|^{4}$ choices of
$(x,a,b,c)\in G^{4}$ (and all 8 arguments of $F$ belong to $A$) and let $\xi$
be a positive quantity. Let $d\in\mathbb{N}$ and let $\varepsilon>0$ be given.
Provided that $\varepsilon$ and $\dim G$ are sufficiently small and large,
respectively, in terms of $c_{0},d$ and $\xi$, the following holds.
Suppose that $F$ respects at most $\varepsilon|G|^{3}$ additive quadruples in
$A$. Then there exists a cost $w_{0}+W$ of a subspace of codimension
$O_{\varepsilon}(1)$ such that, if we choose a subspace $U$ of codimension $d$
uniformly at random, then the probability that the set

$V=\\{x\in A\colon x\in w_{0}+W,F(x)\in U\\}$

is a $(c^{\prime},\delta,\xi)$-approximate quadratic variety for some
$\delta\in[2^{-9}c_{0}^{8}p^{-d},2p^{-d}]$ and
$c^{\prime}\in[c_{0}^{8}2^{-9},2]$, is at least $0.99$.

We remark that the assumption of $F$ respecting very few additive quadruples
is quite natural, as otherwise it becomes an approximate homomorphism instead,
which is much simpler and to which we may apply Theorem 1.3.

Gowers also conjectured that an approximate quadratic variety necessarily has
a large intersection with an exact quadratic variety of comparable size.
Notice the resemblance to Theorem 1.2.

######  Conjecture 1.8.

Fix a prime $p\geq 3$. Let $V\subseteq G$ be a
$(c_{0},\delta,\varepsilon)$-approximate quadratic variety. Suppose that
$\varepsilon$ is sufficiently small in terms of $\delta$ and $c_{0}$. Then
there exists a quadratic variety $Q$ such that $|Q|\leq O_{c_{0}}(|V|)$ and
$|Q\cap V|\geq\Omega_{c_{0}}(|V|)$.

Having discussed approximate quadratic varieties and the motivation for the
definition, we are now ready to state our main result of this paper, which is
the resolution of Conjecture 1.8 with reasonable dependencies on $c_{0}$ and
$\delta$.

######  Theorem 1.9.

There exists an absolute constant $D\geq 1$ such the following holds. Assume
that $p\geq 3$. Let $V\subseteq G$ be a
$(c_{0},\delta,\varepsilon)$-approximate quadratic variety. Suppose that

$\varepsilon\leq(2^{-1}\delta)^{\exp\big{(}\log^{D}(O_{p}(c_{0}^{-1}))\big{)}}.$

Then there exists a quadratic variety $Q$ such that
$|Q|\leq\exp\Big{(}\exp(\log^{D}(O_{p}(c_{0}^{-1})))\Big{)}\cdot|V|$ and
$|Q\cap V|\geq\exp\Big{(}-\exp(\log^{D}(O_{p}(c_{0}^{-1})))\Big{)}\cdot|V|$.

Let us remark that if we only had assumptions $|V|=\delta|G|$ and
$\mathop{\mathbb{E}}_{x,a,b,c}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a,b,c}\mathbbm{1}_{V}(x)=c_{0}\delta^{7}$, the theorem above
would imply that $V$ is either closely related to a quadratic variety or that
$\|\mathbbm{1}_{V}-\delta\|_{\mathsf{U}^{2}}\geq\Omega_{c_{0},\delta}(1)$,
becoming $\Omega_{\delta}(1)$ when $c_{0}\geq\delta$, and proving existence of
weak linear structure. A strong linear structure would be obtaining a non-
trivial Fourier coefficient in the large spectrum of $\mathbbm{1}_{V}$, namely
finding $r\not=0$ such that
$|\widehat{\mathbbm{1}_{V}}(r)|\geq\Omega_{c_{0}}(\delta)$. However, the
example based on Sidon sets shows that for some absolute constant $\theta>0$
we may have $|\widehat{\mathbbm{1}_{V}}(r)|\leq\delta^{1+\theta}$ for all non-
zero $r$, and so one cannot hope for a stronger result in a qualitative sense.
Furthermore, the methods of this paper could likely prove the theorem above in
the case of low characteristic (when $p=2$) as well, but that would require
the use of non-classical polynomials [25] in the final stages of the proof and
a suitable modification of the statement of the result.

Finally, let us also note resemblance to nilspaces, introduced by Antolín
Camarena and Szegedy [2], whose definition we now recall. A _nilspace_ is a
set $X$ together with a collection of sets $\mathsf{C}^{n}(X)\subseteq
X^{\\{0,1\\}^{n}}$, for each non-negative integer $n$, satisfying the
following axioms:

  * (i)

(Composition) For every morphism $\phi\colon\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ and
every $c\in\mathsf{C}^{n}(X)$, we have $c\circ\phi\in\mathsf{C}^{m}(X)$.

  * (ii)

(Ergodicity) $\mathsf{C}^{1}(X)=X^{\\{0,1\\}}$.

  * (iii)

(Corner completion) Let
$c^{\prime}\colon\\{0,1\\}^{n}\setminus\\{(1,\dots,1)\\}\to X$ be such that
every restriction of $c^{\prime}$ to an $(n-1)$-face containing $(0,\dots,0)$
is in $\mathsf{C}^{n-1}(X)$. Then there exists $c\in\mathsf{C}^{n}(X)$ such
that $c(v)=c^{\prime}(v)$ for all $v\not=(1,\dots,1)$.

If every $(k+1)$-corner has a unique completion, then we say that $X$ is a
_$k$ -step nilspace_. (In the theory of nilspaces, one also imposes topology
on cubes, but here we are interested primarily in algebraic aspects, so we
skip such details.) Such nilspaces occur naturally in the higher order Fourier
analysis as they arise in the nilspace approach to the inverse theorems for
uniformity norms.

The key assumption in Theorem 1.9 can be expressed as

$\mathbb{P}(x+a+b+c\in V|x,x+a,x+b,x+c,x+a+b,x+a+c,x+b+c\in V)\geq c_{0}$

so we may think of a an approximate quadratic variety as a combinatorial
counterpart of a 2-step nilspace (though in the case of nilspaces the cube
collections are abstract and thus more general).

### 1.1. Proof overview

Let us begin with some motivation. If $V$ is a
$(c_{0},\delta,\varepsilon)$-approximate quadratic variety, then we expect
that $V$ is a $c_{0}$-dense subset of a quadratic variety $Q=\\{x\in
G\colon\beta(x,x)=0\\}$, where $\beta\colon G\times G\to\mathbb{F}_{p}^{r}$ is
a symmetric bilinear map, and we know that $V$ is very
$\mathsf{U}^{2}$-quasirandom. Furthermore, as we think of parameters
$c_{0},\delta$ and $\varepsilon$ as satisfying the relationship
$c_{0}\gg\delta\gg\varepsilon$, $Q$ has to have density comparable to $\delta$
and to inherit $\mathsf{U}^{2}$-quasirandomness from $V$. Since $Q$ is a
quadratic variety, this in turn implies that $\beta$ is of very high rank. Our
goal now is to identify $\beta$ using $V$. To that end, consider the
intersection $V\cap V-a$. This is a subset of $\\{x\in
G\colon\beta(x+a,x+a)=\beta(x,x)=0\\}=\\{x\in
G\colon\beta(a,x)=-2^{-1}\beta(a,a),\beta(x,x)=0\\}=V\cap\\{x\in
G\colon\beta(a,x)=-2^{-1}\beta(a,a)\\}$. Thus, we expect $V\cap V-a$ to be
$\mathsf{U}^{2}$-quasirandom subset of the subspace coset $\\{x\in
G\colon\beta(a,x)=-2^{-1}\beta(a,a)\\}$. Given
$\mathsf{U}^{2}$-quasirandomness, the set of large values of the convolution
$\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-a}$
should then be equal to the whole subspace $\\{x\in G\colon\beta(a,x)=0\\}$.
This motivates the study of subspaces $W_{a}$ that arise from convolution of
$\mathbbm{1}_{V\cap V-a}$ via methods such as Bogolyubov argument. We expect
these subspaces $W_{a}$ to be related to $\\{x\in G\colon\beta(a,x)=0\\}$.
Hence, we now have a collection of subspaces $W_{a}$, for $a\in G$, of density
comparable to $\delta$. However, this is not an arbitrary family of subspaces.
In fact, the algebraic structure of indices plays an important role. Namely,
if $a_{1},a_{2},a_{3},a_{4}$ form an additive quadruple, thus
$a_{1}+a_{2}=a_{3}+a_{4}$, we have that $x\in W_{a_{1}}\cap W_{a_{2}}\cap
W_{a_{3}}$ implies
$0=\beta(a_{1},x)+\beta(a_{2},x)-\beta(a_{3},x)=\beta(a_{1}+a_{2}-a_{3},x)=\beta(a_{4},x)$
and thus we expect that a relationship along the lines of

$W_{a_{1}}\cap W_{a_{2}}\cap W_{a_{3}}\subseteq W_{a_{4}}$ (1)

holds. The first major step of the proof of Theorem 1.7 achieves this and it
appears as Theorem 3.1 later in the paper.

######  Theorem 1.10 (Step 1).

There exists an absolute constant $D\geq 1$ such that the following holds. Let
$V\subseteq G$ be a $(c_{0},\delta,\varepsilon)$-approximate quadratic variety
and suppose that
$\varepsilon\leq\exp\Big{(}-\log^{D}(2c_{0}^{-1})\Big{)}\delta^{288}$. Then
there exist a quantity $c_{1}$, a set $A\subseteq G$ and a collection of
subspaces $W_{a}\leq G$ indexed by elements $a\in A$ such that

  * (i)

$\exp(-\log^{D}(2c_{0}^{-1}))\leq c_{1}\leq 1$,

  * (ii)

$|A|\geq c_{1}|G|$,

  * (iii)

for each $a\in A$ and $b\in W_{a}$ we have ${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}(b)\geq c_{1}\delta^{15}$,

  * (iv)

$c_{1}\leq\frac{|W_{a}|}{\delta|G|}\leq c_{1}^{-1}$ holds for all $a\in A$,

  * (v)

for $r\in[9]$, for all but at most $D\varepsilon\delta^{-32r}|G|^{r}$ choices
of $(a_{1},\dots,a_{r})\in A^{r}$ we have

$\frac{|W_{a_{1}}\cap W_{a_{2}}\cap\dots\cap W_{a_{r}}|}{\delta^{r}|G|}\leq
c_{1}^{-1},$

and,

  * (vi)

for at least $c_{1}|A|^{6}$ 6-tuples $(b_{1},b_{2},b_{3},x_{2},y_{3},z_{1})\in
A^{6}$ we have that
$b_{1}+b_{2}-b_{3},x_{2}-b_{2}+b_{3},y_{3}+b_{1}-b_{3},b_{1}+b_{2}-z_{1}\in A$
and

$c_{1}\delta^{6}|G|\leq|W_{b_{1}}\cap W_{b_{2}}\cap W_{b_{3}}\cap
W_{b_{1}+b_{2}-b_{3}}\cap W_{x_{2}}\cap W_{x_{2}-b_{2}+b_{3}}\cap
W_{y_{3}}\cap W_{y_{3}+b_{1}-b_{3}}\cap W_{z_{1}}\cap W_{b_{1}+b_{2}-z_{1}}|.$

We remark that ${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}$ stands for an iterated convolution of the function $\mathbbm{1}_{V\cap
V-a}$ with itself, see (4) for a precise definition. Notice that item (vi) is
considerably stronger than what we mentioned in the motivation discussion,
where an approximate version of $W_{b_{1}}\cap W_{b_{2}}\cap
W_{b_{3}}\subseteq W_{b_{1}+b_{2}-b_{3}}$ would be $|W_{b_{1}}\cap
W_{b_{2}}\cap W_{b_{3}}\cap W_{b_{1}+b_{2}-b_{3}}|\geq c_{1}\delta^{3}|G|$. We
shall return to this discrepancy in the discussion of the second major step of
the proof.
We shall refer to a collection of subspaces in satisfying all properties but
(iii) in conclusion of the theorem above as an _approximate quasirandom linear
systems of subspaces_. To explain the terminology, linearity stands for
property (1) for additive quadruples $(a_{1},a_{2},a_{3},a_{4})$,
quasirandomness suggests that intersections of a vast majority subspaces
behave as given by property (v) in the conclusion of the theorem, and all
properties are of an approximate form rather than exact.

