# Some new results on the higher energies I

I.D. Shkredov

Annotation.

We obtain a generalization of the recent Kelley–Meka result on sets avoiding
arithmetic progressions of length three. In our proof we develop the theory of
the higher energies. Also, we discuss the case of longer arithmetic
progressions, as well as a general family of norms, which includes the higher
energies norms and Gowers norms.

##  1 Introduction

The famous Erdős–Turán conjecture [5] asks is it true that for any integer
$k\geqslant 3$ any set of positive integers
$A=\\{n_{1}<n_{2}<\dots<n_{m}<\dots\\}$ satisfying

$\sum_{j=1}^{\infty}\frac{1}{n_{j}}=\infty$ (1)

contains an arithmetic progression of length $k$ (we say $A$ has APk), that is
the sequence of the form $x,x+y,\dots,x+(k-1)y\in A$? This question has a rich
history see, e.g., [8], [9] or [25] and is considered a central one in the
area of classical additive combinatorics due to its connection with many
adjecent fields as combinatorical ergodic theory and graphs/hypergraphs
theory, we just mention some papers [28], [29], [6], [7], [23], [10], [31],
[30], [15], [14] etc. If one defines

$r_{k}(N)=\frac{1}{N}\max\\{|A|~{}:~{}A\subseteq\\{1,\dots,N\\}\,,\quad
A\mbox{ has no APk}\\}\,,$

then the condition (1) means, roughly, that

$r_{k}(N)\ll\frac{1}{\log N\cdot(\log\log N)^{1+\varepsilon}}\,,\quad\quad
N\to\infty$ (2)

for an arbitrary $\varepsilon>0$.

The case of arithmetic progressions of length three was considered to be
special thanks to the Fourier approach of Roth [16], the required information
and references can be found in [2], [13], as well as in [25]. Bloom and Sisask
in [1] proved that $r_{3}(N)\ll(\log N)^{-1-c_{1}}$ for a certain $c_{1}>0$
and hence established conjecture (2) in the case of $k=3$. Recently, Kelley
and Meka [13] achieved a remarkable progress in this question and proved that

$r_{k}(N)\ll\exp(-O((\log N)^{c_{1}}))\,,$

where $c_{1}>0$ is an absolute constant. One of the ideas of paper [13] was to
use the higher energy $\mathsf{E}^{k}_{2}$ and the uniformity relatively to
$\mathsf{E}^{k}_{2}$ (all definitions can be found in Sections 2, 3) with a
growing parameter $k$ to control the number of arithmetic progressions in an
arbitrary set. Namely, bound (2) is an immediate consequence of the following
result (for simplicity we consider the group $\mathbb{F}_{p}^{n}$).

###### Theorem 1

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$ be a set,
$|A|=\delta N$, and $\varepsilon>0$ be a parameter. Then there is a subspace
$V\subseteq{\mathbf{G}}$ and $x\in{\mathbf{G}}$ such that $A\cap(V+x)$ is
$\varepsilon$–uniform relatively to $\mathsf{E}^{k}_{2}$,
$\mu_{V+x}(A)\geqslant\delta$, and

${\rm
codim}V\ll\varepsilon^{-14}k^{4}\mathcal{L}^{3}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\,.$
(3)

The aim of this paper is to generalize Kelley–Meka results to a more wide
additive–combinatorial family of energies $\mathsf{E}^{k}_{l}$ see, e.g.,
[27]. In our regime the parameter $l$ below is $l=O(1)$.

###### Theorem 2

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$ be a set,
$|A|=\delta N$, and $\varepsilon\in(0,1]$ be a parameter. Then there is a
subspace $V\subseteq{\mathbf{G}}$ and $x\in{\mathbf{G}}$ such that
$A\cap(V+x)$ is $\varepsilon$–uniform relatively to $\mathsf{E}^{k}_{l}$,
$\mu_{V+x}(A)\geqslant\delta$ and

${\rm
codim}V\ll\varepsilon^{-28l^{l}}(8l)^{28l^{l}}k^{4}\mathcal{L}^{4l}(\varepsilon)\mathcal{L}^{5l}(\delta)\,.$
(4)

Theorem 4 is interesting in its own right and can be used to solve more
general equations and systems than $x+y=2z$, which corresponds to the case of
AP3. Our forthcoming paper will be devoted to finding some applications of
this result.

Usually the case of the groups $\mathbb{F}_{p}^{n}$ for prime $p$ is
considered to be a model one, see the excellent survey [11] about this theme
and an appropriate generalization for all abelian groups ${\mathbf{G}}$ is
known to be a technical task. We will prove the correspondent result in our
forthcoming paper.

The approach develops the strategy of [8], [9], the method of the higher
energies (see, e.g., [21], [27]) and of course [13]. Also, we extensively use
a brilliant exposition [2], where the Kelley–Meka results were discussed.

In the appendix we discuss the original Erdős–Turán conjecture, that is the
case of longer arithmetic progressions and show that there is series of
difficulties on the conceptual and on the technical levels which make the
question about generalizations of the methods from [13] rather hard. The
author thinks this part is also interesting in its own right because it allows
us to understand the limits of the Kelley–Meka approach. Also, we consider a
general family of norms, which includes, simultaneously, the norms
$\mathsf{E}^{k}_{l}$ above, as well as the classical Gowers norms [9].

##  2 Definitions and preliminaries

Let ${\mathbf{G}}$ be a finite abelian group and denote by $N$ the cardinality
of ${\mathbf{G}}$. We use the same capital letter to denote set
$A\subseteq{\mathbf{G}}$ and its characteristic function
$A:{\mathbf{G}}\to\\{0,1\\}$. Let us define $\mu_{A}(x)=A(x)/|A|$, that is
$\sum_{x\in{\mathbf{G}}}\mu_{A}(x)=1$. Finally, let $f_{A}(x)=A(x)-|A|/N$ be
the balanced function of $A$. Given two sets $A,B\subset{\mathbf{G}}$, define
the sumset of $A$ and $B$ as

$A+B:=\\{a+b~{}:~{}a\in{A},\,b\in{B}\\}\,.$

In a similar way we define the difference sets and higher sumsets, e.g.,
$2A-A$ is $A+A-A$.

Let $f$ be a function from ${\mathbf{G}}$ to $\mathbb{C}.$ We denote the
Fourier transform of $f$ by $\widehat{f},$

$\widehat{f}(\xi)=\sum_{x\in{\mathbf{G}}}f(x)\overline{\chi(x)}\,,$ (5)

where $\chi\in\widehat{{\mathbf{G}}}$ is a character of ${\mathbf{G}}$. We
rely on the following basic identities

$\sum_{x\in{\mathbf{G}}}|f(x)|^{2}=\frac{1}{N}\sum_{\chi\in\widehat{{\mathbf{G}}}}\big{|}\widehat{f}(\chi)\big{|}^{2}\,,$
(6)

and

$f(x)=\frac{1}{N}\sum_{\chi\in\widehat{{\mathbf{G}}}}\widehat{f}(\chi)\chi(x)\,.$
(7)

If

$(f*g)(x):=\sum_{y\in{\mathbf{G}}}f(y)g(x-y)\quad\mbox{ and }\quad(f\circ
g)(x):=\sum_{y\in{\mathbf{G}}}f(y)g(y+x)\,,$

then

$\widehat{f*g}=\widehat{f}\widehat{g}$ (8)

and similar for $f\circ g$. Clearly, $(f*g)(x)=(g*f)(x)$ and $(f\circ
g)(x)=(g\circ f)(-x)$, $x\in{\mathbf{G}}$. The $k$–fold convolution,
$k\in\mathbb{N}$ we denote by $f^{(k)}$, so $f^{(2)}=f*f$ and $f^{(3)}=f*f*f$
for example.

We need some formalism concerning higher convolutions see, e.g., [27]. Let $l$
be a positive integer. Consider two operators ${\mathcal{D}}_{l}$,
${\mathcal{P}}_{l}:{\mathbf{G}}\to{\mathbf{G}}^{l}$ such that for a variable
$x$ one has ${\mathcal{D}}_{l}(x)=(x,\dots,x)\in{\mathbf{G}}^{l}$ and
${\mathcal{P}}_{l}(x)=(x_{1},\dots,x_{l})\in{\mathbf{G}}^{l}$. Notice that
${\mathcal{P}}_{1}(x)={\mathcal{D}}_{1}(x)=x$. In the same way these operators
act on functions, e.g., ${\mathcal{P}}_{l}(f)(x_{1},\dots,x_{l})=f(x_{1})\dots
f(x_{l})$ for $f:{\mathbf{G}}\to\mathbb{C}$ and for
$F:{\mathbf{G}}^{l}\to\mathbb{C}$ one has
${\mathcal{D}}_{l}(F)(x_{1},\dots,x_{l})=F(x_{1},\dots,x_{1})$ if
$x_{1}=\dots=x_{l}$ and zero otherwise. Now given a function
$f:{\mathbf{G}}\to\mathbb{C}$ and a positive integer $l$ define the
generalized convolution

${\mathcal{C}}_{l}(f)(x_{1},\dots,x_{l})=\sum_{z\in{\mathbf{G}}}f(z+x_{1})\dots
f(z+x_{l})=({\mathcal{D}}_{l}({\mathbf{G}})\circ{\mathcal{P}}_{l}(f))(x_{1},\dots,x_{l})$
(9) $:=\sum_{z\in{\mathbf{G}}}f_{x_{1},\dots,x_{l}}(z)\,.$ (10)

In a similar way we can consider
${\mathcal{C}}_{l}(f_{1},\dots,f_{l})(x_{1},\dots,x_{l})$ for any functions
$f_{1},\dots,f_{l}:{\mathbf{G}}\to\mathbb{C}$. One has

${\mathcal{C}}_{l}(f)(x_{1},\dots,x_{l})={\mathcal{C}}_{l}(f)(x_{1}+w,\dots,x_{l}+w)={\mathcal{C}}_{l}(f)((x_{1},\dots,x_{l})+{\mathcal{D}}_{l}(w))$
(11)

for any $w\in{\mathbf{G}}$. Let us emphasise that definitions (9), (10) differ
slightly from the usual one, see, e.g., [27] by a linear change of the
variables. Namely, it is a little bit more traditional to put

$f^{\prime}_{x_{1},\dots,x_{l}}(z)=f_{0,x_{1},\dots,x_{l}}(z)=f(z)f(z+x_{1})\dots
f(z+x_{l})\,,$ (12)

and

${\mathcal{C}}^{\prime}_{l+1}(f)(x_{1},\dots,x_{l})=\sum_{z\in{\mathbf{G}}}f(z)f(z+x_{1})\dots
f(z+x_{l})={\mathcal{C}}_{l}(f)(0,x_{1},\dots,x_{l})\,.$ (13)

Definitions (12), (13) have an advantage that they allow to consider infinite
groups ${\mathbf{G}}$ as well. To this end we use the dual notation
$\|f\|^{kl}_{\mathsf{E}^{k}_{l}}=\bar{\mathsf{E}}^{k}_{l}(f)=N^{-1}\mathsf{E}^{k}_{l}(f)$.
Now having $k,l\geqslant 2$ and a function $f:{\mathbf{G}}\to\mathbb{C}$ one
can consider

$\mathsf{E}^{k}_{l}(f)=\sum_{x_{1},\dots,x_{l}}{\mathcal{C}}^{k}_{l}(f)(x_{1},\dots,x_{l})=\mathsf{E}^{l}_{k}(f)$
(14)

and it was showed in [27, Proposition 30] that for a real function $f$ and
even $k,l$ the formula $(\mathsf{E}^{k}_{l}(f))^{1/kl}$ defines a norm of our
function $f$. If one put $l=1$ in (14), then we formally obtain
$\mathsf{E}_{1}^{k}(f)=N(\sum_{z}f(x))^{k}$ and this is not a norm for any
$k$. Nevertheless, it is convenient to consider the quantities
$\mathsf{E}_{1}^{k}(f)$ sometimes. Notice that $\mathsf{E}^{k}_{l}(f)\geqslant
0$, provided at least one of $k,l$ is even but, nevertheless, it cannot be a
norm in this case, see [27, Sections 4,7]. A general family of norms, which
includes the norms above is considered in the second part of the appendix. In
particular, one can find the discussed properties of the energies
$\mathsf{E}_{l}^{k}(f)$.

