# New Bound for Roth’s Theorem
with Generalized Coefficients

Cédric Pilatte

###### Abstract

We prove the following conjecture of Shkredov and Solymosi: every subset
$A\subset\mathbf{Z}^{2}$ such that $\sum_{a\in
A\setminus\\{0\\}}1/\left\|a\right\|^{2}=+\infty$ contains the three vertices
of an isosceles right triangle. To do this, we adapt the proof of the recent
breakthrough by Bloom and Sisask on sets without three-term arithmetic
progressions, to handle more general equations of the form
$T_{1}a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$ in a finite abelian group $G$, where the
$T_{i}$’s are automorphisms of $G$.

title = New Bound for Roth’s Theorem with Generalized Coefficients, author =
Cédric Pilatte, plaintextauthor = Cedric Pilatte, keywords = Roth’s theorem,
discrete Fourier analysis, Bohr sets, density increments, year=2022,
number=16, received=9 November 2021, published=2 December 2022,
doi=10.19086/da.55553,

[classification=text]

##  1 Introduction

In their 2020 breakthrough paper, Bloom and Sisask [4] improved the best known
upper bound on the largest possible size of a subset of $\\{1,2,\dots,n\\}$
without three-term arithmetic progression. They showed that, if
$A\subset\\{1,2,\dots,n\\}$ has no non-trivial three-term arithmetic
progression, then

$|A|\ll\frac{n}{(\log n)^{1+c}}$

for some absolute constant $c>0$. The best previously available bound was
${n}/{(\log n)^{1-o(1)}}$, which had been obtained in four different ways [7,
2, 3, 8].

Their result received a lot of attention as it settled the first interesting
case of one of Erdős’ most famous conjectures. Erdős conjectured that, if
$A\subset\mathbf{N}$ is such that $\sum_{n\in A}1/n$ diverges, then $A$
contains infinitely many $k$-term arithmetic progressions, for every $k\geq
3$. The result of Bloom and Sisask implies the case $k=3$. The general case
seems to be well beyond the reach of the current techniques.

The theorem of Bloom and Sisask can be applied to the prime numbers to recover
a result of Green in analytic number theory. It is an old result of Van der
Corput that the set of primes contains infinitely many three-term arithmetic
progressions. Much more recently, Green [6] generalized this fact to
relatively dense subsets of the primes. The theorem of Bloom and Sisask gives
a different proof of this, where Chebyshev’s estimate $\pi(x)\gg x/\log{x}$ is
the only fact about the primes that is used.

A three-term arithmetic progression is a solution to the equation
$a_{1}-2a_{2}+a_{3}=0$. In this paper, we generalize the proof of Bloom and
Sisask to deal with equations of the form $T_{1}a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$
for an extended class of coefficients $T_{1}$, $T_{2}$ and $T_{3}$. More
precisely, we prove the following in Section 5.

######  Theorem 1.1.

Let $G$ be a finite abelian group and let $T_{1},T_{2},T_{3}$ be automorphisms
of $G$ such that $T_{1}+T_{2}+T_{3}=0$. If $A$ is a subset of $G$ without non-
trivial solutions111A solution $(a_{1},a_{2},a_{3})\in A^{3}$ is trivial if
$a_{1}=a_{2}=a_{3}$. to the equation

$T_{1}a_{1}+T_{2}a_{2}+T_{3}a_{3}=0,$ (1)

then

$|A|\ll\frac{|G|}{(\log|G|)^{1+c}}$

where $c>0$ is an absolute constant.222In particular, the constant $c$ does
not depend on $G$ or on the coefficients $T_{i}$.

The result [4, Corollary 3.2] of Bloom and Sisask corresponds to the special
case ${T_{1}=T_{2}=\mathrm{Id}_{G}}$ and $T_{3}=-2\,\mathrm{Id}_{G}$ of
Theorem 1.1. Their hypothesis that $G$ has odd order ensures that
$-2\,\mathrm{Id}_{G}$ is an automorphism.

######  Remark 1.2.

The condition $T_{1}+T_{2}+T_{3}=0$ ensures that Eq. 1 is translation-
invariant. It is necessary: for example, if $G=\mathbf{F}_{2}^{n}$,
${T_{1}=T_{2}=T_{3}=\mathrm{Id}_{G}}$ and $A$ is the set of vectors with first
coordinate equal to $1$, then $|A|\asymp|G|$, yet $A$ has no solutions to Eq.
1.