The second major step of the proof is to characterize approximate quasirandom
linear systems of subspaces. Recall that we expect $W_{a}$ to be related to
$\\{x\in G\colon\beta(a,x)=0\\}$. With this in mind, we expect that
$(W_{a})_{a\in A}$ comes from some bilinear map. The second step is to obtain
such a result, which appears as Theorem 4.1. The assumptions of the theorem
are essentially the same as the conclusion of Theorem 1.10.

######  Theorem 1.11 (Step 2).

There exists an absolute constant $D\geq 1$ such that the following holds. Let
$c>0$ and let $d$ be a positive integer. Let $A\subseteq G$ be a set of size
$|A|\geq c|G|$ and let $W_{a}\leq G$ be a subspace of codimension $d$ for each
$a\in A$. Suppose that

$|W_{a_{1}}\cap W_{a_{2}}\cap\dots\cap W_{a_{r}}|\leq Kp^{-rd}|G|$

holds for all but at most $\eta|G|^{r}$ $r$-tuples
$(a_{1},a_{2},\dots,a_{r})\in A^{r}$ for each $r\in[9]$. Assume furthermore
that for at least $c|G|^{6}$ triples
$(a,b_{1},b_{2},b_{3},x_{2},x_{3},y_{1},y_{3},z_{1},z_{2})\in A^{10}$ we have

  * •

$a=b_{1}+b_{2}-b_{3},\,\,x_{3}=x_{2}-b_{2}+b_{3},\,\,y_{1}=y_{3}+b_{1}-b_{3},\,\,z_{2}=b_{1}+b_{2}-z_{1},$

and

  * •

the subspaces

$W_{a}\cap W_{b_{1}}\cap W_{b_{2}}\cap W_{b_{3}}\cap W_{x_{2}}\cap
W_{x_{3}}\cap W_{y_{1}}\cap W_{y_{3}}\cap W_{z_{1}}\cap W_{z_{2}}$

has size at least $K^{-1}p^{-6d}|G|$.

Then, provided $\eta\leq 2^{-31}c^{3}$, there exist parameters
$c^{\prime}\geq\exp\Big{(}-\exp\Big{(}(\log(2c^{-1})+\log_{p}K)^{D}\Big{)}\Big{)}$
and $r\leq\exp\Big{(}(\log(2c^{-1})+\log_{p}K)^{D}\Big{)}$, set
$A^{\prime}\subseteq A$ and a map $\Phi\colon G\times\mathbb{F}_{p}^{d}\to G$,
affine in the first variable and linear in the second, such that
$|A^{\prime}|\geq c^{\prime}|G|$ and for each $a\in A^{\prime}$ we have
$|\operatorname{Im}\Phi(a,\cdot)\cap W_{a}^{\perp}|\geq c^{\prime}p^{d}$.
Moreover, there exists a subspace $\Lambda\leq\mathbb{F}_{p}^{d}$ of dimension
$r$ such that whenever $\lambda\notin\Lambda$ we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,y}\omega\Big{(}\Phi(x,\lambda)\cdot
y\Big{)}\leq\Big{(}\eta{c^{\prime}}^{-2}\Big{)}^{1/2r}.$

Finally, once we have obtained a biaffine map $\Phi$ that parametrizes the
family of subspaces $(W_{a})_{a\in A}$, we have to relate it to the
approximate variety $V$ that we start with. This is the content of the final
major step of the proof. The proposition below appears as Proposition 5.1.

######  Proposition 1.12 (Step 3).

There is an absolute constant $D\geq 1$ such that the following holds. Let
$c,\delta,\varepsilon>0$ and $d\in\mathbb{N}$ be such that $c\leq\delta
p^{d}\leq c^{-1}$. Suppose that $\varepsilon\leq(2^{-1}c\delta)^{D}$. Let
$V\subseteq G$ be a set of density $\delta$ such that
$\|\mathbbm{1}_{V}-\delta\|_{\mathsf{U}^{2}}\leq\varepsilon$. Suppose that we
are also given a subset $A\subseteq G$ of size $|A|\geq c|G|$, a subspace
$W_{a}\leq G$ for each $a\in A$ and a bilinear map $\beta\colon G\times
G\to\mathbb{F}_{p}^{d}$ such that

  * (i)

for each $\lambda\in\mathbb{F}_{p}^{d}\setminus\\{0\\}$ we have
$\operatorname{bias}\lambda\cdot\beta\leq\varepsilon$,

  * (ii)

for each $a\in A$ we have $|W_{a}\cap\\{b\in G\colon\beta(a,b)=0\\}|\geq
cp^{-d}|G|$,

  * (iii)

for each $a\in A$ and $b\in W_{a}$ we have ${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}(b)\geq c\delta^{15}$.

Then there exists a quadratic variety $Q\subseteq G$ of size
$|Q|\leq(2c^{-1})^{D}\delta|G|$ such that $|Q\cap
V|\geq\exp\Big{(}-\log^{D}(2c^{-1})\Big{)}\delta|G|$. Moreover, $Q$ is defined
as $\\{x\in G\colon\gamma(x,x)-\psi(x)=\mu\\}$ for a symmetric bilinear map
$\gamma\colon G\times G\to\mathbb{F}_{p}^{\tilde{d}}$, an affine map
$\psi\colon G\to\mathbb{F}_{p}^{\tilde{d}}$ and
$\mu\in\mathbb{F}_{p}^{\tilde{d}}$, where
$\operatorname{bias}\lambda\cdot\gamma\leq\varepsilon$ for all
$\lambda\not=0$, for some $d-O(\log_{p}(2c^{-1}))\leq\tilde{d}\leq d$.

Let us now say something about the proof of each step.

When it comes to Step 1, we rely heavily on the $\mathsf{U}^{2}$-uniformity
assumption on $V$, which itself is sufficient to prove some properties in the
conclusion of Theorem 1.10. Most basic consequence of uniformity is a standard
fact that intersections of translates of $V$ behave quasirandomly (see Lemma
3.3). We can control more involved expressions as well, for example

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,a_{1},a_{2},a_{3}}\mathbbm{1}_{V\cap
V-a_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{1}}(x)\mathbbm{1}_{V\cap V-a_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{2}}(x)$ $\displaystyle\hskip
56.9055pt=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,y,z,a_{1},a_{2},a_{3}}\mathbbm{1}_{V\cap
V-a_{1}}(y+x)\mathbbm{1}_{V\cap V-a_{1}}(y)\mathbbm{1}_{V\cap
V-a_{2}}(z+x)\mathbbm{1}_{V\cap V-a_{2}}(z)$ $\displaystyle\hskip
56.9055pt=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,y,z,a_{1},a_{2},a_{3}}\mathbbm{1}_{V}(y+x+a_{1})\mathbbm{1}_{V}(y+x)\mathbbm{1}_{V}(y+a_{1})\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(z+x+a_{2})\mathbbm{1}_{V}(z+x)\mathbbm{1}_{V}(z+a_{2})\mathbbm{1}_{V}(z)$

can be shown to be about $\delta^{8}$. If we think of $W_{a}$ as the set of
elements $x$ where $\mathbbm{1}_{V\cap V-a_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{1}}(x)$ is about $\delta^{3}$, then the bound above shows that
$|W_{a_{1}}\cap W_{a_{2}}|$ is $O(\delta^{2}|G|)$ most of the time. A more
involved argument of this form can be used to prove property (v) of Theorem
1.10.
We remark that this resembles the true complexity problem introduced by Gowers
and Wolf [12]. However, these are instances of systems of the true complexity
1, and the arguments of this form that we need in this paper are elementary,
as we are concerned with the $\mathsf{U}^{2}$ norm.
Let us also note that there are expressions that cannot be directly controlled
by $\mathsf{U}^{2}$ norm, but where we can still easily prove correct upper
bounds by neglecting some of the terms. For example, the distribution of
$|V\cap V-a\cap V-b\cap V-a-b|$ depends on the quadratic structure of $V$, but
it is always at most $|V\cap V-a\cap V-b|$ which is about $\delta^{3}|G|$ vast
majority of the time.
However, in order to prove an approximate version of property (1), we need to
use the assumption of having many additive cubes in $V$ and, simplifying
things greatly by assuming that $W_{a}$ is the set of elements $x$ where
$\mathbbm{1}_{V\cap V-a_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{1}}(x)$ is about $\delta^{3}$, we would need a bound of the form

$\displaystyle\Omega_{c_{0}}(\delta^{15})\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},a_{2},a_{3},x}\mathbbm{1}_{V\cap
V-a_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{1}}(x)\mathbbm{1}_{V\cap V-a_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{2}}(x)$ $\displaystyle\hskip 113.81102pt\mathbbm{1}_{V\cap
V-a_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{3}}(x)\mathbbm{1}_{V\cap V-a_{1}-a_{2}+a_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a_{1}-a_{2}+a_{3}}(x).$

Unlike proving upper bounds on expressions that are not directly controllable
by $\mathsf{U}^{2}$ norm, it is surprisingly challenging to prove such a lower
bound. To get the lower bound, we first observe the following _duality_
property of convolutions of indicator functions of intersections

$\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)=\mathbbm{1}_{V\cap V-b}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-b}(a).$

This observation allows us to turn the expression above into

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},a_{2},a_{3},x}\mathbbm{1}_{V\cap
V-x}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x}(a_{1})\mathbbm{1}_{V\cap V-x}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x}(a_{2})$ $\displaystyle\hskip 113.81102pt\mathbbm{1}_{V\cap
V-x}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x}(a_{3})\mathbbm{1}_{V\cap V-x}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x}(a_{1}+a_{2}-a_{3}).$ (2)

Using the properties of $V$, we show that for many $x$, the set $S_{x}$ of
$a\in G$ such that $\mathbbm{1}_{V\cap V-x}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x}(a)\geq\Omega_{c_{0}}(\delta^{3})$ is $\Omega_{c_{0}}(1)$-dense subset of
some subspace $U_{x}$ of size $|U_{x}|\geq\Omega_{c_{0}}(\delta|G|)$ (observe
that this is related to other properties in the conclusion of Theorem 1.10).
Then a lower bound on (2) follows from the fact that $S_{x}$ has at least
$\Omega_{c_{0}}(\delta^{3}|G|^{3})$ additive quadruples. Interestingly, we
needed to use results in the spirit of Bogolyubov argument in this stage of
the proof, as it was essential that we have the subspace structure.
Finally, as we actually need to prove property (vi) of Theorem 1.10 instead of
a simpler variant such as (1), the proof is more involved than the sketch
above, but the sketch indicates the key ideas.

In Step 2, the following observation, stated as Lemma 4.2, is the main tool of
passing from the given family of subspaces to approximate bilinear structure.
Namely, if four subspaces $U_{1},\dots,U_{4}$ of dimension $d$ satisfy
$K^{-1}p^{3d}\leq|U_{i_{1}}+U_{i_{2}}+U_{i_{3}}|$ for any three distinct
indices $i_{1},i_{2}$ and $i_{3}$ and $|U_{1}+U_{2}+U_{3}+U_{4}|\leq Kp^{3d}$,
then for any linear isomorphism $\phi_{4}\colon\mathbb{F}_{p}^{d}\to U_{4}$
there exist linear isomorphisms $\phi_{i}\colon\mathbb{F}_{p}^{d}\to U_{i}$
for $i\in[3]$ such that

$\operatorname{rank}(\phi_{1}+\phi_{2}-\phi_{3}-\phi_{4})\leq O(\log_{p}K).$
(3)

We remark that we do not only say that there are 4 isomorphisms satisfying
(3), but rather that there exist suitable $\phi_{2},\phi_{3}$ and $\phi_{4}$
for _any given_ $\phi_{1}$, which will be crucial in the proof. This fact also
shows that we may think of subspaces satisfying (1) as additive quadruples of
subspaces. Observe that these properties are satisfied by the orthogonal
complements of subspaces in our family (we may easily ensure that all
$W_{a}^{\perp}$ are of same size).
The approach to proving Theorem 1.11 is to find a suitable element $a\in A$,
fix an isomorphism $\theta\colon\mathbb{F}_{p}^{d}\to W_{a}^{\perp}$ and apply
the observation above to additive quadruples in $A$ involving $a$ to get
isomorphisms to other $W_{b}^{\perp}$. This will give us an approximate
homomorphism between $G$ and the space of linear maps
$\operatorname{Hom}(\mathbb{F}_{p}^{d},G)$, where the distance between
elements is measured by rank, from which we shall obtain an exact
homomorphism. We achieve this by using a result of Kazhdan and Ziegler [18]
(see Theorem 2.7), which we reproved using the inverse theorem for Freiman
bihomomorphisms [10, 19] to get quantitative bounds.
It turns out that in this step, the weaker assumption of having many triples
$(b_{1},b_{2},b_{3})$ such that $|W_{b_{1}}\cap W_{b_{2}}\cap W_{b_{3}}\cap
W_{b_{1}+b_{2}-b_{3}}|\geq c_{1}\delta^{3}|G|$ is not sufficiently strong for
the proof to work and that we need more involved configurations of points. The
reason is that in defining the linear isomorphisms $\mathbb{F}_{p}^{d}\to
W_{b}^{\perp}$ we need to ensure that they are well-defined and that these
linear isomorphisms also respect many additive quadruples.