Let $\varepsilon\in(0,1]$ be a real number. We write
$\mathcal{L}(\varepsilon)$ for $\log(2/\varepsilon)$. Let us make a convention
that if a product is taken over an empty set, then it equals one. The signs
$\ll$ and $\gg$ are the usual Vinogradov symbols. When the constants in the
signs depend on a parameter $M$, we write $\ll_{M}$ and $\gg_{M}$. All
logarithms are to base $2$. By $\mathbb{F}_{p}$ denote
$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$. Let us denote by
$[n]$ the set $\\{1,2,\dots,n\\}$.

##  3 Some results on $\mathsf{E}^{k}_{l}$–norms

In this section we obtain some generalization of Kelley–Meka results which
were obtained for $\mathsf{E}_{2}^{k}$–norm to $\mathsf{E}_{l}^{k}$–norm.
Also, we discuss some special properties of such norms. Our results naturally
break down into two cases: uniform and non–uniform.

###  3.1 Uniform sets in the sense of $\mathsf{E}^{k}_{l}$–norm

Let us give the main definition of this subsection.

###### Definition 3

Let ${\mathbf{G}}$ be an abelian group, $A\subseteq{\mathbf{G}}$ be a set,
$|A|=\delta N$, and $\varepsilon>0$ be a parameter. Then we say that $A$ is
$\varepsilon$–uniform relatively to (the energy) $\mathsf{E}^{k}_{l}$ if

$\|f_{A}\|^{kl}_{\mathsf{E}^{k}_{l}}\leqslant\varepsilon^{kl}\delta^{kl}N^{k+l}\,.$
(15)

Usually the number $\varepsilon$ belongs to $(0,1]$ but sometimes
$\varepsilon>1$ and hence one can consider the quantity $\varepsilon$ as the
definition of the energy $\|f_{A}\|^{kl}_{\mathsf{E}^{k}_{l}}$, that is
$\|f_{A}\|^{kl}_{\mathsf{E}^{k}_{l}}:=\varepsilon^{kl}\delta^{kl}N^{k+l}$.
Further by the Hölder inequality, we have

$(\mathsf{E}^{k-1}_{l}(f))^{k}\leqslant(\mathsf{E}^{k}_{l}(f))^{k-1}N^{l}$
(16)

and hence if $A$ is $\varepsilon$–uniform relatively to $\mathsf{E}^{k}_{l}$,
then $A$ is $\varepsilon$–uniform relatively to
$\mathsf{E}^{k^{\prime}}_{l^{\prime}}$ for $k^{\prime}\leqslant k$,
$l^{\prime}\leqslant l$. On the other hand, it is easy to see that a smaller
norm does not control the higher one.

###### Example 4

Let ${\mathbf{G}}=\mathbb{F}_{2}^{n}$, $H<\mathbb{F}_{2}^{n}$,
$\Lambda\subseteq\mathbb{F}_{2}^{n}/H$ be a random set such that
$|\Lambda|=\delta N/|H|$. Also, suppose that $\delta^{2}\gg|H|/N$ and thus
with high probability $\Lambda-\Lambda\approx\mathbb{F}_{2}^{n}/H$. Let $A$ be
the direct sum of $H$ and $\Lambda$, then $|A|=\delta N$. It is easy to see
that for a random $x\in A-A\approx{\mathbf{G}}$ one has
$|A_{x}|\sim\delta^{2}N$ but for $x,y\in H$ one has $A_{x}=A$ and $A_{x,y}=A$.
Thus for any $k\geqslant 2$ the following holds

$\mathsf{E}_{2}^{k}(A)\sim(\delta^{2}N)^{k}N+(\delta
N)^{k}|H|\sim(\delta^{2}N)^{k}N\,,$

provided $|H|\ll\delta^{k}N$ but taking an arbitrary $l$, we see that

$\mathsf{E}^{l}_{3}(A)\sim(\delta^{3}N)^{l}N^{2}+(\delta
N)^{l}|H|^{2}\gg(\delta N)^{l}|H|^{2}\,,$

provided $|H|\gg\delta^{l}N$. It follows that one can take $l=k+1$ and
construct a set $A$ such that $A$ is $\mathsf{E}_{2}^{k}$–uniform but not
$\mathsf{E}_{3}^{k+1}$–uniform.

Now let us obtain the characteristic property of the energy
$\mathsf{E}^{k}_{l}$.

###### Lemma 5

Let $l,k\geqslant 2$ be even numbers, and $A_{j}\subseteq{\mathbf{G}}$,
$j\in[l]$ be sets. Then for any function $g:{\mathbf{G}}\to\mathbb{R}$ one has

$\sum_{x}\prod_{j=1}^{l}(f_{A_{j}}\circ
g)(x)\leqslant\|g\|^{l(1-1/k)}_{1}\|{\mathcal{C}}_{l}(g)\|^{1/k}_{\infty}\cdot\prod_{j=1}^{l}\|f_{A_{j}}\|_{\overline{\mathsf{E}}^{k}_{l}}\,.$
(17)

If the sets $A_{j}$ are the same, then the same is true for even $k$ only.

P r o o f. By the Hölder inequality and [27, Lemma 29] and the duality one has

$\sum_{x}\prod_{j=1}^{l}(f_{A_{j}}\circ
g)(x)=N^{-1}\sum_{|z|=l}{\mathcal{C}}_{l}(f_{A_{1}},\dots,f_{A_{l}})(z){\mathcal{C}}_{l}(g)(z)$
$\leqslant
N^{-1+1/k}\prod_{j=1}^{l}\|f_{A_{j}}\|_{\overline{\mathsf{E}}^{k}_{l}}\cdot\left(\sum_{|z|=l}|{\mathcal{C}}^{k/(k-1)}_{l}(g)(z)|\right)^{1-1/k}\leqslant\prod_{j=1}^{l}\|f_{A_{j}}\|_{\overline{\mathsf{E}}^{k}_{l}}\cdot\|{\mathcal{C}}_{l}(g)\|^{1/k}_{\infty}\|g\|^{l(1-1/k)}_{1}$

This completes the proof. $\hfill\Box$

###### Corollary 6

Let $A,B\subseteq{\mathbf{G}}$ be sets, $|A|=\delta N$, $|B|=\beta N$, and $l$
be a positive integer. Take $k=2\lceil el\log(1/\beta)\rceil$ and suppose that
$A$ is $\varepsilon$–uniform relatively to the energy $\mathsf{E}^{k}_{l}$.
Then

$\sum_{x}(A\circ
B)^{l}(x)\leqslant\delta^{l}|B|^{l}N\cdot\min\\{1.25(1+\varepsilon)^{l},(1+1.25\varepsilon)^{l}\\}\,.$
(18)

P r o o f. Using the formula $A(x)=f_{A}(x)+\delta$, combining with Lemma 5,
we see that the left–hand side of (18) is

$\sum_{j=0}^{l}\binom{l}{j}(\delta|B|)^{l-j}\sum_{x}(f\circ
B)^{j}(x)\leqslant\sum_{j=0}^{l}\binom{l}{j}(\delta|B|)^{l-j}\|f\|^{j}_{\overline{\mathsf{E}}^{k}_{j}}|B|^{j(1-1/k)+1/k}$
$\leqslant\delta^{l}|B|^{l}N\sum_{j=0}^{l}\binom{l}{j}\varepsilon^{j}\beta^{-(j-1)/k}\leqslant\frac{5}{4}\delta^{l}|B|^{l}N(1+\varepsilon)^{l}$

as required and similarly the second bound. This completes the proof.
$\hfill\Box$

Let us remark that, of course, the energy $\mathsf{E}_{2}^{k}$ solely allows
us to control sums from (18) but our task is to obtain the correct power of
$\delta$ and $|B|$ in the right–hand side of this estimate.

We need one more result about uniform set, which is useful for applications.

###### Lemma 7

Let $k,l$ be even numbers, $A_{1},\dots,A_{l}\subseteq{\mathbf{G}}$ be sets,
$|A_{j}|=\delta_{j}N$. Suppose that all $A_{j}$ are $\varepsilon$–uniform
relatively to $\mathsf{E}^{k}_{l}$ and $2l\varepsilon^{k}\leqslant 1$. Then

$\sum_{|x|=l}\left({\mathcal{C}}_{l}(A_{1},\dots,A_{l})(x)-N\prod_{j=1}^{l}\delta_{j}\right)^{k}\leqslant\varepsilon^{k}l2^{kl+1}N^{l+k}\left(\prod_{j=1}^{l}\delta_{j}\right)^{k}\,.$
(19)

P r o o f. Put $\Pi=\prod_{j=1}^{l}\delta_{j}$. Then the left–hand side of
(19) is

$\sigma:=\sum_{|x|=l}\left(\sum_{\emptyset\neq
S\subseteq[l]}{\mathcal{C}}_{l}(f_{1},\dots,f_{l})(x)\right)^{k}=\sum_{|x|=l}\left(\sum_{\emptyset\neq
S\subseteq[l]}F_{S}(x)\right)^{k}\,,$

where for $j\in S$ we put $f_{j}=f_{A_{j}}$ and if $j\notin S$, then
$f_{j}=\delta_{j}$. Using $\varepsilon$–uniformity of all sets $A_{j}$,
combining with [27, Lemma 29], we get

$\|F_{S}\|^{kl}_{\mathsf{E}^{k}_{l}}\leqslant\varepsilon^{|S|k}\Pi^{k}N^{l+k}\,.$

Thus by the Hölder inequality one has

$\sigma\leqslant 2^{kl}\sum_{\emptyset\neq
S\subseteq[l]}\|F_{S}\|^{kl}_{\mathsf{E}^{k}_{l}}\leqslant
2^{kl}\Pi^{k}N^{k+l}((1+\varepsilon^{k})^{l}-1)\leqslant
l2^{kl+1}\varepsilon^{k}\Pi^{k}N^{k+l}$

as required. $\hfill\Box$

Let us consider one more example which shows that one can delete/add a tiny
subset from a non–uniform set to obtain a uniform one. This phenomenon has no
place if we consider the classical uniformity in terms of the Fourier
transform or in terms of Gowers norms [9], say. The reason is normalization
(15), of course.

###### Example 8

Let ${\mathbf{G}}=\mathbb{F}_{2}^{n}$, $H<\mathbb{F}_{2}^{n}$, $|H|=\beta N$,
$\Lambda\subseteq{\mathbf{G}}$ be a random set, $|\Lambda|=\delta N$,
$\beta\leqslant\delta$ and put $A=\tilde{H}\bigsqcup\Lambda$, where
$\tilde{H}=H\setminus\Lambda$. Then with high probability
$|\tilde{H}|\sim\beta(1-\delta)N=\varepsilon|A|$, the set $\Lambda$ is uniform
in any possible sense but $A$ is non $\eta$–uniform set with rather large
$\eta$. Indeed, by Kelley–Meka method [13] or just see Lemma 5, we know that

$\sigma:=\sum_{x}(A\circ A)(x)H(x)=(\delta+(1-\delta)\beta)^{2}\beta
N^{2}+\theta\eta^{2}\delta^{2}\beta N^{2}\,,$ (20)

where $|\theta|\leqslant 4$, say, is a certain number and $A$ is supposed to
be $\mathsf{E}_{2}^{k}$–uniform with $k\sim\mathcal{L}(\beta)$. On the other
hand, the direct calculation shows

$\sigma=|\tilde{H}|^{2}+2\delta\beta^{2}(1-\delta)N^{2}+\delta^{2}|H|N$ (21)

plus a negligible error term. Comparing (20) and (21), we obtain

$\eta^{2}\delta^{3}\varepsilon\gg
2\delta\beta\varepsilon+\beta\varepsilon^{2}-\varepsilon^{2}\delta^{2}\gg\delta\varepsilon^{3}$

and thus $\eta\gg\varepsilon/\delta$ which is much larger than $\varepsilon$
for small $\delta$.