Finally, in Step 3, we first need to show that the given map $\beta$ can be
replaced by a symmetric bilinear map. To that end, we use Green-Tao symmetry
argument [14] to first show that
$(x,y)\mapsto\lambda\cdot(\beta(x,y)-\beta(y,x))$ has small rank for many
vectors $\lambda\in\mathbb{F}_{p}^{d}$. To pass to an exactly symmetric map,
we have to make use of the solution of partition versus analytic problem for
trilinear forms, first proved by Green and Tao [15] in the case of
polynomials, with essentially optimal bounds obtained by Adiprasito, Kazhdan
and Ziegler [1] and by Cohen and Moshkovitz [6]. Note that it the high
characteristic case (when $p\geq 3$) we may usually replace an approximately
bilinear form $\gamma(x,y)$ by a simple mean
$2^{-1}(\gamma(x,y)+\gamma(y,x))$. However, as we are concerned with $\beta$
whose codomain is subspace of somewhat large dimension, we have to use
partition versus analytic problem at some point, even if we use the mean trick
at each coordinate of $\beta$ (see for example Lemma 2.8 in [21]). Once we may
assume that $\beta$ is symmetric, we expect that the approximate variety $V$
comes from a variety defined by $\beta(x,x)+\gamma(x)=\lambda$ for some linear
map $\gamma$, which we still have to identify. To that end, we consider
intersection of $V\cap V-a$ with $\\{x\in G\colon\beta(a,x)=\lambda\\}$ for
various $\lambda$. As it turns out, for many $a$, there is a value
$\lambda(a)$ for which this intersections is almost the whole of $V\cap V-a$.
We then use additional graph-theoretic arguments to show that $\lambda(a)$ is
an approximate homomorphism, from which we may then pass to a linear map and
conclude the proof.

### 1.2. Regularity lemmas are insufficient

A natural approach to proving Theorem 1.9 is to use arithmetic regularity
lemmas of Green and Tao [16]. In this short subsection, we briefly discuss why
such an approach is not helpful for this problem. In this setting, we could in
principle find a bilinear map $\beta\colon G\times G\to\mathbb{F}^{r}$ and
functions $d\colon\mathbb{F}_{p}^{r}\to\mathbb{D}$,
$f_{\text{err}},f_{\text{unif}}\colon G\to\mathbb{D}$ such that

$\mathbbm{1}_{V}(x)=d(\beta(x,x))+f_{\text{err}}(x)+f_{\text{unif}}(x),$

where $f_{\text{err}}$ has small $L^{2}$ norm, $f_{\text{unif}}$ has extremely
small $\|\cdot\|_{\mathsf{U}^{3}}$ norm and

$d(\lambda)=\frac{\mathop{\mathbb{E}}_{x}\mathbbm{1}_{V}(x)\mathbbm{1}(\beta(x,x)=\lambda)}{\mathop{\mathbb{E}}_{x}\mathbbm{1}(\beta(x,x)=\lambda)},$

which comes from projection of $\mathbbm{1}_{V}$ onto layers
$\\{x\colon\beta(x,x)=\lambda\\}$ defined by $\beta$. For simplicity, we
ignore the $\ell^{2}$ error terms and assume that we have perfect
approximation, and also that $\beta$ itself is of high rank (we expect this to
be the case from the expected structure of $V$). In particular, function $d$
simplifies to
$d(\lambda)=|G|^{-1}p^{r}|V\cap\\{x\colon\beta(x,x)=\lambda\\}|$. Observe also
that the high rank of $\beta$ implies that

$(x,a,b,c)\mapsto\Big{(}\beta(x,x),\beta(x+a,x+a),\dots,\beta(x+b+c,x+b+c)\Big{)}$

is equidistributed in $(\mathbb{F}_{p}^{r})^{7}$ (here we listed all points of
additive cube associated with $x,a,b,c$ except $x+a+b+c$), while $\beta$ being
a bilinear form implies the identity

$\displaystyle\beta$
$\displaystyle(x+a+b+c,x+a+b+c)=\beta(x,x)-\beta(x+a,x+a)-\beta(x+b,x+b)-\beta(x+c,x+c)$
$\displaystyle\hskip
28.45274pt+\beta(x+a+b,x+a+b)+\beta(x+a+c,x+a+c)+\beta(x+b+c,x+b+c).$

Using these facts and given that the error terms are negligible, the count of
3-dimensional additive cubes in $V$ essentially becomes

$c_{0}\delta^{7}\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\lambda_{1},\dots,\lambda_{7}\in\mathbb{F}_{p}^{r}}d(\lambda_{1})\dots
d(\lambda_{7})d(\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}-\lambda_{5}-\lambda_{6}-\lambda_{7})=\sum_{\gamma\in\mathbb{F}^{r}}|\hat{d}(\gamma)|^{8}.$

Note that $d(\lambda)\leq 1$ for all $\lambda$ and that

$\sum_{\gamma\in\mathbb{F}^{r}_{p}}|\hat{d}(\gamma)|^{2}=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\lambda\in\mathbb{F}_{p}^{r}}|d(\lambda)|^{2}\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\lambda\in\mathbb{F}_{p}^{r}}d(\lambda)=\frac{1}{|G|}\sum_{\lambda\in\mathbb{F}_{p}^{r}}|V\cap\\{x\colon\beta(x,x)=\lambda\\}|=\frac{|V|}{|G|}=\delta.$

We thus get some $\gamma$ such that
$|\hat{d}(\gamma)|\geq\sqrt[6]{c_{0}}\delta$. We may assume that
$\gamma\not=0$ as the contribution from $\gamma=0$ above is $\delta^{8}$.222In
fact, we expect that $V$ is the union of $\\{\beta=\lambda\\}$ for some
subspace $\Lambda$ such that $|\Lambda|=\delta p^{r}$. If so, we would also
have $d(\lambda)=\mathbbm{1}_{\Lambda}(\lambda)$ and thus
$\hat{d}(\gamma)=\frac{|\Lambda|}{p^{r}}\mathbbm{1}_{\Lambda^{\perp}}(\gamma)=\delta\mathbbm{1}_{\Lambda^{\perp}}(\gamma)$.
Thus,

$\displaystyle\sqrt[6]{c_{0}}\delta\leq$
$\displaystyle\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\lambda\in\mathbb{F}_{p}^{r}}|G|^{-1}p^{r}|V\cap\\{x\colon\beta(x,x)=\lambda\\}|\omega^{-\lambda\cdot\gamma}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\sum_{\lambda\in\mathbb{F}_{p}^{r}}\frac{|V\cap\\{x\colon\beta(x,x)=\lambda\\}|}{|G|}\omega^{-\lambda\cdot\gamma}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\sum_{\mu\in\mathbb{F}_{p}}\sum_{\begin{subarray}{c}\lambda\in\mathbb{F}_{p}^{r}\\\
\text{s.t.
}\gamma\cdot\lambda=\mu\end{subarray}}\frac{|V\cap\\{x\colon\beta(x,x)=\lambda\\}|}{|G|}\omega^{-\mu}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\sum_{\mu\in\mathbb{F}_{p}}\frac{|V\cap\\{x\colon\gamma\cdot\beta(x,x)=\mu\\}|}{|G|}\omega^{-\mu}\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\sum_{\mu\in\mathbb{F}_{p}}\Big{(}\frac{|V\cap\\{x\colon\gamma\cdot\beta(x,x)=\mu\\}|}{|G|}-\delta\Big{)}\omega^{-\mu}\Big{|}.$

Hence, for some $\mu\in\mathbb{F}_{p}$ and codimension 1 quadratic variety
$B=\\{x\colon\gamma\cdot\beta(x,x)=\mu\\}$ we conclude that $V$ gets a density
increment of $p^{-1}\sqrt[6]{c_{0}}\delta$ on $B$. In particular, after $s$
steps, we may only guarantee density
$\delta\Big{(}1+p^{-1}\sqrt[6]{c_{0}}\Big{)}^{s}$ on codimension $s$ quadratic
variety. Proceeding in this fashion, provided $c_{0}$ is sufficiently small,
we expect that we need $s=K\log_{p}\delta^{-1}$ steps in order to get density
$\Omega_{c_{0}}(1)$ on some quadratic variety, where $K$ is arbitrarily large.
But, such a variety would then be of codimension $s$ and would have density
$p^{-s}\leq\delta^{K}$ as our bilinear map is quasirandom, so we cannot obtain
the claimed result in this fashion.

Acknowledgements. I would like to thank Tim Gowers for introducing me to the
problem of proving the structure theorem for approximate quadratic varieties.
This work was supported by the Ministry of Science, Technological Development
and Innovation of the Republic of Serbia through the Mathematical Institute of
the Serbian Academy of Sciences and Arts.

## §2 Preliminaries

Notation. The fixed prime $p$ is assume to be at least 3 throughout the paper
and dependencies on $p$ in bounds are suppressed.
In this paper, we shall frequently consider subsets of products of two vectors
spaces $G$ and $H$. Given $X\subset G\times H$ and an element $x\in G$, we
write
$X_{x\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}=\\{y\in
H\colon(x,y)\in X\\}$ for the _(vertical) slice_ of $X$ in the column indexed
by $x$. Likewise, for an element $y\in H$, we write
$X_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}y}=\\{x\in
G\colon(x,y)\in X\\}$ for the _(horizontal) slice_ of $X$ in the row indexed
by $y$.
For two functions $f,g\colon G\to\mathbb{R}$ we define _convolution_ as
$f\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}g(x)=\mathop{\mathbb{E}}_{y\in
G}f(y+x)\overline{g(x)}$ (note that this is non-standard, as we average over
pairs of elements whose difference is $x$, rather than their sum). We also
need _iterated convolution_ (of _order_ $2k$) of a function $f\colon
G\to\mathbb{R}$, defined as

${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(2k)}f(a)=\mathop{\mathbb{E}}_{x_{1},\dots,x_{2k-1}\in
G}f(x_{1})f(x_{2})\cdots
f(x_{2k-1})f\Big{(}\sum_{\ell\in[2k-1]}(-1)^{\ell+1}x_{i}-a\Big{)}.$ (4)

Note that the order of iterated convolution $2k$ stands for the number of
terms, rather than number of convolutions needed to give the expression above.
Also note that there are no complex conjugates in the expression above. This
is done to simplify the notation for iterated convolution as it will only be
used for real-valued functions in this paper.
The _discrete multiplicative derivative operator_
${\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\ \textbullet}}}\,_{a}$
for shift $a\in G$ is defined by
${\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a}f(x)=f(x+a)\overline{f(x)}$ for functions $f\colon
G\to\mathbb{C}$.
To save writing in situations where we have many indices of variables
appearing in predictable patterns, we use the following convention. Instead of
denoting a sequence of length $m$ by $(x_{1},\dots,x_{m})$, we write
$x_{[m]}$, and for $I\subset[m]$ we write $x_{I}$ for the subsequence with
indices in $I$.

We need a few auxiliary results. The first one is a robust version of
Bogolyubov-Ruzsa lemma, which is essentially due to Schoen and Sisask and
builds upon the work of Sanders.

######  Theorem 2.1.

Let $A\subset G$ be a subset having at least $\alpha|A|^{3}$ additive
quadruples. Then there exists a subspace $V\subseteq 2A-2A$ of size
$|V|\geq\exp\Big{(}-O\Big{(}\log^{O(1)}(2c^{-1})\Big{)}\Big{)}|A|$ such that
the following holds. Every $y\in V$ can be expressed as
$y=a_{1}+a_{2}-a_{3}-a_{4}$ with $a_{1},a_{2},a_{3},a_{4}\in A$ in at least
$\alpha^{O(1)}|A|^{3}$ many ways.

###### Proof.

Apply Balog-Szemerédi-Gowers theorem to find a subset $A^{\prime}\subseteq A$
such that $|A^{\prime}|\geq\Omega(\alpha^{O(1)}|A|)$ and
$|A^{\prime}+A^{\prime}|\leq O(\delta^{-O(1)}|A^{\prime}|)$. The theorem
follows from results of Sanders (Theorem A.2 for arbitrary prime $p$ instead
of $p=2$ in [23]) and Schoen and Sisask (Theorem 5.1 in [24]).∎

A closely related result is the inverse theorem for approximate homomorphisms,
which we already stated in the introduction as Theorem 1.3. We make use of a
more efficient version, which is proved using Balog-Szemerédi-Gowers theorem
and Sanders’s results on Bogolyubov-Ruzsa lemma [23].

######  Theorem 2.2.

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_{p}$. Let
$A\subseteq G$ be a subset and let $\phi\colon A\to H$ be a map which respects
at least $c|G|^{3}$ additive quadruples in $A$. Then there exists an affine
map $\Phi\colon G\to H$ such that $\phi=\Phi$ holds for at least
$\exp\Big{(}-\log^{O(1)}(2c^{-1})\Big{)}$ points in $A$.

Recall that the _bias_ of a multilinear form $\phi\colon
G^{k}\to\mathbb{F}_{p}$ is defined as

$\operatorname{bias}\phi=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{1},\dots,x_{k}}\omega^{\phi(x_{1},\dots,x_{k})},$

where $\omega=\exp\Big{(}\frac{2\pi i}{p}\Big{)}$. This quantity is a measure
of how far from being quasirandom the given form $\phi$ is. We need the
inverse theorem for biased trilinear forms. First results in this spirit were
proved by Green and Tao [15] for the case of polynomials and a multilinear
variant was proved by Bhowmick and Lovett [5]. The version below, which has
essentially optimal bounds, is due to Adiprasito, Kazhdan and Ziegler [1] and
due to Cohen and Moshkovitz [6].

######  Theorem 2.3 (Inverse theorem for biased trilinear forms).

Suppose that $\phi\colon G\times G\times G\to\mathbb{F}_{p}$ is a trilinear
form such that $\operatorname{bias}\phi\geq c$. Then there exists a positive
integer $r\leq O(\log_{p}c^{-1})$, linear forms $\alpha_{1},\dots,$
$\alpha_{r},$ $\beta_{1},\dots,$ $\beta_{r},$ $\gamma_{1},\dots,$
$\gamma_{r}\colon G\to\mathbb{F}_{p}$ and bilinear forms
$\alpha^{\prime}_{1},\dots,$ $\alpha^{\prime}_{r},$
$\beta^{\prime}_{1},\dots,$ $\beta^{\prime}_{r},$ $\gamma^{\prime}_{1},\dots,$
$\gamma^{\prime}_{r}\colon G\times G\to\mathbb{F}_{p}$ such that

$\phi(x,y,z)=\sum_{i\in[r]}\alpha_{i}(x)\alpha^{\prime}_{i}(y,z)+\beta_{i}(y)\beta_{i}^{\prime}(x,z)+\gamma_{i}(z)\gamma^{\prime}_{i}(x,y)$

holds for all $x,y,z\in G$.

Next, we need a result on the number of certain arrangements of points
(related to additive quadruples) inside dense subsets of vector spaces.

######  Lemma 2.4.

Let $G$ be a finite-dimensional vector space over $\mathbb{F}_{p}$ and let
$A\subset G$ be a subset of density $c$. Then

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}\in
G}\mathbbm{1}_{A}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(x_{2}-b_{2}+b_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{A}(z_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-z_{1})\geq
c^{32}.$

###### Proof.

Since the density of $A$ is $c$ we have333In the first step we prove the
standard fact that the number of additive quadruples in a set in $G$ of
density $c$ is at least $c^{4}|G|^{3}$. We could have simply stated that fact,
but since the rest of the proof uses identical method, we opted to start from
density assumption.

$\displaystyle
c^{4}\leq\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{A}(x)\Big{)}^{4}=\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x,y}\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(y)\Big{)}^{2}=$
$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{d}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+d)\Big{)}\Big{)}^{2}\hskip
56.9055pt\text{(change of variables)}$ $\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{d}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{A}(x)\mathbbm{1}_{A}(x+d)\Big{)}^{2}\hskip
56.9055pt\text{(by Cauchy-Schwarz)}$ $\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3}}\mathbbm{1}_{A}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{3}).$

Let us make a change of variables and use $a=b_{1}+b_{2}-b_{3}$ instead of
$b_{3}$. Then

$\displaystyle c^{8}\leq$
$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{2}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\Big{)}^{2}=\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{2}}\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\Big{)}\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{2}}\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\Big{)}^{2}\hskip
56.9055pt\text{(by Cauchy-Schwarz)}$ $\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{2},x_{2}}\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a)$
$\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{2},x_{2}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a).$

Apply Cauchy-Schwarz inequality another time to get

$\displaystyle c^{16}\leq$
$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{2},x_{2}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(x_{2})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(b_{1}+x_{2}-a)\Big{)}\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{2},x_{2}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(x_{2})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(b_{1}+x_{2}-a)\Big{)}^{2}$
$\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{2},x_{2}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(x_{2})\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},y_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(y_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(y_{1}+b_{2}-a)\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{1}+x_{2}-a)$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{2},x_{2},y_{1}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{1})\mathbbm{1}_{A}(y_{1}+b_{2}-a),$

where we omitted term $\mathbbm{1}_{A}(y_{1}+x_{2}-a)$ in the last line, which
is fine as all terms take values in the interval $[0,1]$.
Make another change of variables and use $y_{3}=y_{1}+b_{2}-a$ instead of
$y_{1}$ so we get

$c^{16}\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{2},x_{2},y_{3}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{1}+b_{2}-a)\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+a-b_{2}).$

Make a further change of variables and use $b_{3}=b_{1}+b_{2}-a$ instead of
$b_{2}$. Thus

$\displaystyle c^{32}\leq$
$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{3},x_{2},y_{3}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{3}+a-b_{1})\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\Big{)}^{2}$
$\displaystyle=$
$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{3},x_{2},y_{3}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(y_{3})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{3}+a-b_{1})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\Big{)}\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{3},x_{2},y_{3}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(y_{3})\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{3}+a-b_{1})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\Big{)}^{2}$
$\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{3},x_{2},y_{3}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(y_{3})\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},z_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{3}+a-b_{1})\mathbbm{1}_{A}(b_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{A}(z_{1})\mathbbm{1}_{A}(b_{3}+a-z_{1})\mathbbm{1}_{A}(z_{1}+x_{2}-a)\mathbbm{1}_{A}(y_{3}+z_{1}-b_{3})$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b_{1},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{A}(a)\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{3}+a-b_{1})\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(b_{1}+x_{2}-a)$
$\displaystyle\hskip
227.62204pt\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{A}(z_{1})\mathbbm{1}_{A}(b_{3}+a-z_{1})$

where we used Cauchy-Schwarz inequality in the first inequality above and
omitted terms $\mathbbm{1}_{A}(z_{1}+x_{2}-a)$ and
$\mathbbm{1}_{A}(y_{3}+z_{1}-b_{3})$ in the second. We make the final change
of variables and use $b_{2}=b_{3}+a-b_{1}$ instead of $a$. Thus

$\displaystyle c^{32}\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(x_{2}-b_{2}+b_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{A}(z_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-z_{1}).\qed$

We also need the fact that linear maps of very high rank between spaces of
same dimension can be efficiently modified to yield an isomorphism. The proof
is elementary.

######  Lemma 2.5.

Let $A$ and $B$ be two vector spaces of dimension $d$ and let $\phi\colon A\to
B$ be a linear map of rank $d-\ell$. Then there exists an isomorphism
$\psi\colon A\to B$ such that $\operatorname{rank}(\phi-\psi)\leq\ell$.

###### Proof.

By the rank-nullity theorem, the kernel $K$ of $\phi$ has dimension $\ell$.
Let $U$ be an arbitrary subspace such that $A=K\oplus U$, and let $\pi\colon
A\to K$ be the resulting projection onto $K$. Then $I=\phi(U)$ is the image of
the map $\phi$. Let $B=I\oplus V$ for some subspace $V$. Then $\dim V=d-\dim
I=\ell=\dim K$, so there exists a linear isomorphism $\theta\colon K\to V$.
Define $\psi=\phi+\theta\circ\pi$. We claim that $\psi$ is isomorphism. It
suffices to prove that $\psi$ is injective. To that end, let $x\in A$ be such
that $\psi(x)=0$. Then $\phi(x)+\theta(\pi(x))=0$. However, $\phi(x)\in I$ and
$\theta(\pi(x))\in V$, so as $I\cap V=0$, it follows that $\phi(x)=0$ and
$\theta(\pi(x))=0$. Hence, $x\in K$ and $\pi(x)=0$, so $x=0$, as desired.∎

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_{p}$ and
let $A\subseteq G$ be a subset $G\times H$. Let $\phi\colon A\to K$ be a map.
We say that $\phi$ _respects_ all horizontal additive quadruples if for all
$y\in H$ and $x_{1},x_{2},x_{3},x_{4}\in G$ such that
$x_{1}+x_{2}=x_{3}+x_{4}$ and $(x_{i},y)\in A$ for $i\in[4]$ we have
$\phi(x_{1},y)+\phi(x_{2},y)=\phi(x_{3},y)+\phi(x_{4},y)$. Analogously, we say
that $\phi$ _respects_ all vertical additive quadruples if the same condition
holds with the roles of $G$ and $H$ reversed. If $\phi$ respects all
horizontal and vertical additive quadruples we say that $\phi$ is a _Freiman
bihomomorphism_. It turns out that global biaffine maps are essentially the
only sources of Freiman bihomomorphisms. This theorem was first proved in [10]
by Gowers and the author and the bounds were later improved by Lovett
(personal communication), and by Kim, Li and Tidor [19] by optimizing some
steps in the original proof.

######  Theorem 2.6.

Let $A\subseteq G$ be a subset $G\times H$ of density $c$. Suppose that
$\phi\colon G\times H\to K$ is a Freiman bihomomorphism. Then there eixsts a
biaffine map $\Phi\colon G\times H\to K$ such that $\Phi=\phi$ holds for at
least $c^{\prime}|G||H|$ points in $A$, where
$c^{\prime}=\exp\Big{(}-\exp(O(\log^{O(1)}c^{-1}))\Big{)}$.

In this paper, we shall use the following corollary.

######  Corollary 2.7.

Let $G,H,K$ be finite-dimensional vector spaces over $\mathbb{F}_{p}$ and let
$A\subseteq G$ be a subset of density $c$ and let for each $a\in A$ be given a
linear map $\phi_{a}\colon H\to K$. Suppose that for at least $c|G|^{3}$
additive quadruples $(a_{1},a_{2},a_{3},a_{4})\in A^{4}$ we have

$\operatorname{rank}\Big{(}\phi_{a_{1}}+\phi_{a_{2}}-\phi_{a_{3}}-\phi_{a_{4}}\Big{)}\leq
r.$ (5)

Then there exists a map $\Phi\colon G\times H\to K$, which is affine in the
first coordinate and linear in the second, such that
$\operatorname{rank}(\Phi(a,\cdot)-\phi_{a})\leq\exp\Big{(}(\log
c^{-1}+r)^{O(1)}\Big{)}$ for at least $\exp\Big{(}-\exp\Big{(}(\log
c^{-1}+r)^{O(1)}\Big{)}\Big{)}|G|$ of $a\in A$, where $\Phi(a,\cdot)$ stands
for the map from $H$ to $K$ given by $b\mapsto\Phi(a,b)$.

Let us remark that a similar result was proved by Kazhdan and Ziegler in [18]
by using the inverse theorem for $\mathsf{U}^{4}$ norm. Unlike the result
above, Kazhdan and Ziegler assume that bound (5) holds for all choices of
additive quadruples in $G$. It is very likely that their proof can be modified
to work in this setting as well, but it turns out that property (5) almost
implies that $\phi$ is a Freiman bihomomorphism, so applying Theorem 2.6 gives
a shorter deduction. This is no surprise as Theorem 2.6 was initially proved
in order to give a quantitative inverse theorem for $\mathsf{U}^{4}$ norm.