Similarly, one can show that deleting a subspace $H$ from a random set
$\Lambda$, $|H|=\varepsilon|A|$, $H$ lives on the first coordinates, say, we
obtain a non $\eta$–uniform set with $\eta\gg 1$ thanks to the equality
$\sum_{x}H(x)(A\circ A)(x)=0$.

###  3.2 Non–uniformity and almost periodicity

The aim of this subsection is to obtain Sanders’ almost periodicity result for
higher convolutions, see Lemma 38 below.

At the beginning we want to transfer a lower bound for the energy
$\mathsf{E}^{k}_{l}(f_{A})$ to the largeness of the energy
$\mathsf{E}^{k}_{l}(A)$. We follow a more simple method from [2] which differs
from the approach of [13] by some logarithms. The dependence on $l$ in the
first multiple in (23) is, probably, can be improved significantly (also, see
Remark 20 from the appendix) but in our regime $l=O(1)$ and thus it is not so
critical for us.

###### Lemma 9

Let $A\subseteq{\mathbf{G}}$ be a set, $|A|=\delta N$ and $\varepsilon>0$ be a
parameter, $\varepsilon_{*}:=\min\\{\varepsilon,1\\}$. Suppose that for an odd
$k\geqslant 5$ one has

$\mathsf{E}_{l}^{k}(f_{A})=\varepsilon^{lk}\delta^{lk}N^{l+k}\,,$ (22)

and that for $k_{*}=O(kl\varepsilon^{-l}_{*}\mathcal{L}(\varepsilon_{*}))$ the
set $A$ is $\frac{\varepsilon\varepsilon^{l-1}_{*}}{8l}$–uniform relatively to
$\mathsf{E}^{k_{*}}_{l-1}$. Then there is an even $k_{1}\leqslant k_{*}$ such
that

$\mathsf{E}_{l}^{k_{1}}(A)\geqslant\left(1+\frac{\varepsilon\varepsilon^{l-1}_{*}}{8l}\right)^{lk_{1}}\delta^{lk_{1}}N^{l+k_{1}}\,.$
(23)

P r o o f. Write $f(x)=f_{A}(x)$. We have

$\varepsilon^{lk}\delta^{lk}N^{l+k}\leqslant\sum_{|x|=l}{\mathcal{C}}^{k-1}_{l}(f)(x)|{\mathcal{C}}_{l}(f)(x)|+\sum_{|x|=l}{\mathcal{C}}^{k}_{l}(f)(x)=2\sum_{|x|=l}{\mathcal{C}}^{k-1}_{l}(f)(x)\cdot\max\\{0,{\mathcal{C}}_{l}(f)(x)\\}\,,$

and thus considering the set $P=\\{x~{}:~{}{\mathcal{C}}_{l}(f)(x)\geqslant
0\\}$, we get

$\sum_{x\in P}{\mathcal{C}}^{k}_{l}(f)(x)\geqslant
2^{-1}\varepsilon^{lk}\delta^{lk}N^{l+k}\,.$ (24)

Now let consider the subset of the set $P$

$P_{\varepsilon}:=\\{x~{}:~{}{\mathcal{C}}_{l}(f)(x)\geqslant\frac{3}{4}\varepsilon^{l}\delta^{l}N\\}\,.$

Then we have

$\sum_{x\notin
P_{\varepsilon}}{\mathcal{C}}^{k}_{l}(f)(x)\leqslant\left(\frac{3}{4}\varepsilon^{l}\delta^{l}N\right)^{k}N^{l}\leqslant
2^{-2}\varepsilon^{lk}\delta^{lk}N^{l+k}\,.$ (25)

Combining (24), (25) and using the Hölder inequality, we obtain

$|P_{\varepsilon}|\mathsf{E}^{2k}_{l}(f)\geqslant
2^{-4}\varepsilon^{2lk}\delta^{2lk}N^{2l+2k}\,.$ (26)

By the norm property of $\mathsf{E}^{2k}_{l}(f)$ for positive functions, see
[27, Proposition 30] one has

$\mathsf{E}^{2k}_{l}(f)\leqslant\left(\|A\|_{\mathsf{E}^{2k}_{l}}+\|\delta\|_{\mathsf{E}^{2k}_{l}}\right)^{2kl}\leqslant(2+\varepsilon/2)^{2kl}\delta^{2kl}N^{l+2k}$

otherwise there is nothing to prove with $k_{1}=2k$ and much larger
$\varepsilon$. Thus we derive from (26) that
$|P_{\varepsilon}|\geqslant(2\varepsilon_{*}/5)^{2kl}N^{l}$. Now

${\mathcal{C}}_{l}(A)(x)={\mathcal{C}}_{l}(f+\delta)(x)=\delta^{l}N+{\mathcal{C}}_{l}(f)+\sum_{S\subseteq[l]~{}:~{}1\leqslant|S|<l}\delta^{l-|S|}{\mathcal{C}}_{|S|}(f)(x_{S})=\delta^{l}N+{\mathcal{C}}_{l}(f)+\mathcal{E}(x)\,,$

where for a set $S\subseteq[l]$ the vector $x_{S}$ has coordinates $x_{j}$,
$j\in S$. By the triangle inequality for $L_{k_{1}}$–norm, we have

$(\mathsf{E}^{k_{1}}_{l}(A))^{1/k_{1}}=\|{\mathcal{C}}_{l}(A)\|_{k_{1}}=\|{\mathcal{C}}_{l}(f+\delta)\|_{k_{1}}\geqslant\|\delta^{l}N+{\mathcal{C}}_{l}(f)\|_{k_{1}}-\|\mathcal{E}\|_{k_{1}}\,.$
(27)

Using our bound for the cardinality of the set $P_{\varepsilon}$, we get

$\|\delta^{l}N+{\mathcal{C}}_{l}(f)\|^{k_{1}}_{k_{1}}\geqslant\sum_{x\in
P_{\varepsilon}}(\delta^{l}N+{\mathcal{C}}_{l}(f))^{k_{1}}(x)\geqslant(2\varepsilon_{*}/5)^{2kl}N^{l}\cdot(1+3\varepsilon^{l}/4)^{k_{1}}\delta^{lk_{1}}N^{k_{1}}$
$\geqslant(1+\varepsilon^{l}/2)^{k_{1}}\delta^{lk_{1}}N^{k_{1}+l}\,,$ (28)

provided $k_{1}\geqslant
20kl\varepsilon^{-l}_{*}\mathcal{L}(\varepsilon_{*})$. On the other hand, by
our assumption the set $A$ is $\zeta:=\frac{\varepsilon^{l}}{8l}$–uniform
relatively to $\mathsf{E}^{k_{*}}_{j}$ for all $j<l$ and $k_{*}=k_{1}$. It
follows that

$\|\mathcal{E}\|_{k_{1}}\leqslant\sum_{S\subseteq[l]~{}:~{}1\leqslant|S|<l}\delta^{l-|S|}N^{\frac{l-|S|}{k_{1}}}\|f\|^{|S|}_{\mathsf{E}^{k_{1}}_{|S|}}\leqslant\delta^{l}N^{\frac{l+k_{1}}{k_{1}}}((1+\zeta)^{l}-1)\,.$
(29)

Combining (27), (28) and (29), we obtain

$\mathsf{E}^{k_{1}}_{l}(A)\geqslant\delta^{lk_{1}}N^{k_{1}+l}\left(2+\varepsilon^{l}/2-(1+\zeta)^{l})\right)^{k_{1}}\geqslant\delta^{lk_{1}}N^{k_{1}+l}\left(1+\frac{\varepsilon\varepsilon^{l-1}_{*}}{8l}\right)^{lk_{1}}$

as required. $\hfill\Box$

Now we use duality (14) to obtain an appropriate version of multi–dimensional
version of the Balog–Szemerédi–Gowers theorem as was done in [20] (also, see
[26, Theorem 17]). Thanks to duality (76) one can show that a similar result
takes place for more general energies $\mathcal{E}^{k}_{s,t}$, see the
appendix. Of course, in this case one needs to replace
${\mathcal{C}}_{|x|}(\cdot)(x)$ to ${\mathcal{C}}_{|x||z|}(\cdot)(x\oplus z)$
or ${\mathcal{C}}_{|y||z|}(\cdot)(y\oplus z)$ and use symmetries (78) instead
of the symmetry (11) below.

###### Lemma 10

Let $A\subseteq{\mathbf{G}}$ be a set, $|A|=\delta N$ and $\varepsilon>0$,
$\eta\in(0,1/2)$ be parameters. Suppose that for some integers $k,l\geqslant
2$ with $kl\geqslant 4\varepsilon^{-1}_{*}\mathcal{L}(\eta)$ one has

$\mathsf{E}^{k}_{l}(A)\geqslant(1+\varepsilon)^{lk}\delta^{lk}N^{l+k}\,.$ (30)

Define the set

$S=\\{|x|=l~{}:~{}{\mathcal{C}}_{l}(A)(x)\geqslant(1+\varepsilon/4)^{l}\delta^{l}N\\}\,.$
(31)

Then there is a set $B$ such that

$N^{-1}\sum_{|x|=l}S(x){\mathcal{C}}_{l}(B)(x)\geqslant(1-2\eta)|B|^{l}\,,$
(32)

and $|B|>2^{-1/(l-1)}(1+\varepsilon)^{k}\delta^{k}N$.

P r o o f. For any set $S\subseteq{\mathbf{G}}^{l}$ with the property
$S(x+{\mathcal{D}}_{l}(t))=S(x)$, $t\in{\mathbf{G}}$, $x\in{\mathbf{G}}^{l}$,
we have

$\sum_{|x|=l}S(x){\mathcal{C}}^{k}_{l}(A)(x)=\sum_{|x|=l}S(x)\sum_{|z|=k}A^{k}(z+{\mathcal{D}}_{k}(x_{1}))\dots
A^{k}(z+{\mathcal{D}}_{k}(x_{l}))$
$=\sum_{|z|=k}\,\sum_{|x|=l}S(x)A_{z}(x_{1})\dots
A_{z}(x_{l})=N^{-1}\sum_{|z|=k}\,\sum_{|x|=l}S(x){\mathcal{C}}_{l}(A_{z})(x)\,,$
(33)

where we have made the change of the variables $x_{j}\to x_{j}+t$ in the last
formula. Clearly, we have from identity (11) and definition (31) that
$S(x+{\mathcal{D}}_{l}(t))=S(x)$ for and $t\in{\mathbf{G}}$ and
$x\in{\mathbf{G}}^{l}$ and thus the argument above can be applied for the set
$S$ as well. Thus using the definition of the set $S$, as well as the
conditions (30) and $kl\geqslant 4\varepsilon^{-1}_{*}\log(4/\eta)$, we get

$\sum_{x\notin
S}{\mathcal{C}}^{k}_{l}(A)(x)\leqslant(1+\varepsilon/4)^{lk}\delta^{lk}N^{k+l}\leqslant
2^{-2}\eta(1+\varepsilon)^{lk}\delta^{lk}N^{k+l}\leqslant
2^{-2}\eta\mathsf{E}^{k}_{l}(A)\,.$ (34)

Now define the set

$\Omega=\left\\{|z|=k~{}:~{}|A_{z}|\geqslant
2^{-1/(l-1)}(1+\varepsilon)^{k}\delta^{k}N\right\\}\,.$

By the definition of the set $\Omega$, we derive

$\sum_{z\notin\Omega}|A_{z}|^{l}\leqslant(\max_{z\notin\Omega}|A_{z}|)^{l-1}|A|^{k}N\leqslant
2^{-1}(1+\varepsilon)^{lk}\delta^{lk}N^{l+k}\leqslant
2^{-1}\mathsf{E}^{k}_{l}(A)\,.$ (35)

In view of bounds (34), (35) one has

$N^{-1}\sum_{z\in\Omega}\left(\sum_{x\in
S}{\mathcal{C}}_{l}(A_{z})(x)-\eta^{-1}\sum_{x\notin
S}{\mathcal{C}}_{l}(A_{z})(x)\right)\geqslant
2^{-1}\sum_{|z|=k}|A_{z}|^{l}-2^{-1}\mathsf{E}^{k}_{l}(A)=0\,.$ (36)

Hence there is $z\in\Omega$ such that inequality (32) holds for $B=A_{z}$ and
$|B|>2^{-1/(l-1)}(1+\varepsilon)^{k}\delta^{k}N$ as required. $\hfill\Box$

Now we need an analogue of the almost periodicity result [4] (also, see [17],
[18], [19] and, especially, [22, Theorem 3.2]) for the higher convolutions.
This theme is rather well–known and thus we give just a scheme of the proof
emphasizing the necessary distinctions we must make. For the convolution
${\mathcal{C}}_{|x||z|}(x\oplus z)$ a similar result takes place, see Lemma 84
from the appendix.