###### Proof.

Consider map $\psi\colon A\times H\to K$ given by $\psi(a,x)=\phi_{a}(x)$.
Since each $\phi_{a}$ is linear, the map $\psi$ respects all vertical additive
quadruples. When it comes to horizontal additive quadruples, given any
additive quadruple $(a_{1},a_{2},a_{3},a_{4})\in A^{4}$ such that

$\operatorname{rank}\Big{(}\phi_{a_{1}}+\phi_{a_{2}}-\phi_{a_{3}}-\phi_{a_{4}}\Big{)}\leq
r,$

it follows that the codimension of kernel of
$\phi_{a_{1}}+\phi_{a_{2}}-\phi_{a_{3}}-\phi_{a_{4}}$ is at most $r$, so we
have at least $p^{-r}|H|$ of elements $y\in H$ for which $\psi$ respects
horizontal additive quadruple
$\Big{(}(a_{1},y),(a_{2},y),(a_{3},y),(a_{4},y)\Big{)}$. By averaging and
using Theorem 2.2 we may pass to a subset $B\subseteq A\times H$, of size
$|B|\geq\exp\Big{(}-(\log c^{-1}+r)^{O(1)}\Big{)}|G||H|$, on which $\psi$ is a
Freiman bihomomorphism. By Theorem 2.6 there exists a further subset
$B^{\prime}\subseteq B$ and biaffine map $\Phi\colon G\times H\to K$ such that
$\Phi=\psi$ on $B^{\prime}$, with $|B^{\prime}|\geq c_{1}|G||H|$ where
$c_{1}\geq\exp\Big{(}-\exp\Big{(}(\log c^{-1}+r)^{O(1)}\Big{)}\Big{)}$. Define
$\tilde{\Phi}(x,y)=\Phi(x,y)-\Phi(x,0)$, which is linear in $y$. Take now any
$a\in A$ for which
$|B^{\prime}_{a\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}|\geq\frac{c_{1}}{2}|H|$.
For each $y\in
B^{\prime}_{a\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}$
we have $\Phi(a,y)=\phi_{a}(y)$. By Cauchy-Schwarz inequality, we have
$\tilde{\Phi}(a,y-y^{\prime})=\Phi(a,y)-\Phi(a,y^{\prime})=\phi_{a}(y)-\phi_{a}(y^{\prime})=\phi_{a}(y-y^{\prime})$
for at least $\frac{c_{1}^{2}}{4}|H|^{2}$ choices of $(y,y^{\prime})$. Thus,
$\operatorname{rank}\Big{(}\tilde{\Phi}(a,\cdot)-\phi_{a}\Big{)}\leq\log_{p}(4c_{1}^{-2})$,
for each such $a$, of which there are at least $\frac{c_{1}}{2}|G|$.∎

## §3 From approximate quadratic varieties to approximate quasirandom linear
systems of subspaces

In this section, we begin the study of approximate quadratic varieties. Our
goal is to obtain an approximate quasirandom linear systems of subspaces that
is closely related to the given quadratic variety. The main result of this
section is the following theorem.

######  Theorem 3.1.

There exists an absolute constant $D\geq 1$ such that the following holds. Let
$V\subseteq G$ be a $(c_{0},\delta,\varepsilon)$-approximate quadratic variety
and suppose that
$\varepsilon\leq\exp\Big{(}-\log^{D}(2c_{0}^{-1})\Big{)}\delta^{288}$. Then
there exist a quantity $c_{1}$, a set $A\subseteq G$ and a collection of
subspaces $W_{a}\leq G$ indexed by elements $a\in A$ such that

  * (i)

$\exp(-\log^{D}(2c_{0}^{-1}))\leq c_{1}\leq 1$,

  * (ii)

$|A|\geq c_{1}|G|$,

  * (iii)

for each $a\in A$ and $b\in W_{a}$ we have ${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}(b)\geq c_{1}\delta^{15}$,

  * (iv)

$c_{1}\leq\frac{|W_{a}|}{\delta|G|}\leq c_{1}^{-1}$ holds for all $a\in A$,

  * (v)

for $r\in[9]$, for all but at most $D\varepsilon\delta^{-32r}|G|^{r}$ choices
of $(a_{1},\dots,a_{r})\in A^{r}$ we have

$\frac{|W_{a_{1}}\cap W_{a_{2}}\cap\dots\cap W_{a_{r}}|}{\delta^{r}|G|}\leq
c_{1}^{-1},$

and,

  * (vi)

for at least $c_{1}|A|^{6}$ 6-tuples $(b_{1},b_{2},b_{3},x_{2},y_{3},z_{1})\in
A^{6}$ we have that
$b_{1}+b_{2}-b_{3},x_{2}-b_{2}+b_{3},y_{3}+b_{1}-b_{3},b_{1}+b_{2}-z_{1}\in A$
and

$c_{1}\delta^{6}|G|\leq|W_{b_{1}}\cap W_{b_{2}}\cap W_{b_{3}}\cap
W_{b_{1}+b_{2}-b_{3}}\cap W_{x_{2}}\cap W_{x_{2}-b_{2}+b_{3}}\cap
W_{y_{3}}\cap W_{y_{3}+b_{1}-b_{3}}\cap W_{z_{1}}\cap W_{b_{1}+b_{2}-z_{1}}|.$

Throughout the section $V,c_{0},\delta$ and $\varepsilon$ will be fixed and
$V$ will be a $(c_{0},\delta,\varepsilon)$-approximate quadratic variety.

### 3.1. Elementary estimates

We frequently need control expressions such as
$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b,x}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{a,b}\mathbbm{1}_{V}(x)$ using the fact that the
$\mathsf{U}^{2}$ norm of the difference $\mathbbm{1}_{V}-\delta$ is small. In
order to be efficient we record the following lemma which says that if we can
find two variables in such an expression that both appear in a single
occurrence of $\mathbbm{1}_{V}$, then we may replace that term with $\delta$.

######  Lemma 3.2.

Let $f_{1},\dots,f_{r}\colon G\to\mathbb{D}$ be functions and let
$\lambda_{i,j}\in\mathbb{F}_{p}$ be coefficients for $i\in[0,r]$, $j\in[s]$.
Suppose that for some distinct indices $a,b\in[s]$ we have
$\lambda_{i,a}\lambda_{i,b}$ non-zero if and only if $i=0$. Then

$\Big{|}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{1},\dots,x_{s}}\mathbbm{1}_{V}\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}\prod_{i\in[r]}f_{i}\Big{(}\sum_{j\in[s]}\lambda_{i,j}x_{j}\Big{)}\Big{)}-\delta\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{1},\dots,x_{s}}\prod_{i\in[r]}f_{i}\Big{(}\sum_{j\in[s]}\lambda_{i,j}x_{j}\Big{)}\Big{)}\Big{|}\leq\varepsilon.$

###### Proof.

Write $g(x)=\mathbbm{1}_{V}(x)-\delta$. Let $I$ be the set of all indices
$i\in[r]$ such that $\lambda_{i,a}\not=0$. Standard applications of Cauchy-
Schwarz inequality give

$\displaystyle\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{1},\dots,x_{s}}g\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}\prod_{i\in[r]}f_{i}\Big{(}\sum_{j\in[s]}\lambda_{i,j}x_{j}\Big{)}\Big{|}^{4}\leq$
$\displaystyle\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{[s]\setminus\\{a\\}}}\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{a}}g\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}\prod_{i\in
I}f_{i}\Big{(}\sum_{j\in[s]}\lambda_{i,j}x_{j}\Big{)}\Big{|}^{2}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{[s]},y}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{\lambda_{0,a}y}g\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}\prod_{i\in
I}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{\lambda_{i,a}y}f_{i}\Big{(}\sum_{j\in[s]}\lambda_{i,j}x_{j}\Big{)}\Big{|}^{2}$
$\displaystyle\leq$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{[s]\setminus\\{b\\}},y}\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{b}}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{\lambda_{0,a}y}g\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x_{[s]},y,z}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{\lambda_{0,a}y}{\Delta\\!\\!\\!\\!\\!\hbox{\raisebox{1.29167pt}{\tiny\
\textbullet}}}\,_{\lambda_{0,b}z}g\Big{(}\sum_{j\in[s]}\lambda_{0,j}x_{j}\Big{)}=\|g\|_{\mathsf{U}^{2}}^{4}.\qed$

Note that we may still use this lemma if a term $\mathbbm{1}_{V}$ has a
variable $x_{j}$ that occurs only at that place as we can introduce another
dummy variable $y$ and make a change of variables $x_{j}\mapsto x_{j}+y$.

We now use Lemma 3.2 to prove some easy properties of the set $V$. The most
basic consequence of $\mathsf{U}^{2}$ uniformity is that translates of $V$
have roughly the same intersection size.

######  Claim 3.3.

Let $r$ be a positive integer. We have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},\dots,a_{r}}\Big{|}|V\cap(V-a_{1})\cap(V-a_{2})\cap\dots\cap(V-a_{r})|-\delta^{r+1}|G|\Big{|}^{2}\leq
4(r+1)\varepsilon.$

In particular, for all but at most $4(r+1)\varepsilon^{1/2}|G|^{r}$ $r$-tuples
$(a_{1},\dots,a_{r})\in G$ we have
$\Big{|}|V\cap(V-a_{1})\cap(V-a_{2})\cap\dots\cap(V-a_{r})|-\delta^{r+1}|G|\Big{|}\leq\varepsilon^{1/4}|G|$.

###### Proof.

Consider the expression

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},\dots,a_{r}}$
$\displaystyle\Big{|}|V\cap(V-a_{1})\cap(V-a_{2})\cap\dots\cap(V-a_{r})|-\delta^{r+1}|G|\Big{|}^{2}$
$\displaystyle=$
$\displaystyle|G|^{2}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},\dots,a_{r}}\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a_{1})\dots\mathbbm{1}_{V}(x+a_{r})-\delta^{r+1}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle|G|^{2}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}a_{1},\dots,a_{r}\\\
x,y\end{subarray}}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a_{1})\dots\mathbbm{1}_{V}(x+a_{r})\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(y+a_{1})\dots\mathbbm{1}_{V}(y+a_{r})$
$\displaystyle\hskip
199.16928pt-2\delta^{r+1}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a_{1})\dots\mathbbm{1}_{V}(x+a_{r})+\delta^{2r+2}\Big{)}.$
(6)

Applying Lemma 3.2 $2r+2$ times to terms $\mathbbm{1}_{V}(x+a_{i})$ using $x$
and $a_{i}$ for $i\in[r]$, $\mathbbm{1}_{V}(y+a_{i})$ using $y$ and $a_{i}$
for $i\in[r]$, $\mathbbm{1}_{V}(x)$ using $x$ and $\mathbbm{1}_{V}(y)$ using
$y$ in that order,444Note that the first $2r$ applications of Lemma 3.2 have
turned terms $\mathbbm{1}_{V}(x+a_{i})$ and $\mathbbm{1}_{V}(y+a_{i})$ into
copies of $\delta$ and after those $2r$ steps $x$ and $y$ only appear in terms
$\mathbbm{1}_{V}(x)$ and $\mathbbm{1}_{V}(y)$. we see that

$\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}a_{1},\dots,a_{r}\\\
x,y\end{subarray}}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a_{1})\dots\mathbbm{1}_{V}(x+a_{r})\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(y+a_{1})\dots\mathbbm{1}_{V}(y+a_{r})-\delta^{2r+2}\Big{|}\leq(2r+2)\varepsilon.$

Applying Lemma 3.2 to the simpler expression
$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a_{1},\dots,a_{r},x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a_{1})\dots\mathbbm{1}_{V}(x+a_{r})$
in a similar fashion, we conclude that expression (6) is at most
$(4r+4)\varepsilon$. The second part of the conclusion follows after
averaging.∎

Next, we give an upper bound on intersections
$|V\cap(V-a)\cap(V-b)\cap(V-a-b)|$. We may no longer give a lower bound for
the following reason. Recall that we expect $V$ to be closely related to a
quadratic variety $Q=\\{x\in G\colon\beta(x,x)=0\\}$, defined by some high
rank symmetric bilinear map $\beta\colon G\times G\to\mathbb{F}_{p}^{r}$. If
$x\in V\cap(V-a)\cap(V-b)\cap(V-a-b)$ then
$\beta(x,x)=\beta(x+a,x+a)=\beta(x+b,x+b)=\beta(x+a+b,x+a+b)=\lambda$, so
$\beta(a,b)=0$. Notice also that $Q\cap Q-a\cap Q-b\subseteq Q-a-b$. Thus, we
expect that the size of intersection $|V\cap(V-a)\cap(V-b)\cap(V-a-b)|$ is
about $\delta^{3}|G|$ when $\beta(a,b)=0$, and significantly smaller
otherwise.