###### Lemma 11

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $l$ be an integer and
$\epsilon\in(0,1]$ be a real parameter. Also, let $B\subseteq{\mathbf{G}}$ be
a set, $|B|=\beta N$, and $f:{\mathbf{G}}^{l}\to[-1,1]$ be a function. Then
there is a subspace $V\leqslant{\mathbf{G}}$ with

$\mathrm{codim}V\ll\epsilon^{-2}l\mathcal{L}^{2}(\beta)\mathcal{L}^{2}(\epsilon\beta^{l})$
(37)

and such that

$\left|\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B*\mu_{V}))(x)-\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B))(x)\right|\leqslant\epsilon|B|^{l+1}\,.$
(38)

P r o o f. We begin with a rather general argument which takes place in any
abelian group ${\mathbf{G}}$. Let $k\geqslant 2$ be an integer parameter and
$q\geqslant 2$ be a real parameter. Applying the Croot–Sisask lemma [4], [18]
(clearly, one has
$|{\mathcal{D}}_{l}(B)+{\mathcal{D}}_{l}({\mathbf{G}})|\leqslant\beta^{-1}|{\mathcal{D}}_{l}(B)|$,
so the set ${\mathcal{D}}_{l}(B)$ has the small doubling), we find a set
$T\subseteq{\mathbf{G}}$,
$|T|\geqslant|B|\exp(-O(\epsilon^{-2}qk^{2}\log(1/\beta)))$ and such that for
any $t\in kT$ the following holds

$\sum_{|x|=l}\left|(f\circ{\mathcal{D}}_{l}(B))(x+{\mathcal{D}}_{l}(t))-\sum_{|x|=l}(f\circ{\mathcal{D}}_{l}(B))(x)\right|^{q}$
$\leqslant\left(\frac{\epsilon}{4}\right)^{q}\|f\|_{q}^{q}|B|^{q}\leqslant\left(\frac{\epsilon}{4}\right)^{q}|B|^{q}N^{l}\,.$
(39)

Fixing $t\in kT$ and using the Hölder inequality, combining with estimate
(39), we get

$\left|\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B))(x+{\mathcal{D}}_{l}(t))-\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B))(x)\right|\leqslant\frac{\epsilon}{4}|B|N^{l/q}|B|^{l(1-1/q)}$
$=\frac{\epsilon}{4}\beta^{-l/q}|B|^{l+1}\leqslant\frac{\epsilon}{2}|B|^{l+1}\,,$
(40)

where we have taken $q=Cl\log(1/\beta)$ for a sufficiently large constant
$C>0$. It follows that

$\left|\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B*\mu^{(k)}_{T}))(x)-\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B))(x)\right|\leqslant
2^{-1}\epsilon|B|^{l+1}\,.$ (41)

Let us analyze the sum
$\sigma:=|\sum_{|x|=l}f(x)(B^{l}\circ{\mathcal{D}}_{l}(B*\mu^{(k)}_{T}))(x)|$
from (41). Clearly,
$\widehat{\mu}_{T}(r_{1},\dots,r_{l})=|T|^{-1}\widehat{T}(r_{1}+\dots+r_{l})$
and thus

$\sigma\leqslant\frac{|B|}{|T|^{k}N^{l}}\sum_{z}|\widehat{T}(z)|^{k}\sum_{r_{1}+\dots+r_{l}=z}|\widehat{f}(r_{1},\dots,r_{l})||\widehat{B}(r_{1})|\dots|\widehat{B}(r_{l})|\,.$
(42)

As usual let us estimate the last sum over $z\in{\rm Spec\,}_{c}(T)$, where
$c\in(0,1]$ is a parameter and over $z\notin{\rm Spec\,}_{c}(T)$, where we
have defined

${\rm Spec\,}_{c}(T)=\\{z\in{\mathbf{G}}~{}:~{}|\widehat{T}(z)|\geqslant
c|T|\\}\,.$

By the definition of the set ${\rm Spec\,}_{c}(T)$, the Hölder inequality and
the Parseval identity, we have

$\sigma_{1}:=\frac{|B|}{|T|^{k}N^{l}}\sum_{z\notin{\rm
Spec\,}_{c}(T)}|\widehat{T}(z)|^{k}\sum_{r_{1}+\dots+r_{l}=z}|\widehat{f}(r_{1},\dots,r_{l})||\widehat{B}(r_{1})|\dots|\widehat{B}(r_{l})|$
$\leqslant\frac{c^{k}|B|}{N^{l}}\sum_{r_{2},\dots,r_{l}}|\widehat{B}(r_{2})|\dots|\widehat{B}(r_{l})|\sum_{z}|\widehat{f}(z-r_{2}+\dots+r_{l},\dots,r_{l})||\widehat{B}(z-r_{2}+\dots+r_{l})|$
$\leqslant\frac{c^{k}|B|^{3/2}}{N^{l-1}}\sum_{r_{2},\dots,r_{l}}|\widehat{B}(r_{2})|\dots|\widehat{B}(r_{l})|\left(\sum_{a}|\widehat{f}_{a}(r_{2},\dots,r_{l})|^{2}\right)^{1/2}\,,$

where $f_{a}(x_{2},\dots,x_{l})=f(a,x_{2},\dots,x_{l})$. Using the Hölder
inequality and the Parseval formula one more time, we derive

$\sigma_{1}\leqslant\frac{c^{k}|B|^{(l+2)/2}}{N^{(l-1)/2}}\left(\sum_{r_{2},\dots,r_{l}}\sum_{a}|\widehat{f}_{a}(r_{2},\dots,r_{l})|^{2}\right)^{1/2}\leqslant
c^{k}|B|^{(l+2)/2}N^{l/2}\leqslant 2^{-1}\epsilon|B|^{l+1}\,.$

Here we have taken $c=1/2$ and $k=\lceil
2\mathcal{L}(\epsilon\beta^{l})\rceil$, say. For the sum over $z\in{\rm
Spec\,}_{c}(T)$ we use the Chang lemma [3] about the dimension of the spectrum
and find a subspace $V$ such that (38) takes place and

$\mathrm{codim}V\ll\log(N/|T|)\ll\epsilon^{-2}l\mathcal{L}^{2}(\beta)k^{2}\ll\epsilon^{-2}l\mathcal{L}^{2}(\beta)\mathcal{L}^{2}(\epsilon\beta^{l})\,,$

see details in [4], [18], [19] or in [26, Section 5]. This completes the
proof. $\hfill\Box$

##  4 Some generalizations of Kelley–Meka results

Using the density increment Kelley–Meka [13] (or just repeat the calculations
of the previous section, combining forthcoming Proposition 47 in the case
$l=2$) obtained the following result.

###### Theorem 12

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$ be a set,
$|A|=\delta N$, and $\varepsilon>0$ be a parameter. Then there is a subspace
$V\subseteq{\mathbf{G}}$ and $x\in{\mathbf{G}}$ such that $A\cap(V+x)$ is
$\varepsilon$–uniform relatively to $\mathsf{E}^{k}_{2}$,
$\mu_{V+x}(A)\geqslant\delta$, and

${\rm
codim}V\ll\varepsilon^{-14}k^{4}\mathcal{L}^{3}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\cdot\mathcal{L}^{4}(\varepsilon)\,.$
(43)

###### Remark 13

Actually, we formulate Theorem 43 in the form of Bloom–Sisask [2]. Kelley–Meka
[13] obtained this result without $\mathcal{L}^{4}(\varepsilon)$ in (43).

We generalize Theorem 43 for the higher energies $\mathsf{E}^{k}_{l}$.

###### Theorem 14

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$ be a set,
$|A|=\delta N$, and $\varepsilon\in(0,1]$ be a parameter. Then there is a
subspace $V\subseteq{\mathbf{G}}$ and $x\in{\mathbf{G}}$ such that
$A\cap(V+x)$ is $\varepsilon$–uniform relatively to $\mathsf{E}^{k}_{l}$,
$\mu_{V+x}(A)\geqslant\delta$ and

${\rm
codim}V\ll\varepsilon^{-28l^{l}}(8l)^{28l^{l}}k^{4}\mathcal{L}^{4l}(\varepsilon)\mathcal{L}^{5l}(\delta)\,.$
(44)

Now we are ready to obtain our driving result about the density increment. As
always we will apply Proposition 47 in an iterative way and we see that
estimate (47) allows us to do it in at most
$O(\varepsilon^{-1}\mathcal{L}(\delta))$ times.

###### Proposition 15

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$, $|A|=\delta
N$, $\varepsilon>0$ be a real number and $k,l\geqslant 2$ be positive
integers, $kl\gg\varepsilon^{-1}\mathcal{L}(\varepsilon)$. Suppose that

$\mathsf{E}^{k}_{l}(A)\geqslant(1+\varepsilon)^{kl}\delta^{kl}N^{k+l}\,,$ (45)

and that $A$ is $\varepsilon/5$–uniform relatively to
$\mathsf{E}^{k_{*}}_{l-1}$ for an even $k_{*}=O(kl\mathcal{L}(\delta))$. Then
there is a subspace $V\subseteq{\mathbf{G}}$ such that

$\mathrm{codim}V\ll\varepsilon^{-2}l^{3}k^{4}\mathcal{L}^{2}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\,,$
(46)

and for a certain $x\in{\mathbf{G}}$ one has

$|A\cap(V+x)|\geqslant(1+\varepsilon/8)\delta|V|\,.$ (47)

P r o o f. Applying Lemma 10 for the energy $\mathsf{E}^{k}_{l}(A)$ with the
parameters $\epsilon=\varepsilon$, $\eta=\varepsilon/30$, we construct the set

$S=\\{|x|=l~{}:~{}{\mathcal{C}}_{l}(A)(x)\geqslant(1+\varepsilon/4)^{l}\delta^{l}N\\}$

and such that for a certain set $B\subseteq{\mathbf{G}}$,
$|B|>2^{-1/(l-1)}(1+\varepsilon)^{k}\delta^{k}N:=\beta N$ the following holds

$N^{-1}\sum_{|x|=l}S(x){\mathcal{C}}_{l}(B)(x)\geqslant(1-2\eta)|B|^{l}\,.$
(48)

We have $kl\geqslant 4\varepsilon^{-1}\mathcal{L}(\eta)$ and thus Lemma 10 can
be applied indeed. Using formulae (11) and making the required change of the
variables (in (9) we put, consequently, $z\to z-x_{1}$) one can see that (48)
is equivalent to

$\sum_{|x|=l-1}\bar{S}(x)(B^{l-1}\circ{\mathcal{D}}_{l-1}(B))(x)\geqslant(1-2\eta)|B|^{l}\,,$
(49)

where $\bar{S}\subseteq{\mathbf{G}}^{l-1}$ is a certain set which is
constructed via the set $S$, see formulae (11), (13). Now we apply Lemma 38
with $f=\bar{S}$, $B=B$, $\epsilon=\eta$, and $l=l-1$. By this result and
inequality (49) we find a subspace $V\subseteq{\mathbf{G}}$ such that the
co–dimension of $V$ is controlled by estimate (37) and

$\sum_{|x|=l-1}\bar{S}(x)(B^{l-1}\circ{\mathcal{D}}_{l-1}(B*\mu_{V}))(x)\geqslant(1-3\eta)|B|^{l}\,.$
(50)