######  Claim 3.4.

For all but at most $12\varepsilon^{1/2}|G|^{2}$ pairs of elements $(a,b)\in
G^{2}$ we have
$\Big{|}|V\cap(V-a)\cap(V-b)|-\delta^{3}|G|\Big{|}\leq\varepsilon^{1/4}|G|$.
In particular
$|V\cap(V-a)\cap(V-b)\cap(V-a-b)|\leq(\delta^{3}+\varepsilon^{1/4})|G|$ holds
for all but at most $12\varepsilon^{1/2}|G|^{2}$ pairs of elements $(a,b)\in
G^{2}$.

###### Proof.

Consider

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\Big{|}\frac{1}{|G|}|V\cap(V-a)\cap(V-b)|-\delta^{3}\Big{|}^{2}=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}a,b\\\
x,y\end{subarray}}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+b)\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(y+a)\mathbbm{1}_{V}(y+b)$
$\displaystyle\hskip
56.9055pt-2\delta^{3}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+b)+\delta^{6}.$

Applying Lemma 3.2 6 times for terms $\mathbbm{1}_{V}(x+a)$ using $x$ and $a$,
$\mathbbm{1}_{V}(x+b)$ using $x$ and $b$, $\mathbbm{1}_{V}(x)$ using $x$,
$\mathbbm{1}_{V}(y+a)$ using $y$ and $a$, $\mathbbm{1}_{V}(y+b)$ using $y$ and
$b$ and $\mathbbm{1}_{V}(y)$ using $y$ in that order, we see that

$\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}a,b\\\
x,y\end{subarray}}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+b)\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(y+a)\mathbbm{1}_{V}(y+b)-\delta^{6}\Big{|}\leq
6\varepsilon.$

Similar argument for
$\mathop{\mathbb{E}}_{x,a,b}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+b)$
shows that the expression above is at most $12\varepsilon$. The first part of
the claim now follows.
For the second part of the claim simply note that
$|V\cap(V-a)\cap(V-b)\cap(V-a-b)|\leq|V\cap(V-a)\cap(V-b)|$.∎

We need one more claim of this form. Notice that we have 8 $\mathbbm{1}_{V}$
terms in the expression below, yet the upper bound is about $\delta^{5}$,
which is due to algebraic dependencies between arguments of $\mathbbm{1}_{V}$.

######  Claim 3.5.

For all but at most $20\sqrt{\varepsilon}|G|^{4}$ of $(a,d_{1},d_{2},d_{3})\in
G^{4}$ we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}+a)\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{1}-d_{2}+a)\mathbbm{1}_{V}(x+d_{3})\mathbbm{1}_{V}(x+d_{3}+a)\leq\delta^{5}+\sqrt[4]{\varepsilon}.$
(7)

Having terms $\mathbbm{1}_{V}(x+d_{2})\mathbbm{1}_{V}(x+d_{2}+a)$ instead of
$\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{1}-d_{2}+a)$ would be more
natural, but we opted for this formulation as we get this expression when the
claim is applied.

###### Proof.

Notice that the expression in question is at most

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{3}).$

We now show that

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,d_{1},d_{2},d_{3}}\Big{|}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{3})-\delta^{5}\Big{|}^{2}\leq
20\varepsilon.$

By (by now) usual argument based on Lemma 3.2 we get that

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,d_{1},d_{2},d_{3}}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{3})\Big{)}^{2}$
$\displaystyle\hskip
56.9055pt=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}a,d_{1},d_{2},d_{3}\\\
x,y\end{subarray}}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{V}(y)\mathbbm{1}_{V}(y+a)\mathbbm{1}_{V}(y+d_{1})\mathbbm{1}_{V}(y+d_{1}-d_{2})\mathbbm{1}_{V}(y+d_{3})$

differs from $\delta^{10}$ by at most $10\varepsilon$. Namely, we apply Lemma
3.2 for term $\mathbbm{1}_{V}(x+d_{3})$ using $x$ and $d_{3}$, for term
$\mathbbm{1}_{V}(x+d_{1}-d_{2})$ using $x$ and $d_{2}$, for term
$\mathbbm{1}_{V}(x+d_{1})$ using $x$ and $d_{1}$, for term
$\mathbbm{1}_{V}(x+a)$ using $x$ and $a$, for term $\mathbbm{1}_{V}(x)$ using
$x$, in that order, and similarly for terms involving $y$. Similarly,

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,d_{1},d_{2},d_{3},x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{3})$

differs from $\delta^{5}$ by at most $5\varepsilon$. The claim follows after
averaging.∎

### 3.2. Additive structure

In this subsection show that the set of large values iterated convolutions of
$\mathbbm{1}_{V\cap V-a}$ has significant additive structure for many $a\in
G$. We begin with an important technical definition.
Let $\eta>0$ be a parameter. We say that an element $a\in G$ is _$\eta$
-regular_ if it satisfies

  * (i)

$\frac{1}{2}\delta^{2}|G|\leq|V\cap V-a|\leq 2\delta^{2}|G|$,

  * (ii)

for all but at most $\eta|G|$ of $b\in G$ we have

$|V\cap(V-a)\cap(V-b)\cap(V-a-b)|\leq 2\delta^{3}|G|,$

and

  * (iii)

for all but at most $\eta|G|^{3}$ of triples $(d_{1},d_{2},d_{3})\in G$ we
have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+a)\mathbbm{1}_{V}(x+d_{1})\mathbbm{1}_{V}(x+d_{1}+a)\mathbbm{1}_{V}(x+d_{1}-d_{2})\mathbbm{1}_{V}(x+d_{1}-d_{2}+a)\mathbbm{1}_{V}(x+d_{3})\mathbbm{1}_{V}(x+d_{3}+a)\leq
2\delta^{5}.$

Let us show that regular elements are abundant.

######  Proposition 3.6.

Suppose that $\varepsilon\leq 2^{-4}\delta^{20}$. Then all but at most
$40\sqrt[4]{\varepsilon}|G|$ elements are $\varepsilon^{1/4}$-regular.

###### Proof.

By Claim 3.3, provided that $\varepsilon\leq 2^{-4}\delta^{8}$, we have

$\Big{|}|V\cap(V-a)|-\delta^{2}|G|\Big{|}\leq\frac{1}{2}\delta^{2}|G|$

for all but at most $8\varepsilon^{1/2}|G|$ elements $a\in G$. Next, provided
that $\varepsilon\leq\delta^{12}$, it follows from Claim 3.4 that

$|V\cap(V-a)\cap(V-b)\cap(V-a-b)|\leq 2\delta^{3}|G|$ (8)

holds for all but at most $12\varepsilon^{1/2}|G|^{2}$ pairs of elements
$(a,b)\in G^{2}$. Hence, the number of $a$ such that (8) fails for at least
$\varepsilon^{1/4}|G|$ elements $b\in G$ is at most $12\varepsilon^{1/4}|G|$.
Finally, using Claim 3.5 and provided $\varepsilon\leq\delta^{20}$, we
conclude that the number of elements $a$ such that (7) fails for at least
$\sqrt[4]{\varepsilon}|G|^{3}$ triples $(d_{1},d_{2},d_{3})\in G^{3}$ is at
most $20\varepsilon^{1/4}|G|$. The proposition now follows.∎

The following proposition finds desired additive structure if we are given a
regular element.

######  Proposition 3.7.

Let $\eta>0$ and let $a\in G$ be an $\eta$-regular element. Let $P\subseteq G$
be a set and let $c^{\prime}>0$ be such that

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{P}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq c^{\prime}\delta^{7}.$ (9)

Suppose that $\eta\leq 2^{-41}{c^{\prime}}^{8}\delta^{8}$. Then there exists a
subset $\tilde{P}\subseteq P$ which contains at least
$2^{-42}{c^{\prime}}^{8}\delta^{3}|G|^{3}$ additive quadruples, has size
$\frac{c^{\prime}\delta}{16}|G|\leq|\tilde{P}|\leq
2^{9}{c^{\prime}}^{-2}\delta|G|$ and every element $b\in\tilde{P}$ satisfies

$\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\geq\frac{1}{8}c^{\prime}\delta^{3}.$

###### Proof.

By assumption (ii) of $\eta$-regularity, for all but at most $\eta|G|$ of
$b\in G$ we have

$\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)=\mathop{\mathbb{E}}_{x}\mathbbm{1}_{V\cap(V-a)}(x+b)\mathbbm{1}_{V\cap(V-a)}(x)=\frac{1}{|G|}|V\cap(V-a)\cap(V-b)\cap(V-a-b)|\leq
2\delta^{3}.$ (10)

By removing $b$ for elements $b\in G$ when this inequality fails, we may find
a subset $P^{\prime}\subseteq P$ of size $|P^{\prime}|\geq|P|-\eta|G|$ such
that (10) holds for all $b\in P^{\prime}$. From (9) we deduce that

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{P^{\prime}}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq c^{\prime}\delta^{7}-\eta.$

Let $\tilde{P}\subseteq P^{\prime}$ be the set of all elements $b\in
P^{\prime}$ such that $\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\geq\frac{1}{8}c^{\prime}\delta^{3}$. Then

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{P^{\prime}\setminus\tilde{P}}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq$
$\displaystyle\frac{1}{8}c^{\prime}\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{P^{\prime}\setminus\tilde{P}}(b)\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-a}(b)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}c^{\prime}\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-a}(b)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8}c^{\prime}\delta^{3}\Big{(}\frac{|V\cap
V-a|}{|G|}\Big{)}^{2}\leq\frac{1}{2}c^{\prime}\delta^{7},$

where we used assumption (i) of $\eta$-regularity in the last step.
Hence

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{\tilde{P}}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq\frac{1}{2}c^{\prime}\delta^{7}-\eta,$

which, since $\tilde{P}\subseteq P^{\prime}$, also gives
$|\tilde{P}|\geq\Big{(}\frac{c^{\prime}\delta}{8}-\eta\delta^{-6}\Big{)}|G|\geq\frac{c^{\prime}\delta}{16}|G|$,
since $\eta\leq\frac{c^{\prime}\delta^{7}}{16}$. The same argument as the one
above and the fact that $\tilde{P}\subseteq P^{\prime}$ imply

$\frac{|\tilde{P}|}{|G|}\Big{(}\frac{1}{8}c^{\prime}\delta^{3}\Big{)}^{2}\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{\tilde{P}}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq
2\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\leq 8\delta^{7}$

so $|\tilde{P}|\leq 2^{9}{c^{\prime}}^{-2}\delta|G|$.

We say that $b$ is a _popular difference_ if $b\in\tilde{P}$. Consider the
bipartite graph $\Gamma$ with both vertex classes $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ being a copy of $V\cap(V-a)$ and edges between
$x\in\mathcal{C}_{1}$ and $y\in\mathcal{C}_{2}$ if $x-y$ is a popular
difference. By a _cycle of length 4_ we mean an ordered quadruple of vertices
$v_{1},v_{2},v_{3},v_{4}$ such that $v_{1}v_{2},\dots,v_{4}v_{1}$ are
edges.555It would be more precise to call these quadruples homomorphisms from
$C_{4}$ to the given graph. This is a graph of density

$\frac{\sum_{b\in\tilde{P}}\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)|G|}{|V\cap(V-a)|^{2}}\geq\frac{\frac{c^{\prime}\delta}{16}|G|\cdot\frac{1}{8}c^{\prime}\delta^{3}|G|}{4\delta^{4}|G|^{2}}\geq
2^{-9}{c^{\prime}}^{2}$

so it has $2^{-36}{c^{\prime}}^{8}|V\cap(V-a)|^{4}$ cycles of length 4. Notice
that each such cycle gives an additive quadruple in $\tilde{P}$: namely if
$x_{1},x_{2}$ are vertices in class $\mathcal{C}_{1}$ and $y_{1}$ and $y_{2}$
are vertices in class $\mathcal{C}_{2}$, then $p_{i,j}=x_{i}-y_{j}$ is a
popular difference and $p_{1,1}+p_{2,2}=p_{1,2}+p_{2,1}$. On the other hand,
if we fix an additive quadruple $(d_{1},d_{2},d_{3},d_{2}+d_{3}-d_{1})$ of
popular differences it gives

$\sum_{y}\mathbbm{1}_{V\cap(V-a)}(y)\mathbbm{1}_{V\cap(V-a)}(y+d_{1})\mathbbm{1}_{V\cap(V-a)}(y+d_{1}-d_{2})\mathbbm{1}_{V\cap(V-a)}(y+d_{3})$

cycles of length 4. This quantity is bounded above by $2\delta^{5}|G|$ for all
but at most $\eta|G|^{3}$ of triples $(d_{1},d_{2},d_{3})\in G$ by assumption
(iii) of $\eta$-regularity, while it is always at most $|G|$. Let $Q$ be the
number of additive quadruples in $\tilde{P}$. By double-counting we get

$2^{-40}{c^{\prime}}^{8}\delta^{8}|G|^{4}\leq
2^{-36}{c^{\prime}}^{8}|V\cap(V-a)|^{4}\leq Q\cdot
2\delta^{5}|G|+\eta|G|^{3}\cdot|G|.$

Since $\eta\leq 2^{-41}{c^{\prime}}^{8}\delta^{8}$, we conclude that $Q\geq
2^{-42}{c^{\prime}}^{8}\delta^{3}|G|^{3}$, as desired.∎

Let $\xi>0$ be a parameter to be chosen later (it will be a function of $c$)
and let $A^{\xi}\subseteq G$ be the set of all elements $a\in G$ which are
$\varepsilon^{1/4}$-regular and satisfy

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq\xi\delta^{7}.$

We first show that $A^{\xi}$ is large.