By the definition of the set $S$ (and hence $\bar{S}$), we have
${\mathcal{C}}_{l}(A)(x)\geqslant(1+\varepsilon/4)^{l}\delta^{l}N$ for any
$x\in\bar{S}$ and hence inequality (50) give us

$(1-3\eta)(1+\varepsilon/4)^{l}\delta^{l}|B|^{l}N\leqslant\sum_{|x|=l-1}(A^{l-1}\circ{\mathcal{D}}_{l-1}(A))(x)(B^{l-1}\circ{\mathcal{D}}_{l-1}(B*\mu_{V}))(x)$
$=\sum_{\alpha}(A*B*\mu_{V})(\alpha)\sum_{|x|=l-1}A^{l-1}(x)B^{l-1}(x+{\mathcal{D}}_{l-1}(\alpha))=\sum_{\alpha}(A*B*\mu_{V})(\alpha)(A\circ
B)^{l-1}(\alpha)$ $\leqslant|B|\|A*\mu_{V}\|_{\infty}\sum_{\alpha}(A\circ
B)^{l-1}(\alpha)\,.$ (51)

Now by our assumption $A$ is $\varepsilon/5$–uniform relatively to
$\mathsf{E}^{k_{*}}_{l-1}$ and a certain even
$k_{*}=O(kl\mathcal{L}(\delta))$. Using Corollary 18, we derive

$\|A*\mu_{V}\|_{\infty}\geqslant\delta(1-3\eta)(1+\varepsilon/4)\geqslant\delta(1+\varepsilon/8)\,.$

Finally, thanks to (37), we get

${\rm
codim}V\ll\varepsilon^{-2}l\mathcal{L}^{2}(\beta)\mathcal{L}^{2}(\varepsilon\beta^{l})\ll\varepsilon^{-2}l^{3}k^{4}\mathcal{L}^{2}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\,.$
(52)

This completes the proof. $\hfill\Box$

Now we can prove our new Theorem 44.

As always the proof follows the density increment scheme and our aim is to
construct a shift of a subspace $V_{\varepsilon}(l,k)$ where the set $A$ is
$\varepsilon$–uniform relatively to $\mathsf{E}_{l}^{k}$. Also, to obtain
Theorem 44 we use induction on parameter $l\geqslant 2$ and the first step of
the induction for $l=2$ and an arbitrary $k$ one can use either Kelley–Meka
Theorem 43 or the arguments of Section 3 (one can check that we do not need
any uniformity conditions in this case), combining with Proposition 47. Now
let $l\geqslant 3$ and suppose that the set $A$ is not $\varepsilon$–uniform
relatively to $\mathsf{E}^{k}_{l}$ for a certain $k$ because otherwise there
is nothing to prove. Of course (see, e.g., inequality (16)), one can take $k$
to be a sufficiently large number and we choose $k$ such that $k\gg
l\varepsilon^{-l}\mathcal{L}(\varepsilon)$. Put
$\varepsilon_{l}=\frac{\varepsilon^{l}}{8l}$. We can freely assume that our
set $A$ is $\varepsilon_{l}/5$–uniform relatively to
$\mathsf{E}^{k_{*}}_{l-1}$ with
$k_{*}=k_{*}(l)=O(kl^{2}\varepsilon^{-l}\mathcal{L}(\varepsilon)\mathcal{L}(\delta))$
in a shift of a subspace $V_{\varepsilon_{l}/5}(l-1,k_{*}(l))$ thanks to
Theorem 43 in the case $l=3$ or by the induction assumption for larger $l$.
Now we apply Lemma 23 and find
$k_{1}=O(kl\varepsilon^{-l}\mathcal{L}(\varepsilon))$ such that

$\mathsf{E}_{l}^{k_{1}}(A)\geqslant\left(1+\varepsilon_{l}\right)^{lk_{1}}\delta^{lk_{1}}N^{l+k_{1}}\,.$

After that we use Proposition 47 with $\varepsilon=\varepsilon_{l}$. One can
check that $kl\gg\varepsilon_{l}^{-1}\mathcal{L}(\varepsilon_{l})$ and that
$A$ is sufficiently uniform set (to apply our proposition) relatively to
$\mathsf{E}^{k_{*}}_{l-1}$ thanks to our choice of $k_{*}$. Estimate (47)
implies that the procedure must stop after
$O(\varepsilon^{-1}_{l}\mathcal{L}(\delta))$ number of steps and thus the
final co–dimension is

${\rm codim}V_{\varepsilon}(l,k)\ll\varepsilon^{-1}_{l}\mathcal{L}(\delta){\rm
codim}V_{\varepsilon_{l}/5}(l-1,k_{*}(l))\ll
l\varepsilon^{-l}\mathcal{L}(\delta){\rm
codim}V_{\varepsilon_{l}/5}(l-1,kl^{2}\varepsilon^{-l}\mathcal{L}(\varepsilon)\mathcal{L}(\delta))\,.$

Put $L=l!$. Solving the functional inequality above and using (43), we get

${\rm codim}V_{\varepsilon}(l,k)\ll
L\varepsilon^{-l(l-1)/2}\mathcal{L}^{l-2}(\delta){\rm
codim}V_{(\varepsilon/8l)^{2L}}(2,k(l!)^{2}\varepsilon^{-l(l-1)/2}(\mathcal{L}(\varepsilon)\mathcal{L}(\delta))^{l-2})$
(53)
$\ll\varepsilon^{-28l^{l}}(8l)^{28l^{l}}k^{4}\mathcal{L}^{4l}(\varepsilon)\mathcal{L}^{5l}(\delta)$

as required. Actually, one can see that the number of steps of our algorithm
is at most $O(\varepsilon^{-1}_{l}\mathcal{L}(\delta))$ (due to every time we
increase the density $\delta$ to $\delta(1+\varepsilon_{l}/8)$ and hence we do
not need the first multiple in (53)). Nevertheless, it gives us a bound of the
same sort. This concludes the proof of the theorem. $\hfill\Box$

Theorem 44 allows us to find a shift of subspace where our set $A$ is uniform
relatively to $\mathsf{E}^{k}_{l}$. A modification of this argument allows to
prove more and this is important for applications. Actually, for any
relatively dense $A\subseteq\mathbb{F}_{p}^{n}$, $|A|=\delta N$ there is a
partition of $\mathbb{F}_{p}^{n}$ onto shifts of subspaces $V_{j}$ such that

$A=\left(\bigsqcup_{j}A\cap(V_{j}+x_{j})\right)\bigsqcup\Omega$ (54)

such that $|\Omega|=\omega N$, each set $(A-x_{j})$ is $\varepsilon$–uniform
in $V_{j}$ relatively to $\mathsf{E}^{k}_{l}$ and ${\rm
codim}V_{j}=O_{\varepsilon,\omega,\delta}(1)$. Such results are not new see,
e.g., [12] and [11], [24], [25]. The idea is to replace $L_{\infty}$–increment
to $L_{q}$–increment with a controllable parameter $q$. In this section we
obtain the driving proposition, which we will use in the future paper.

###### Proposition 16

Let ${\mathbf{G}}=\mathbb{F}_{p}^{n}$, $A\subseteq{\mathbf{G}}$, $|A|=\delta
N$, $\varepsilon\in(0,1/2)$ be a real number and $k,l\geqslant 2$ be positive
integers, and $k\gg l\varepsilon^{-l}\mathcal{L}(\varepsilon)$. Suppose that

$\mathsf{E}^{k}_{l}(f_{A})=\varepsilon^{kl}\delta^{kl}N^{k+l}\,,$ (55)

and that $A$ is $2^{-11l}l^{-l}\varepsilon^{l^{2}}$–uniform with respect to
$\mathsf{E}^{k_{*}}_{l-1}$ for an even
$k_{*}=O(kl^{2}\varepsilon^{-l}\mathcal{L}(\varepsilon)\mathcal{L}(\delta))$.
Then there is a subspace $V\subseteq{\mathbf{G}}$ such that

$\mathrm{codim}V\ll\varepsilon^{-2l}l^{5}k^{4}\mathcal{L}^{2}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\,,$
(56) $N^{-1}\sum_{x}(f_{A}*\mu_{V})^{l}(x)\geqslant
2\left(\frac{\varepsilon\delta}{2^{7}}\right)^{l}\,.$ (57)

In particular,

$N^{-1}\sum_{x}(A*\mu_{V})^{l}(x)\geqslant\delta^{l}\left(1+\frac{\varepsilon^{l}}{2^{7l}}\right)\,.$
(58)

Further suppose that $A$ is $2^{-11}\varepsilon^{l}/l^{2}$–uniform with
respect to $\mathsf{E}^{k_{*}}_{l-1}$ for an even
$k_{*}=O(kl^{2}\varepsilon^{-l}\mathcal{L}(\varepsilon)\mathcal{L}(\delta))$.
Then for a certain even $q$, $q\ll l^{3}\mathcal{L}(\varepsilon\delta)$ and
$l>2$ the following holds

$N^{-1}\sum_{x}(f_{A}*\mu_{V})^{q}(x)\geqslant(4\delta)^{q}\,.$ (59)

In particular,

$N^{-1}\sum_{x}(A*\mu_{V})^{q}(x)\geqslant(2\delta)^{q}\,.$ (60)

P r o o f. Using Lemma 23 we find $k_{1}\ll
kl\varepsilon^{-l}\mathcal{L}(\varepsilon)$ such that

$\mathsf{E}^{k_{1}}_{l}(A)\geqslant\left(1+\frac{\varepsilon\varepsilon^{l-1}_{*}}{8l}\right)^{k_{1}l}\delta^{k_{1}l}N^{k_{1}+l}=(1+\varepsilon_{l})^{k_{1}l}\delta^{k_{1}l}N^{k_{1}+l}\,.$

Let $\zeta=2^{-8l}\varepsilon^{l}_{l}$, $\zeta_{1}=2^{-8}\varepsilon_{l}/l$
and for any positive integer $n$ we put
$S(n)=N^{-1}\sum_{\alpha}(f_{A}*\mu_{V})^{n}(\alpha)$. We use the same
argument as in Proposition 47 (the parameter $\eta$ equals
$\varepsilon_{l}/30$) before inequality (51). One has

$(1-3\eta)(1+\varepsilon_{l}/4)^{l}\delta^{l}|B|^{l}N=\delta|B|\sum_{\alpha}(A\circ
B)^{l-1}(\alpha)+\sum_{\alpha}(f_{A}*\mu_{V})(\alpha)((A\circ B)^{l-1}\circ
B)(\alpha)=\sigma_{1}+\sigma_{2}\,.$ (61)

Here we have the required upper bound for the dimension of $V$ similar to (52)

${\rm
codim}V\ll\varepsilon^{-2}_{l}l\mathcal{L}^{2}(\beta)\mathcal{L}^{2}(\varepsilon_{l}\beta^{l})\ll\varepsilon^{-2l}l^{5}k^{4}\mathcal{L}^{2}(\delta)\mathcal{L}^{2}(\varepsilon\delta)\,.$

We need to consider the cases $l=2$ and $l>2$ separately. In the case $l=2$ we
follow the method from [13]. Clearly, if $l=2$, then
$\sigma_{1}=\delta|B|^{2}|A|$ and hence

$\frac{7\varepsilon_{2}}{18}\delta^{2}|B|^{2}N\leqslant\sum_{\alpha}((f_{A}\circ
f_{A})*\mu_{V})(\alpha)(B\circ B)(\alpha)\,.$ (62)

Using the Fourier transform twice and the fact that
$\widehat{\mu}_{V}(r)=V^{\perp}(r)$, we get

$\frac{7\varepsilon_{2}}{18}\delta^{2}|B|^{2}N\leqslant
N^{-1}\sum_{r}\widehat{\mu}_{V}(r)|\widehat{f}_{A}(r)|^{2}|\widehat{B}(r)|^{2}\leqslant
N^{-1}|B|^{2}\sum_{r}\widehat{\mu}_{V}(r)|\widehat{f}_{A}(r)|^{2}=|B|^{2}\sum_{\alpha}(f_{A}\circ
f_{A})(\alpha)\mu_{V}(\alpha)\,.$

It remains to notice that $\mu_{V}\circ\mu_{V}=\mu_{V}$. Let us give another
proof which does not use the Fourier approach. Using the identity
$\mu_{V}\circ\mu_{V}=\mu_{V}$ we estimate the right–hand side of (62) as

$\|((f_{A}\circ
f_{A})*(\mu_{V}\circ\mu_{V}))(\alpha)\|_{\infty}\cdot\sum_{\alpha}(B\circ
B)(\alpha)=|B|^{2}\sum_{x}(f_{A}*\mu_{V})^{2}(x)$

as required.

Now consider the general case. To calculate the sum $\sigma_{1}$ we apply
Corollary 18 and our assumption on $l$ to derive

$(\delta|B|)^{-1}\sigma_{1}\leqslant\delta^{l-1}|B|^{l-1}N(1+1.25\zeta)^{l-1}\leqslant\delta^{l-1}|B|^{l-1}N(1+2\zeta
l)\,.$ (63)

In view of our choice of $\eta$ and
$\zeta\leqslant\zeta_{1}=2^{-8}\varepsilon_{l}/l$ it gives us

$2^{-3}\varepsilon_{l}l\delta^{l}|B|^{l}N\leqslant((1-3\eta)(1+\varepsilon_{l}l/4)-1-2\zeta
l)\delta^{l}|B|^{l}N\leqslant\sigma_{2}\,.$ (64)

Applying the Hölder inequality several times for even $l$, we derive

$\sigma_{2}=\sum_{\alpha}(f_{A}*\mu_{V})(\alpha)((A\circ B)^{l-1}\circ
B)(\alpha)$
$\leqslant\left(\sum_{\alpha}(f_{A}*\mu_{V})^{l}(\alpha)\right)^{1/l}\left(\sum_{\alpha}((A\circ
B)^{l-1}\circ B)^{l/(l-1)}(\alpha)\right)^{1-1/l}$ (65)
$\leqslant|B|S^{1/l}(l)N^{1/l}\left(\sum_{\alpha}(A\circ
B)^{l}(\alpha)\right)^{1-1/l}N^{1-1/l}\,.$ (66)

Now using Corollary 18 and recalling estimate (64), we see that

$2^{-4l+2}l^{l}\varepsilon^{l}_{l}\delta^{l}\leqslant
S(l)=N^{-1}\sum_{\alpha}(f_{A}*\mu_{V})^{l}(\alpha)$ (67)

as required. For odd $l$ the argument is the similar, one has

$\sum_{\alpha}|(f_{A}*\mu_{V})^{l}(\alpha)|\leqslant\sum_{\alpha}(f_{A}*\mu_{V})^{l-1}(\alpha)((f_{A}*\mu_{V})(\alpha)+2\delta)=NS(l)+2\delta\sum_{\alpha}(f_{A}*\mu_{V})^{l-1}(\alpha)$
$\leqslant NS(l)+2\zeta^{l-1}\delta^{l}N\leqslant
NS(l)+2^{-4l}l^{l}\varepsilon^{l}_{l}\delta^{l}N$

and hence we have obtain a lower bound for $S(l)$ similar to (67). Thus we
have proved (57). To derive (58), we write

$N^{-1}\sum_{\alpha}(A*\mu_{V})^{l}(\alpha)=S(l)+\delta^{l}+\sum_{0<j<l}\binom{l}{j}\delta^{l-j}\sum_{\alpha}(f_{A}*\mu_{V})^{j}(\alpha)$

and apply Lemma 23

$N^{-1}\sum_{\alpha}(A*\mu_{V})^{l}(\alpha)\geqslant
S(l)+\delta^{l}-2l\delta^{l}\zeta\geqslant\delta^{l}(1+2^{-4l}l^{l}\varepsilon^{l}_{l})$

as required.

Now let us obtain (59) and (60). Returning to (61) and (64), we have

$\sigma_{2}=\sum_{j=1}^{l-1}\binom{l-1}{j}(\delta|B|)^{l-1-j}\sum_{\alpha}((f_{A}\circ
B)^{j}\circ B)(\alpha)(f_{A}*\mu_{V})(\alpha)$
$=\sum_{j=1}^{l-1}\binom{l-1}{j}(\delta|B|)^{l-1-j}\sigma(j)\,.$ (68)

For even $j$ we can apply formula (65) with $l=q$ again, then the estimate
$\|((f_{A}\circ B)^{j}\circ B)(\alpha)\|_{\infty}\leqslant|B|^{j+1}$, as well
as Lemma 5 with the parameter $j<l$ and derive

$\sigma^{q}(j)\leqslant|B|^{q+j}\left(\sum_{\alpha}(f_{A}\circ
B)^{j}(\alpha)\right)^{q-1}\cdot\sum_{\alpha}(f_{A}*\mu_{V})^{q}(\alpha)$
$\leqslant\zeta^{j(q-1)}_{1}\delta^{j(q-1)}|B|^{(j+1)q}N^{q}\cdot S(q)$ (69)

For odd $j$, we write

$\sigma(j)=\sum_{\alpha}((f_{A}\circ B)^{j-1}\cdot(A\circ B)\circ
B)(\alpha)(f_{A}*\mu_{V})(\alpha)-\delta|B|\sigma(j-1)=\sigma_{*}(j)-\delta|B|\sigma(j-1)$

and thus we need to estimate $\sigma_{*}(j)$ because in $\sigma(j-1)$ the
argument $j-1$ is even. To bound the sum $\sigma_{*}(j)$ we use the same
argument as in (69) (and the identity $(A\circ B)(\alpha)=(f_{A}\circ
B)(\alpha)+\delta|B|$, of course) to show that

$\sigma^{q}_{*}(j)\leqslant|B|^{q+j}\left(\sum_{\alpha}(f_{A}\circ
B)^{j-1}(\alpha)(A\circ
B)(\alpha)\right)^{q-1}\cdot\sum_{\alpha}(f_{A}*\mu_{V})^{q}(\alpha)$
$\leqslant 2^{q-1}\zeta^{j(q-1)}_{1}\delta^{j(q-1)}|B|^{(j+1)q}N^{q}\cdot
S(q)\,.$ (70)

Thus in view of (64), (69), (70) as well as our choice of $q$, we get

$2^{-5}\varepsilon_{l}l\delta\leqslant
l((1+\zeta_{1})^{l-1}-1)S(q)^{1/q}\leqslant 2\zeta_{1}l^{2}S(q)^{1/q}$

and hence

$(4\delta)^{q}\leqslant N^{-1}\sum_{\alpha}(f_{A}*\mu_{V})^{q}(\alpha)$ (71)

as required. To get (60) just use the identity
$(f_{A}*\mu_{V})(\alpha)=(A*\mu_{V})(\alpha)-\delta$ and apply the binomial
formula to (71). This completes the proof. $\hfill\Box$

###### Remark 17

At first sight it looks strange that estimates (59), (60) do not depend on
$\varepsilon$ but we just pay for this taking larger codimension in (56).

##  5 Appendix

###  5.1 On longer progressions

We need a generalization of norms (14). Having vectors
$x=(x_{1},\dots,x_{s})\in{\mathbf{G}}^{s}$ and
$y=(y_{1},\dots,y_{t})\in{\mathbf{G}}^{t}$ (we write that $|x|=s$ and $|y|=t$)
define its “Minkowski” sum as $x\oplus y\in{\mathbf{G}}^{st}$, where the
components of $x\oplus y$ are $x_{i}+y_{j}$, $i\in[s]$, $j\in[t]$ (and
similarly for higher sums). Put

$\mathcal{E}^{k}_{s,t}(f)=\sum_{|x|=s}\,\sum_{|y|=t}{\mathcal{C}}^{k}_{st}(f)(x\oplus
y)=\sum_{|x|=s}\,\sum_{|y|=t}\,\sum_{|z|=k}{\mathcal{P}}_{stk}(f)(x\oplus
y\oplus z)\,.$ (72)

In these terms

$\mathsf{E}^{k}_{l}(f)=\sum_{|x|=l}\,\sum_{|y|=k}\,{\mathcal{P}}_{kl}(f)(x\oplus
y)\,.$ (73)

For even $k,s,t$ and a real function $f$ one has
$\mathcal{E}^{k}_{s,t}(f)\geqslant\mathsf{E}^{k}_{s}(f),\mathsf{E}^{k}_{t}(f)\geqslant
0$ and the triangle inequality for $\mathcal{E}^{k}_{s,t}$ can be obtained
exactly as in [27, Appendix] or just see Subsection 5.2 (but of course one
needs an additional application of the Hölder inequality due to we have the
longer sum in (72) than in (73)). Thus $(\mathcal{E}^{k}_{s,t}(f))^{1/kst}$
defines a norm of $f:{\mathbf{G}}\to\mathbb{R}$ in the case of even $k,s,t$.
Notice that similar to $\mathsf{E}^{k}_{l}(f)$ the quantity
$\mathcal{E}^{k}_{s,t}(f)\geqslant 0$, provided at least one of $k,s,t$ is
even but, nevertheless, it is not always a norm in this case, see [27, Section
4]. By some symmetricity reasons (see, e.g., formulae (78) below) we make a
normalization and put

$\|f\|_{\mathcal{E}^{k}_{s,t}}:=(|{\mathbf{G}}|^{-2}\mathcal{E}^{k}_{s,t}(f))^{1/kst}:=(\bar{\mathcal{E}}^{k}_{s,t}(f))^{1/kst}$

for $f:{\mathbf{G}}\to\mathbb{R}$. Clearly, one has

$\mathcal{E}^{k}_{s,t}(f)=\sum_{|x|=s}\,\sum_{|z|=k}{\mathcal{C}}^{t}_{s}(f_{z})(x)=\sum_{|x|=s}\,\sum_{|z|=k}{\mathcal{C}}^{t}_{sk}(f)(x\oplus
z)\,,$ (74)

and

$\mathcal{E}^{k}_{s,t}(f)=\sum_{|y|=t}\,\sum_{|z|=k}{\mathcal{C}}^{s}_{t}(f_{z})(y)=\sum_{|y|=t}\,\sum_{|z|=k}{\mathcal{C}}^{s}_{tk}(f)(y\oplus
z)\,.$ (75)

Thus we have the duality relation similar to (14)

$\mathcal{E}^{k}_{t,s}(f)=\mathcal{E}^{k}_{s,t}(f)=\mathcal{E}^{t}_{s,k}(f)=\mathcal{E}^{s}_{t,k}(f)\,.$
(76)

Also, let us remark that the expectations over $x\oplus y$ of the generalized
convolution of any real function $f:{\mathbf{G}}\to\mathbb{R}$ is connected
with the higher energies

$\sum_{|x|=s}\,\sum_{|y|=t}{\mathcal{C}}_{st}(f)(x\oplus
y)=N\mathsf{E}^{t}_{s}(f)\,.$ (77)

In particular, the expectation above is always non–negative if $s$ or $t$ is
an even number and we see immediately that the duality (14) takes place.
Formula (77) can be proved directly or it follows from (74), (75) and the fact
that $\mathcal{E}^{k}_{t,1}(f)=N\mathsf{E}^{k}_{t}(f)$. Finally, notice that
in contrast to ${\mathcal{C}}_{l}(x)$ the function ${\mathcal{C}}_{st}(x\oplus
y)$ enjoys even two symmetries, namely,

${\mathcal{C}}_{st}(f)(x\oplus
y)={\mathcal{C}}_{st}(f)((x+{\mathcal{D}}_{s}(w_{1}))\oplus
y)={\mathcal{C}}_{st}(f)(x\oplus(y+{\mathcal{D}}_{t}(w_{2})))$ (78)

for any $w_{1},w_{2}\in{\mathbf{G}}$. It gives, in particular,

${\mathcal{C}}_{st}(f)(x\oplus
y)={\mathcal{C}}_{st}(f)((x-{\mathcal{D}}_{s}(x_{1}))\oplus(y-{\mathcal{D}}_{t}(y_{1})))=N^{2}{\mathcal{C}}^{\prime}_{st}(f)(w)\,,$
(79)

where $|w|=st-1$ and, more concretely, $w_{ij}=(x_{i}-x_{1})+(y_{j}-y_{1})$,
$i\in[s]$, $j\in[t]$ and $(i,j)\neq(1,1)$.