######  Claim 3.8.

Suppose that $\xi\leq c_{0}/2$ and $\varepsilon\leq
2^{-32}c_{0}^{4}\delta^{28}$. We have $|A^{\xi}|\geq\frac{c_{0}}{40}|G|$ and

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\mathbbm{1}_{G\setminus
A^{\xi}}(a)\Big{(}\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq\xi\delta^{7}+40\sqrt[4]{\varepsilon}.$ (11)

###### Proof.

Write $A=A^{\xi}$ to simplify the notation. The fact that $V$ is a
$(c_{0},\delta,\varepsilon)$-approximate quadratic variety implies that

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq c_{0}\delta^{7}.$ (12)

Let $I$ be the set of elements $a\in G$ which are not
$\varepsilon^{1/4}$-regular. Proposition 3.6 implies $|I|\leq
40\sqrt[4]{\varepsilon}|G|$. Their contribution to (12) is

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\mathbbm{1}_{I}(a)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a}\mathbbm{1}_{I}(a)\leq
40\sqrt[4]{\varepsilon}.$

Furthermore, the contribution from $G\setminus(I\cup A)$ is

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\mathbbm{1}_{I^{c}\cap
A^{c}}(a)\Big{(}\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a}\mathbbm{1}_{I^{c}\cap
A^{c}}(a)\bigg{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\bigg{)}\leq\xi\delta^{7},$

from which we obtain (11).
To prove the bounds on $|A|$, notice first that if $a$ is
$\varepsilon^{1/4}$-regular then

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\frac{|V\cap(V-a)\cap(V-b)\cap(V-a-b)|^{2}}{|G|^{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt[4]{\varepsilon}+2\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\frac{|V\cap(V-a)\cap(V-b)\cap(V-a-b)|}{|G|}$
$\displaystyle\leq$
$\displaystyle\sqrt[4]{\varepsilon}+2\delta^{3}\frac{|V\cap
V-a|^{2}}{|G|^{2}}\leq\sqrt[4]{\varepsilon}+8\delta^{7}\leq 10\delta^{7},$

provided $\varepsilon\leq\delta^{28}$, where we used assumption (ii) of
$\sqrt[4]{\varepsilon}$-regularity in the second line and assumption (i) of
$\sqrt[4]{\varepsilon}$-regularity in the third line. Combining (11) and (12)
we get

$10\delta^{7}\frac{|A|}{|G|}\geq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a}\mathbbm{1}_{A}(a)\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq(c_{0}-\xi)\delta^{7}-40\sqrt[4]{\varepsilon}\geq\frac{c_{0}\delta^{7}}{4},$

using the assumptions on $\xi$ and $\varepsilon$ in the last inequality. ∎

As a corollary, we get lower bounds on the number of occurrences of another
configuration in $V$.

######  Corollary 3.9.

Provided $\varepsilon\leq 2^{-200}c_{0}^{8}\delta^{28}$, we have

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-b_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{1}}(w)\,\,\mathbbm{1}_{V\cap V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)\,\,\mathbbm{1}_{V\cap V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)\,\,\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w)\geq\exp\Big{(}-O\Big{(}\log^{O(1)}(2c_{0}^{-1})\Big{)}\Big{)}\delta^{36}.$

###### Proof.

In this proof, we set temporarily $A=A^{c_{0}/2}$. For each $w\in A$ we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a}\Big{(}\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(a)\Big{)}^{2}\geq\frac{c_{0}}{2}\delta^{7}.$

By Claim 3.8 we have $|A|\geq\frac{c}{40}|G|$. For each $w\in A$, apply
Proposition 3.7 with set $P=G$ to obtain $\tilde{P}_{w}\subseteq G$ which
contains at least $2^{-50}c_{0}^{8}\delta^{3}|G|^{3}$ additive quadruples, has
size $|\tilde{P}_{w}|\leq 2^{11}c_{0}^{-2}\delta|G|$ and every element
$a\in\tilde{P}_{w}$ satisfies $\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(a)\geq\frac{1}{16}c_{0}\delta^{3}$. Theorem 2.1 applies to give a
subspace666Averaging over elements $a_{2},a_{3},a_{4}$ in conclusion of
Theorem 2.1 we get a coset instead of a subspace for $S_{w}$, but then we
simply take a subspace contatining $S_{w}$ with dimension greater by 1.
$S_{w}$ such that $|S_{w}|\geq c_{1}\delta|G|$ and $|\tilde{P}_{w}\cap
S_{w}|\geq c_{1}|S_{w}|$, where
$c_{1}\geq\exp\Big{(}-O\Big{(}\log^{O(1)}(2c_{0}^{-1})\Big{)}\Big{)}$. Write
$\tilde{P}^{\prime}_{w}=\tilde{P}_{w}\cap S_{w}$.

Note that

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{\tilde{P}_{w}}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{\tilde{P}_{w}}(b_{1})\mathbbm{1}_{\tilde{P}_{w}}(b_{2})\mathbbm{1}_{\tilde{P}_{w}}(b_{3})\mathbbm{1}_{\tilde{P}_{w}}(x_{2})\mathbbm{1}_{\tilde{P}_{w}}(x_{2}-b_{2}+b_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{\tilde{P}_{w}}(y_{3})\mathbbm{1}_{\tilde{P}_{w}}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{\tilde{P}_{w}}(z_{1})\mathbbm{1}_{\tilde{P}_{w}}(b_{1}+b_{2}-z_{1})$
$\displaystyle\hskip
56.9055pt\geq\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\frac{|S_{w}|^{6}}{|G|^{6}}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}\in
S_{w}}\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(b_{1})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(b_{2})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(b_{3})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(x_{2})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(x_{2}-b_{2}+b_{3})$
$\displaystyle\hskip
170.71652pt\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(y_{3})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(z_{1})\mathbbm{1}_{\tilde{P}^{\prime}_{w}}(b_{1}+b_{2}-z_{1})\Big{)}\geq(c_{1}\delta)^{6}\cdot
c_{1}^{32},$

where we used Lemma 2.4 for each $\tilde{P}^{\prime}_{w}$ and $S_{w}$ in the
last step.

However, every element $a\in\tilde{P}_{w}$ satisfies $\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(a)\geq\frac{1}{16}c\delta^{3}$ so we conclude that

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{\tilde{P}_{w}}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(b_{1}+b_{2}-b_{3})$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{\tilde{P}_{w}}(b_{1})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(b_{1})\mathbbm{1}_{\tilde{P}_{w}}(b_{2})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(b_{2})\mathbbm{1}_{\tilde{P}_{w}}(b_{3})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(b_{3})$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{\tilde{P}_{w}}(x_{2})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(x_{2})\mathbbm{1}_{\tilde{P}_{w}}(x_{2}-b_{2}+b_{3})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(x_{2}-b_{2}+b_{3})$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{\tilde{P}_{w}}(y_{3})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(y_{3})\mathbbm{1}_{\tilde{P}_{w}}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(y_{3}+b_{1}-b_{3})$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{\tilde{P}_{w}}(z_{1})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(z_{1})\mathbbm{1}_{\tilde{P}_{w}}(b_{1}+b_{2}-z_{1})\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(b_{1}+b_{2}-z_{1})\geq 2^{-40}c_{0}^{10}c_{1}^{38}\delta^{36}.$ (13)

Finally, notice

$\displaystyle\mathbbm{1}_{V\cap V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-w}(t)=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V\cap
V-w}(x+t)\mathbbm{1}_{V\cap
V-w}(x)=\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V}(x+t)\mathbbm{1}_{V}(x+t+w)\mathbbm{1}_{V}(x)\mathbbm{1}_{V}(x+w)$
$\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{x}\mathbbm{1}_{V-t}(x)\mathbbm{1}_{V-t-w}(x)\mathbbm{1}_{V}(x)\mathbbm{1}_{V-w}(x)=\frac{|V\cap(V-t)\cap(V-w)\cap(W-w-t)|}{|G|},$

which is symmetric in $w$ and $t$, and thus $\mathbbm{1}_{V\cap
V-w}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-w}(t)=\mathbbm{1}_{V\cap V-t}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-t}(w)$.
Using this identity 10 times in (13) completes the proof.∎

Next, we combine Proposition 3.7 with standard additive combinatorial
arguments to approximate the set of large values of
$b\mapsto\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-a}(b)$
by a union of large subspaces for each $a\in A^{\xi}$.

######  Proposition 3.10.

Suppose that $\varepsilon\leq 2^{-500}\xi^{32}\delta^{32}$. There exist
positive integers $M,K\leq\exp\Big{(}\log^{O(1)}(2\xi^{-1})\Big{)}$ such that
the following holds. For each $a\in A^{\xi}$, we may find $m=m_{a}\leq M$
pairs of subspaces $U_{a,1}\leq U^{\prime}_{a,1},U_{a,2}\leq
U^{\prime}_{a,2},\dots,U_{a,m}\leq U^{\prime}_{a,m}$, such that

  * (i)

for each $i\in[m]$ the bounds
$K^{-1}\delta|G|\leq|U_{a,i}|\leq|U^{\prime}_{a,i}|\leq K\delta|G|$ hold,

  * (ii)

for each $i\in[m]$ we have $|U_{a,i}|=p^{-1}|U^{\prime}_{a,i}|$,

  * (iii)

writing $\tilde{U}_{a}=\cup_{i\in[m]}U^{\prime}_{a,i}$, we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{G\setminus\tilde{U}_{a}}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq\frac{\xi}{100}\delta^{7},$ (14)

and,

  * (iv)

for each $b\in\cup_{i\in[m]}U_{a,i}$ the number of ordered 8-tuples
$(x_{1},x_{2},\dots,x_{8})\in(V\cap(V-a))^{8}$ such that
$\sum_{i\in[8]}(-1)^{i}x_{i}=b$ is at least
$(\xi/2)^{O(1)}\delta^{15}|G|^{7}$.

Note that property (iv) in the conclusion can be expressed more succinctly for
each $b\in\cup_{i\in[m]}U_{a,i}$ using iterated convolution notation as

$\displaystyle(\xi/2)^{O(1)}\delta^{15}|G|^{7}\leq$
$\displaystyle\sum_{x_{1},\dots,x_{8}\in G}\mathbbm{1}_{V\cap
V-a}(x_{1})\dots\mathbbm{1}_{V\cap
V-a}(x_{8})\mathbbm{1}\Big{(}\sum_{i\in[8]}(-1)^{i}x_{i}=b\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{x_{1},\dots,x_{7}\in G}\mathbbm{1}_{V\cap
V-a}(x_{1})\dots\mathbbm{1}_{V\cap V-a}(x_{7})\mathbbm{1}_{V\cap
V-a}\Big{(}x_{1}-x_{2}+x_{3}-x_{4}+x_{5}-x_{6}+x_{7}+b\Big{)}$
$\displaystyle=$ $\displaystyle|G|^{7}{\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}(-b).$

Since each $U_{a,i}$ is symmetric, we have ${\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}}^{(8)}\mathbbm{1}_{V\cap
V-a}(b)\geq(\xi/2)^{O(1)}\delta^{15}$ for all $b\in\cup_{i\in[m]}U_{a,i}$.