Now we are ready to obtain our counting lemma. Let us write $L(x,y)=\alpha
x+\beta y+\gamma$ for a non–trivial linear form. We say that two forms are
non–proportional if their coefficients are not proportional. Given a real
number $q>1$ put $q^{*}=\frac{q}{q-1}$.

###### Theorem 18

Let $N$ be a prime number and $k=4$, $l_{1},l_{2}\geqslant 2$ be positive
integers. Also, let $f_{1},\dots,f_{k}:\mathbb{Z}/N\mathbb{Z}\to\mathbb{R}$ be
functions and $L_{1},\dots,L_{k}$ be non–proportional linear forms such that
$L_{2},\dots,L_{k}$ depend on both variables. Then

$\left|\sum_{x,y}f_{1}(L_{1}(x,y))\dots
f_{k}(L_{k}(x,y))\right|\leqslant\|f_{1}\|_{l^{*}_{1}}\|f_{2}\|_{l^{*}_{2}}\|f_{3}\|_{\mathcal{E}^{2}_{l_{1},l_{2}}}\|f_{4}\|_{\mathcal{E}^{2}_{l_{1},l_{2}}}\,.$
(80)

P r o o f. Let $\sigma$ be the left–hand side of (80). Without loss of
generality one can assume that $L_{j}(x,y)=\alpha_{j}x+\beta_{j}y$, $j\in[k]$.
Consider the nonzero form $L_{1}$ and suppose for concreteness that
$\alpha_{1}\neq 0$. Changing the variables $\alpha_{1}x+\beta_{1}y\to x$, we
obtain

$\sigma=\sum_{x,y}f_{1}(x)f_{2}(\tilde{L}_{2}(x,y))\dots
f_{k}(\tilde{L}_{k}(x,y))\,,$ (81)

where here and below we write
$\tilde{L}_{j}(x,y)=L_{j}(x,y)=\alpha_{j}x+\beta_{j}y$ and the coefficients
$\alpha_{j}$, $\beta_{j}$ may change from line to line. Anyway one can check
that all new forms $\tilde{L}_{2},\dots,\tilde{L}_{k}$ in (81) are nonzero and
non–proportional. Moreover, by assumption the initial forms
$L_{2},\dots,L_{k}$ depend on both variables and we see that the new forms in
(81) depend on both variables as well. Now we use the Hölder inequality and
get

$(\sigma/\|f_{1}\|_{l^{*}_{1}})^{l_{1}}\leqslant\sum_{x}\left(\sum_{y}f_{2}(L_{2}(x,y))\dots
f_{k}(L_{k}(x,y))\right)^{l_{1}}$
$=\sum_{x,y}{\mathcal{P}}_{l_{1}}(f_{2})(L_{2}({\mathcal{D}}_{l_{1}}(x),{\mathcal{P}}_{l_{1}}(y)))\dots\mathcal{P}_{l_{1}}(f_{k})(L_{k}({\mathcal{D}}_{l_{1}}(x),{\mathcal{P}}_{l_{1}}(y)))\,.$

Notice that we have decreased the number of our linear forms (but increased
the number of variables). Now let us make the changing of the variables
similar to above, namely,
$\alpha_{1}{\mathcal{D}}_{l_{1}}(x)+\beta_{1}{\mathcal{P}}_{l_{1}}(y)\to{\mathcal{P}}_{l_{1}}(y)$
and again one can easily check that we preserve all conditions on our linear
forms $L_{3},\dots,L_{k}$. Thus one has

$(\sigma/\|f_{1}\|_{l^{*}_{1}})^{l_{1}}\leqslant\sum_{y}{\mathcal{P}}_{l_{1}}(f_{2})({\mathcal{P}}_{l_{1}}(y))\sum_{x}{\mathcal{P}}_{l_{1}}(f_{3})(L_{3}({\mathcal{D}}_{l_{1}}(x),{\mathcal{P}}_{l_{1}}(y)))\dots\mathcal{P}_{l_{1}}(f_{k})(L_{k}({\mathcal{D}}_{l_{1}}(x),{\mathcal{P}}_{l_{1}}(y)))$

and using the Hölder inequality one more time, as well as the obvious identity

$\left(\sum_{y}{\mathcal{P}}^{l^{*}_{2}}_{l_{1}}(f_{2})({\mathcal{P}}_{l_{1}}(y))\right)^{l_{2}-1}=\|f_{2}\|^{l_{1}l_{2}}_{l^{*}_{2}}\,,$
(82)

we derive

$(\sigma/\|f_{1}\|_{l^{*}_{1}}\|f_{2}\|_{l^{*}_{2}})^{l_{1}l_{2}}$
$\leqslant\sum_{x,y}{\mathcal{P}}_{l_{1}l_{2}}(f_{3})(L_{3}({\mathcal{P}}_{l_{2}}{\mathcal{D}}_{l_{1}}(x),{\mathcal{D}}_{l_{2}}{\mathcal{P}}_{l_{1}}(y))){\mathcal{P}}_{l_{1}l_{2}}(f_{4})(L_{4}({\mathcal{P}}_{l_{2}}{\mathcal{D}}_{l_{1}}(x),{\mathcal{D}}_{l_{2}}{\mathcal{P}}_{l_{1}}(y)))\,.$
(83)

Now let us analyse the right–hand side of formula (83). First of all, it is
easy to see that there are $l_{2}$ different variables $x_{i}$ and $l_{1}$
different variables $y_{j}$ in (83). Secondly, take the form $L_{k-1}$ (for
$L_{k}$ the argument is the same) and notice that it depends on
$\alpha_{k-1}x_{i}+\beta_{k-1}y_{j}$, $i\in[l_{2}]$, $j\in[l_{1}]$ and that
every such expression appears exactly once. Now introducing two more variables
$z,w$ such that $x_{i}\to x_{i}+z$, $y_{j}\to y_{j}+w$ and then replacing
$z,w$ to other variables $Z,W$, where $Z=\alpha_{k-1}z+\beta_{k-1}w$,
$W=\alpha_{k}z+\beta_{k}w$ (this change of the variables is allowable because
the forms $L_{k-1}$, $L_{k}$ are not proportional), we arrive to the
quantities ${\mathcal{C}}_{l_{1}l_{2}}(f_{k-1})$,
${\mathcal{C}}_{l_{1}l_{2}}(f_{k})$ in 0 (83). Writing
$x=(x_{1},\dots,x_{l_{2}})$, $y=(y_{1},\dots,y_{l_{1}})$, we have finally

$(\sigma/\|f_{1}\|_{l^{*}_{1}}\|f_{2}\|_{l^{*}_{2}})^{l_{1}l_{2}}\leqslant
N^{-2}\sum_{\vec{x},\vec{y}}{\mathcal{C}}_{l_{1}l_{2}}(f_{k-1})(\alpha_{k-1}\cdot
x\oplus\beta_{k-1}\cdot y){\mathcal{C}}_{l_{1}l_{2}}(f_{k})(\alpha_{k}\cdot
x\oplus\beta_{k}\cdot y)\,.$

Using the Hölder inequality the last time, as well as the fact that
$\alpha_{k-1},\alpha_{k},\beta_{k-1},\beta_{k}\neq 0$, we obtain

$(\sigma/\|f_{1}\|_{l^{*}_{1}}\|f_{2}\|_{l^{*}_{2}})^{l_{1}l_{2}}$
$\leqslant\left(N^{-2}\sum_{|x|=l_{2},\,|y|=l_{1}}{\mathcal{C}}^{2}_{l_{1}l_{2}}(f_{k-1})(x\oplus
y)\right)^{1/2}\left(N^{-2}\sum_{|x|=l_{2},\,|y|=l_{1}}{\mathcal{C}}^{2}_{l_{1}l_{2}}(f_{k})(x\oplus
y)\right)^{1/2}$
$=\|f_{k-1}\|^{l_{1}l_{2}}_{\mathcal{E}^{2}_{l_{1},l_{2}}}\cdot\|f_{k}\|^{l_{1}l_{2}}_{\mathcal{E}^{2}_{l_{1},l_{2}}}$

as required. $\hfill\Box$

###### Remark 19

One can check that for any $l_{1},\dots,l_{k-2}$ one has

$\frac{1}{l^{*}_{1}}+\frac{1}{l^{*}_{2}}+\frac{1}{l_{1}}+\frac{1}{l_{2}}=2$

and hence the right–hand side of bound (80) has the correct order in $N$.
Similarly, taking $f_{j}(x)=f_{A}(x)$, $j\in[4]$ be the balanced function of a
set $A$ and $l_{1}\sim l_{2}\sim\mathcal{L}(\delta)$ we see that the
dependence on $\delta$ is also correct.

###### Remark 20

As we have said in the previous remark the optimal dependence on the
parameters $l_{1},l_{2}$ in Theorem 80 is $l_{1}\sim
l_{2}\sim\mathcal{L}(\delta)$. Suppose that the dependence on $\varepsilon$ in
Lemma 23 and in all statements below is almost optimal, say, $c\varepsilon$
for a constant $c\in(0,1)$. Thanks to the induction scheme of the proof, it
gives us the multiple $c^{\mathcal{L}(\delta)}=\delta^{-C}$ for a certain
$C>0$ in codimention of the subspace $V$, where our set $A$ is uniform. But
$\delta^{-C}$ is more or less that usual Gowers’ method gives to us and hence
we have no special gain. Thus there are considerable difficulties to extend
the Kelley–Meka method for more complicated objects than arithmetic
progressions of length three even on the technical level.

We conclude the appendix showing that the convolutions
${\mathcal{C}}_{st}(f)(x\oplus y)$, $|x|=s$, $|y|=t$ enjoy the almost
periodicity properties similar to the ordinary convolutions
${\mathcal{C}}_{s}(f)(x)$. For $x=(x_{1},\dots,x_{r})$ let us write for
convenience $\bar{x}=(0,x_{1},\dots,x_{r})$.

###### Lemma 21

Let $\varepsilon\in(0,1]$ be a real number, $s,t,q\geqslant 2$ be positive
integers, $l:=st-1$, $B\subseteq{\mathbf{G}}$, $|B|=\beta N$ and
$F:{\mathbf{G}}^{l}\to\mathbb{R}$. Then there is a set
$T\subseteq{\mathbf{G}}$,
$|T|\geqslant|B|\exp(-O(\varepsilon^{-2}q\log(1/\beta)))$ and such that for
any $t\in T$ one has

$\sum_{|x|=s-1}\,\sum_{|y|=t-1}\left|(F\circ{\mathcal{D}}_{l}(B+t))(\bar{x}\oplus\bar{y})-(F\circ{\mathcal{D}}_{l}(B))(\bar{x}\oplus\bar{y})\right|^{q}$
$\leqslant\varepsilon^{q}|B|^{q-1}\sum_{|y|=t-1}{\mathcal{C}}^{\prime}_{t}(|F|^{q})^{s-1}(y)\cdot{\mathcal{C}}^{\prime}_{t}(B,|F|^{q},\dots,|F|^{q})(y)\,.$
(84)

P r o o f. We choose $k=O(\varepsilon^{-2}q)$ random points
$b_{1},\dots,b_{k}\in B$ uniformly and independently and let
$Z_{j}((\bar{x}\oplus\bar{y})=F((\bar{x}\oplus\bar{y})+{\mathcal{D}}_{l}(b_{j}))-(F\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})$.
Clearly, the random variables $Z_{j}$ are independent, have zero expectation
and their variances do not exceed
$(|F|^{2}\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})$. By the
Khintchine inequality for sums of independent random variables,

$\|\sum_{j=1}^{k}Z_{j}(\bar{x}\oplus\bar{y})\|_{L_{p}(\mu^{k}_{B})}\ll(|F|^{2}\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})^{1/2}\,.$

Raising the last inequality to the power $q$, dividing by $k^{q}$, summing
over $\bar{x}\oplus\bar{y}$, and using the Hölder inequality, which gives
$(|F|^{2}\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})^{q/2}\leqslant(|F|^{q}\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})$,
we get that

$\sum_{|x|=s-1}\,\sum_{|y|=t-1}\int\left|\frac{1}{k}\sum_{j=1}^{k}F((\bar{x}\oplus\bar{y})+{\mathcal{D}}_{l}(b_{j}))-(F\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})\right|^{q}d\mu_{B}^{k}(x_{1},\dots,x_{k})$
$\ll(qk^{-1})^{q/2}\sum_{|x|=s-1}\,\sum_{|y|=t-1}(|F|^{q}\circ{\mathcal{D}}_{l}(\mu_{B}))(\bar{x}\oplus\bar{y})$
$=(qk^{-1})^{q/2}|B|^{-1}\sum_{|y|=t-1}{\mathcal{C}}^{\prime}_{t}(|F|^{q})^{s-1}(y)\cdot{\mathcal{C}}^{\prime}_{t}(B,|F|^{q},\dots,|F|^{q})(y)\,.$

After that we repeat the argument from [4], [18], [19] and [26, Theorem 15].
This completes the proof. $\hfill\Box$

###  5.2 On a family of norms

In this section we define a very general family of norms, which includes the
norms $\mathsf{E}^{k}_{l}$, $\mathcal{E}^{k}_{s,t}$ above, as well as the
classical Gowers norms [9]. As the reader can see we do not use the Fourier
approach in our proofs below.

Let ${\mathbf{G}}$ be an abelian group, $r,k_{1},\dots,k_{r}\geqslant 2$ be
integers and $f:{\mathbf{G}}\to\mathbb{R}$ be an arbitrary function. Let
$K=\prod_{j=1}^{r}k_{j}$, $B=[k_{1}]\times\dots\times[k_{r}]$ and write
$x_{1}=(x^{(1)}_{1},\dots,x^{(k_{1})}_{1}),\dots,x_{r}=(x^{(1)}_{r},\dots,x^{(k_{r})}_{r})$.
Also, for $\omega\in B$ we write $\omega=(\omega_{1},\dots,\omega_{r})$.
Define

$\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r}|=k_{r}}\,{\mathcal{P}}_{K}(f)(x_{1}\oplus\dots\oplus
x_{r})$ $=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r}|=k_{r}}\,\prod_{\omega\in
B}f(x^{(\omega_{1})}_{1}+\dots+x^{(\omega_{r})}_{r})\,.$ (85)

The case $r=2$ corresponds to $\mathsf{E}^{k}_{l}$–norm, $r=3$ is just
$\mathcal{E}^{k}_{s,t}$–norms and for $k_{1}=\dots=k_{r}=2$, we obtain Gowers’
$U^{k}$ norms (up to some normalizations). In a similar way one can define the
multi–scalar product for the quantity $\|\cdot\|_{E_{k_{1},\dots,k_{r}}}$ as
was done in [9], namely, having any functions $(f^{\omega})_{\omega\in B}$, we
write

$\langle
f^{\omega}\rangle_{E_{k_{1},\dots,k_{r}}}=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r}|=k_{r}}\,\prod_{\omega\in
B}f^{\omega}(x^{(\omega_{1})}_{1}+\dots+x^{(\omega_{r})}_{r})\,.$ (86)

It is easy to see that if $K$ is an even number, then

$\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}\geqslant 0\,.$ (87)

Indeed, let, say, $k_{r}$ be an even number, then we can write $x_{r}$ as
$x_{r}=(x^{\prime}_{r},x^{\prime\prime}_{r})$, where
$|x^{\prime}_{r}|=|x^{\prime\prime}_{r}|=k_{r}/2$ and whence

$\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r-1}|=k_{r-1}}\,\left(\sum_{|x^{\prime}_{r}|=k_{r}/2}{\mathcal{P}}_{K/2}(f)(x_{1}\oplus\dots\oplus
x_{r-1}\oplus x^{\prime}_{r})\right)^{2}\geqslant 0\,.$

Also, let us remark the inductive property of the norm
$E_{k_{1},\dots,k_{r}}$. For concreteness, we take the $r$th coordinate and
obtain from definition (85) that

$\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}=\sum_{|z|=k_{r}}\|f_{z}\|^{K/k_{r}}_{E_{k_{1},\dots,k_{r-1}}}\,.$
(88)

Let us make a simple remark concerning $E_{k_{1},\dots,k_{r}}$–norm.

###### Lemma 22

Let $f:{\mathbf{G}}\to\mathbb{R}$ be a function. Suppose that there is
$j\in[r]$ such that $k_{j}$ is even and $K/k_{j}$ is also even. Then
$\|f\|_{E_{k_{1},\dots,k_{r}}}=0$ iff $f\equiv 0$.

P r o o f. Without loosing of the generality assume that $j=r$. Write

$\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r-1}|=k_{r-1}}\,\left(\sum_{z}{\mathcal{P}}_{K/k_{r}}(f)(x_{1}\oplus\dots\oplus
x_{r-1}\oplus z)\right)^{k_{r}}=0\,.$

Since $k_{r}$ is an even number, it follows that, in particular,
$\sum_{z}f^{K/k_{r}}(z)=0$ (we have taken $x_{1}=\dots=x_{r-1}=0$ in the last
formula) and hence $f\equiv 0$. This completes the proof. $\hfill\Box$

Now let us show that the multi–scalar product is controlled via
$E_{k_{1},\dots,k_{r}}$–norm.

###### Lemma 23

Let $r\geqslant 2$ be a positive integer, $k_{1},\dots,k_{r}\geqslant 2$ be
even integers and $f^{\omega}:{\mathbf{G}}\to\mathbb{R}$, $\omega\in B$ be any
functions. Then

$|\langle f^{\omega}\rangle_{E_{k_{1},\dots,k_{r}}}|\leqslant\prod_{\omega\in
B}\|f^{\omega}\|_{E_{k_{1},\dots,k_{r}}}\,.$ (89)

P r o o f. We write

$\langle
f^{\omega}\rangle_{E_{k_{1},\dots,k_{r}}}=\sum_{|x_{1}|=k_{1}}\,\dots\sum_{|x_{r-1}|=k_{r-1}}\,\left(\sum_{x^{(1)}_{r}}\,\prod_{\omega\in
B,\,\omega_{r}=1}f^{\omega}(x^{(\omega_{1})}_{1}+\dots+x^{(\omega_{r-1})}_{r-1}+x^{(1)}_{r})\right)$
$\dots\left(\sum_{x^{(k_{1})}_{r}}\,\prod_{\omega\in
B,\,\omega_{r}=k_{r}}f^{\omega}(x^{(\omega_{1})}_{1}+\dots+x^{(\omega_{r-1})}_{r-1}+x^{(k_{r})}_{r})\right)\,.$

After that apply the Hölder inequality (here we have used the fact that
$k_{r}$ is an even number) and we arrive to the new $k_{r}$ families of
functions. Take any of them, say, $(\tilde{f}^{\omega})$, $\omega\in B$ and
notice that

$\tilde{f}^{\omega}=\tilde{f}^{\omega^{\prime}}=f^{\omega}$

for all $\omega=(\omega_{1},\dots,\omega_{r})$,
$\omega^{\prime}=(\omega^{\prime}_{1},\dots,\omega^{\prime}_{r})$ with
$(\omega_{2},\dots,\omega_{r})=(\omega^{\prime}_{2},\dots,\omega^{\prime}_{r})$.
In particular, the family $(\tilde{f}^{\omega})$, $\omega\in B$ has $K/k_{r}$
different functions. Now we use the same argument for all remaining variables
$x_{1},\dots,x_{r-1}$ subsequently changing the families $(f^{\omega})$,
$\omega\in B$. One can easily see that after all these $r$ steps we arrive to
$K$ families consisting of single functions $f^{\omega}$, $\omega\in B$ (just
thanks to the fact that any two points of our box $B$ can be reached by a path
in the directions of coordinate axes). This is equivalent to inequality (89)
and we complete the proof. $\hfill\Box$

Finally, we are ready to obtain the main result of this section. Let us write
$(k_{1},\dots,k_{r})\leqslant(m_{1},\dots,m_{t})$ if the first vector is
lexigraphically smaller than the second one (i.e., $r\leqslant t$ and
$k_{j}\leqslant m_{j}$, $j\in[r]$). Also, put

$\|f\|^{K}_{\bar{E}_{k_{1},\dots,k_{r}}}=N^{-(k_{1}+\dots+k_{r})}\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}\,.$
(90)

Thus for any $f:{\mathbf{G}}\to[-1,1]$ one has
$\|f\|_{\bar{E}_{k_{1},\dots,k_{r}}}\leqslant 1$.

###### Theorem 24

Let $r\geqslant 2$ be a positive integer, $k_{1},\dots,k_{r}\geqslant 2$ be
even integers and $f:{\mathbf{G}}\to\mathbb{R}$ be a function. Then formula
(85) defines a norm of $f$. Further if
$(k_{1},\dots,k_{r})\leqslant(m_{1},\dots,m_{t})$, then

$\|f\|_{\bar{E}_{k_{1},\dots,k_{r}}}\leqslant\|f\|_{\bar{E}_{m_{1},\dots,m_{t}}}\,.$
(91)

P r o o f. Take two functions $f,g:{\mathbf{G}}\to\mathbb{R}$. In view of
Lemma 89, we have

$\|f+g\|^{K}_{E_{k_{1},\dots,k_{r}}}=\langle
f+g\rangle_{E_{k_{1},\dots,k_{r}}}\leqslant\sum_{j=1}^{k}\binom{K}{j}\|f\|^{j}_{E_{k_{1},\dots,k_{r}}}\|g\|^{K-j}_{E_{k_{1},\dots,k_{r}}}=(\|f\|_{E_{k_{1},\dots,k_{r}}}+\|g\|_{E_{k_{1},\dots,k_{r}}})^{K}$

and we have obtained the triangle inequality for $E_{k_{1},\dots,k_{r}}$. By
estimate (87) we know that our quantity $\|f\|_{\bar{E}_{k_{1},\dots,k_{r}}}$
is non–negative. Also, Lemma 22 guaranties that
$\|f\|_{E_{k_{1},\dots,k_{r}}}=0$ iff $f\equiv 0$. Thus indeed formula (85)
defines a norm of $f$.

It remains to obtain (91). Let $M=\prod_{j=1}^{t}m_{j}$,
$B^{\prime}=[m_{1}]\times\dots\times[m_{t}]$, $S=\sum_{j=1}^{r}k_{j}$ and
$S^{\prime}=\sum_{j=1}^{t}m_{j}$. Consider the family of functions
$(g^{\omega})_{\omega\in B^{\prime}}$ such that for $\omega\in B$ one has
$g^{\omega}=f^{\omega}$ and let $g^{\omega}\equiv 1$ otherwise. It is easy to
see that

$\|f\|^{K}_{\bar{E}_{k_{1},\dots,k_{r}}}N^{S}=\|f\|^{K}_{E_{k_{1},\dots,k_{r}}}=N^{S-S^{\prime}}\cdot\langle
g^{\omega}\rangle_{E_{m_{1},\dots,m_{t}}}$

Using the last formula, Definition (90), as well as Lemma 89, we obtain

$\|f\|^{K}_{\bar{E}_{k_{1},\dots,k_{r}}}N^{S}\leqslant
N^{S-S^{\prime}}\|f\|^{K}_{E_{m_{1},\dots,m_{t}}}(N^{S^{\prime}/M})^{M-K}=N^{S-S^{\prime}K/M}\cdot
N^{S^{\prime}K/M}\|f\|^{K}_{\bar{E}_{m_{1},\dots,m_{t}}}$
$=N^{S}\|f\|^{K}_{\bar{E}_{m_{1},\dots,m_{t}}}$

as required. $\hfill\Box$

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