###### Proof of Proposition 3.10.

Fix $a\in A^{\xi}$. We use an iterative argument, where at each step we find
additional subspaces $U_{a,i}$ and $U^{\prime}_{a,i}$ until (14) is satisfied.
Suppose that so far we have subspaces $U_{a,1}\leq
U^{\prime}_{a,1},U_{a,2}\leq U^{\prime}_{a,2},\dots,U_{a,i-1}\leq
U^{\prime}_{a,i-1}$, for some $i\geq 1$. Assume that (14) fails with the
current definition of $\tilde{U}_{a}$. Let $P\subseteq
G\setminus\tilde{U}_{a}$ be the set of all elements $b\in
G\setminus\tilde{U}_{a}$ such that $\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\geq\frac{\xi}{1000}\delta^{3}$. Then we have

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{G\setminus(\tilde{U}_{a}\cup
P)}(b)\Big{(}\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\leq$
$\displaystyle\frac{\xi}{1000}\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{G\setminus(\tilde{U}_{a}\cup
P)}(b)\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap V-a}(b)$
$\displaystyle\leq$
$\displaystyle\frac{\xi}{1000}\delta^{3}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\leq\frac{\xi}{250}\delta^{7},$

where we used assumption (i) of regularity for $a$. Since (14) fails, we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b}\mathbbm{1}_{P}(b)\Big{(}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\Big{)}^{2}\geq\frac{\xi}{200}\delta^{7}.$

By Proposition 3.7, provided $\sqrt[4]{\varepsilon}\leq
2^{-105}\xi^{8}\delta^{8}$, there exists a subset $\tilde{P}\subseteq P$ such
that $|\tilde{P}|\leq 2^{25}\xi^{-2}\delta|G|$ and $\tilde{P}$ has at least
$2^{-106}\xi^{8}\delta^{3}|G|^{3}$ additive quadruples. By Theorem 2.1, there
exists a subspace $U_{a,i}\leq G$ of size
$|U_{a,i}|\geq\exp\Big{(}-\log^{O(1)}(2\xi^{-1})\Big{)}\delta|G|$ such that
for each $v\in U_{a,i}$ there are at least $(\xi/2)^{O(1)}\delta^{3}|G|^{3}$
quadruples $(b_{1},b_{2},b_{3},b_{4})\in\tilde{P}^{4}$ such that
$b_{1}+b_{2}-b_{3}-b_{4}=v$, from which the last property in the conclusion
follows. (Note that this condition also implies
$|U_{a,i}|\leq\exp\Big{(}-\log^{O(1)}(2\xi^{-1})\Big{)}\delta|G|$.) In
particular, by averaging over $b_{2},b_{3},b_{4}$, we obtain an element $t\in
G$ such that $|t+U_{a,i}\cap
P|\geq\exp\Big{(}-\log^{O(1)}(2\xi^{-1})\Big{)}\delta|G|$. We now set
$U^{\prime}_{a,i}=U_{a,i}+\langle t\rangle$.
By regularity of $a$, we have $\mathop{\mathbb{E}}_{b}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\leq 4\delta^{4}$, so the total number of $b\in G$ such that
$\mathbbm{1}_{V\cap V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)\geq\frac{\xi}{1000}\delta^{3}$ is at most
$(2\xi^{-1})^{O(1)}\delta|G|$. The procedure terminates after at most
$\exp\Big{(}\log^{O(1)}(2\xi^{-1})\Big{)}$ steps, since we cover at least
$\exp\Big{(}-\log^{O(1)}(2\xi^{-1})\Big{)}\delta|G|$ of such elements $b$ by
$U^{\prime}_{a,i}$ that were not covered by $\tilde{U}_{a}$ previously.∎

By the way we defined the set $A^{\xi}$ and unions of subspaces
$\tilde{U}_{a}$, we deduce the following approximation property.

######  Claim 3.11.

Provided $\xi\leq c_{0}/2$ and $\varepsilon\leq 2^{-500}\xi^{32}\delta^{32}$,
we have

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\Big{(}1-\mathbbm{1}_{A^{\xi}}(a)\mathbbm{1}_{\tilde{U}_{a}}(b)\Big{)}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)^{2}\leq 2\xi\delta^{7}+40\sqrt[4]{\varepsilon}.$

###### Proof.

The expression in question is the sum of

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\Big{(}1-\mathbbm{1}_{A^{\xi}}(a)\Big{)}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)^{2}$ (15)

and

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{a,b}\mathbbm{1}_{A^{\xi}}(a)\Big{(}1-\mathbbm{1}_{\tilde{U}_{a}}(b)\Big{)}\mathbbm{1}_{V\cap
V-a}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-a}(b)^{2}$ (16)

which we estimate in turn. The inequality (11) shows that the expression in
(15) is at most $\xi\delta^{7}+40\sqrt[4]{\varepsilon}$. On the other hand, by
properties in the conclusion of Proposition 3.10, we get that the expression
in (16) is at most $\frac{\xi}{100}\delta^{7}$.∎

### 3.3. Finding approximate quasirandom linear systems of subspaces

The following proposition is the key step towards obtaining an approximate
quasirandom linear system of subspaces.

######  Proposition 3.12.

There is an absolute constant $D\geq 1$ such that the following holds. Write
$A=A^{\xi}$. Provided $\xi\leq\exp(-D\log^{D}(2c_{0}^{-1}))$ and
$\varepsilon\leq 2^{-500}\xi^{32}\delta^{288}$, we have

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}b_{1},b_{2},b_{3}\\\
x_{2},y_{3},z_{1}\end{subarray}}\mathbbm{1}_{A}(b_{1}+b_{2}-b_{3})\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{A}(b_{2})\mathbbm{1}_{A}(b_{3})\mathbbm{1}_{A}(x_{2})\mathbbm{1}_{A}(x_{2}-b_{2}+b_{3})\mathbbm{1}_{A}(y_{3})\mathbbm{1}_{A}(y_{3}+b_{1}-b_{3})\mathbbm{1}_{A}(z_{1})\mathbbm{1}_{A}(b_{1}+b_{2}-z_{1})$
$\displaystyle\hskip
56.9055pt|\tilde{U}_{b_{1}+b_{2}-b_{3}}\cap\tilde{U}_{b_{1}}\cap\tilde{U}_{b_{2}}\cap\tilde{U}_{b_{3}}\cap\tilde{U}_{x_{2}}\cap\tilde{U}_{x_{2}-b_{2}+b_{3}}\cap\tilde{U}_{y_{3}}\cap\tilde{U}_{y_{3}+b_{1}-b_{3}}\cap\tilde{U}_{z_{1}}\cap\tilde{U}_{b_{1}+b_{2}-z_{1}}|$
$\displaystyle\hskip
369.88582pt\geq\exp(-D\log^{D}(2c_{0}^{-1}))\delta^{6}|G|.$

###### Proof.

Let us begin by applying Corollary 3.9. Thus

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-b_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{1}}(w)\,\,\mathbbm{1}_{V\cap V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)\,\,\mathbbm{1}_{V\cap V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)\,\,\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)$ $\displaystyle\hskip 56.9055pt\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w)\geq c_{1}\delta^{36},$

where $c_{1}\geq\exp\Big{(}-O\Big{(}\log^{O(1)}(2c_{0}^{-1})\Big{)}\Big{)}$.
In the next step of the proof, we show that we may restrict attention to
elements $b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}$ such that

$b_{1}+b_{2}-b_{3},b_{1},b_{2},b_{3},x_{2},x_{2}-b_{2}+b_{3},y_{3},y_{3}+b_{1}-b_{3},z_{1},b_{1}+b_{2}-z_{1}\in
A$

and

$w\in\tilde{U}_{b_{1}+b_{2}-b_{3}},\tilde{U}_{b_{1}},\tilde{U}_{b_{2}},\tilde{U}_{b_{3}},\tilde{U}_{x_{2}},\tilde{U}_{x_{2}-b_{2}+b_{3}},\tilde{U}_{y_{3}},\tilde{U}_{y_{3}+b_{1}-b_{3}},\tilde{U}_{z_{1}},\tilde{U}_{b_{1}+b_{2}-z_{1}}.$

To see that, we give an upper bound for

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},b_{2},b_{3},x_{2},y_{3},z_{1}}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{w}\Big{(}1-\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{\tilde{U}_{b_{1}}}(w)\Big{)}\mathbbm{1}_{V\cap
V-b_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{V\cap V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)\,\,\mathbbm{1}_{V\cap V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)\,\,\mathbbm{1}_{V\cap V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)\,\,\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w).$ (17)

Using Cauchy-Schwarz inequality (note below that
$1-\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{\tilde{U}_{b_{1}}}(w)\in\\{0,1\\}$ so it
equals its square), this expression is at most

$\displaystyle\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},w}\Big{(}1-\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{\tilde{U}_{b_{1}}}(w)\Big{)}\mathbbm{1}_{V\cap
V-b_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{1}}(w)^{2}\Big{)}^{1/2}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},w}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{2},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,$ $\displaystyle\hskip
28.45274pt\mathbbm{1}_{V\cap V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)\,\,\mathbbm{1}_{V\cap V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)\,\,\mathbbm{1}_{V\cap V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)\,\,\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w)\Big{)}^{2}\Big{)}^{1/2}.$ (18)

Claim 3.11 implies

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},w}\Big{(}1-\mathbbm{1}_{A}(b_{1})\mathbbm{1}_{\tilde{U}_{b_{1}}}(w)\Big{)}\mathbbm{1}_{V\cap
V-b_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{1}}(w)^{2}\leq 2\xi\delta^{7}+40\sqrt[4]{\varepsilon}.$ (19)

When it comes to the second long sum in (18), note that

$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},w}\Big{(}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{2},b_{3},x_{2},y_{3},z_{1}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)\,\,\mathbbm{1}_{V\cap V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w)\Big{)}^{2}$ $\displaystyle=$
$\displaystyle\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{b_{1},w}\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern
0.0pt\mathbb{E}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{\begin{subarray}{c}b_{2},b_{3},x_{2},y_{3},z_{1}\\\
b^{\prime}_{2},b^{\prime}_{3},x^{\prime}_{2},y^{\prime}_{3},z^{\prime}_{1}\end{subarray}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-b_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{2}}(w)\,\,\mathbbm{1}_{V\cap V-b_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-x_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x_{2}-b_{2}+b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-y_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y_{3}+b_{1}-b_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-z_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-z_{1})}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-b^{\prime}_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-b^{\prime}_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-b^{\prime}_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b^{\prime}_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-b^{\prime}_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-b^{\prime}_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-x^{\prime}_{2}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-x^{\prime}_{2}}(w)\,\,\mathbbm{1}_{V\cap
V-(x^{\prime}_{2}-b^{\prime}_{2}+b^{\prime}_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(x^{\prime}_{2}-b^{\prime}_{2}+b^{\prime}_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-y^{\prime}_{3}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-y^{\prime}_{3}}(w)$ $\displaystyle\hskip 28.45274pt\mathbbm{1}_{V\cap
V-(y^{\prime}_{3}+b_{1}-b^{\prime}_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(y^{\prime}_{3}+b_{1}-b^{\prime}_{3})}(w)\,\,\mathbbm{1}_{V\cap
V-z^{\prime}_{1}}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-z^{\prime}_{1}}(w)\,\,\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-z^{\prime}_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-z^{\prime}_{1})}(w).$

To evade unnecessary writing, note that each of the 18 convolutions above
gives rise to additional dummy variable in further expansion. Let us name
these variables $v_{1},\dots,v_{9}$ and $v^{\prime}_{1},\dots,v^{\prime}_{9}$
in this order. Thus, $\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b_{2}-b_{3})}(w)$ expands as

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}}\displaylimits_{v_{1}}\mathbbm{1}_{V}(v_{1})\mathbbm{1}_{V}(v_{1}+b_{1}+b_{2}-b_{3})\mathbbm{1}_{V}(v_{1}+w)\mathbbm{1}_{V}(v_{1}+b_{1}+b_{2}-b_{3}+w)$

and the last term $\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-z^{\prime}_{1})}\mathop{\ast}\limits^{\vbox
to-1.50694pt{\kern-3.46594pt\hbox{$\bm{-}$}\vss}}\mathbbm{1}_{V\cap
V-(b_{1}+b^{\prime}_{2}-z^{\prime}_{1})}(w)$ expands as

$\mathop{\mathchoice{\vbox{\hbox{$\mathbb{E}$}}}{\kern 0.0pt\mathbb{E}}{\kern
0.0pt\mathbb{E}}{\kern