We will deduce the following corollary, which generalizes [4, Corollary 1.2]
to higher dimensions and matrix coefficients. It is also a strengthening of
[1, Theorem 2.21].

######  Corollary 1.3.

Let $M_{1},M_{2},M_{3}$ be nonsingular $d\times d$ matrices with integer
coefficients such that $M_{1}+M_{2}+M_{3}=0$. If $A\subset\mathbf{Z}^{d}$
satisfies

$\sum_{a\in A\setminus\\{0\\}}\frac{1}{\left\|a\right\|^{d}}=+\infty,$

then $A$ contains infinitely many non-trivial solutions to the equation
$M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0$.

Using Corollary 1.3, we are able to prove a conjecture of Shkredov and
Solymosi [10, Conjecture 2].

######  Example 1.4.

If a subset $A$ of the square lattice satisfies $\sum_{a\in
A\setminus\\{0\\}}{1}/{\left\|a\right\|^{2}}=+\infty$, then there are
infinitely many isosceles right triangles whose vertices are in $A$.

We also obtain the following aesthetic result.

######  Example 1.5.

If a subset $A$ of the hexagonal lattice satisfies $\sum_{a\in
A\setminus\\{0\\}}{1}/{\left\|a\right\|^{2}}=+\infty$, then $A$ contains
infinitely many equilateral triangles.

Examples 1.5 and 1.4 are special cases of the following corollary.

######  Corollary 1.6.

Let $\Lambda\subset\mathbf{C}$ be a lattice of the form
$\Lambda=\omega_{1}\mathbf{Z}\oplus\omega_{2}\mathbf{Z}$, such that
$\omega_{i}\Lambda\subset\Lambda$ for $i=1,2$. Let $T$ be any triangle with
vertices in $\Lambda$. If $A\subset\Lambda$ is such that

$\sum_{a\in A\setminus\\{0\\}}\frac{1}{|a|^{2}}=+\infty,$

then there are infinitely many triangles, with vertices in $A$, which are
directly similar333Two triangles are directly similar if there is an
orientation-preserving similitude of the plane mapping one to the other. to
$T$.

###### Proof.

The orientation-preserving similitudes of the plane are exactly the
transformations of the form $z\mapsto uz+v$ with $u,v\in\mathbf{C}$, $u\neq
0$. Let $p_{1},p_{2}$ and $p_{3}$ be the (distinct) vertices of $T$.

Finding triangles in $A$ that are directly similar to $T$ is equivalent to
solving the system of equations

$\left\\{\begin{array}[]{c}up_{1}+v=a_{1}\\\ up_{2}+v=a_{2}\\\
up_{3}+v=a_{3}\end{array}\right.$

for $u\in\mathbf{C}\setminus\\{0\\}$, $v\in\mathbf{C}$ and
$a_{1},a_{2},a_{3}\in A$. This system is equivalent to the single equation

$(p_{3}-p_{2})a_{1}+(p_{1}-p_{3})a_{2}+(p_{2}-p_{1})a_{3}=0,$

to be solved for distinct $a_{1},a_{2},a_{3}\in A$.

Define $M_{1}$, $M_{2}$ and $M_{3}$ to be the matrices corresponding to
multiplication by $p_{3}-p_{2}$, $p_{1}-p_{3}$ and $p_{2}-p_{1}$ in the
$\mathbf{Z}$-basis $(\omega_{1},\omega_{2})$ of $\Lambda$. These matrices sum
to zero, are nonsingular as the $p_{i}$’s are distinct, and have integer
coefficients since $p_{i}\omega_{j}\in\Lambda$ for all $i,j$. We conclude by
Corollary 1.3. ∎

######  Remark 1.7.

It is believed that Example 1.4 can be extended significantly: a conjecture of
Graham states that, if $A\subset\mathbf{Z}^{2}$ is such that $\sum_{a\in
A\setminus\\{0\\}}{1}/{\left\|a\right\|^{2}}=+\infty$, then $A$ contains
infinitely many axes-parallel squares [5, Conjecture 8.4.6]. The difficulty of
Graham’s conjecture is comparable to that of Erdős’ conjecture on arithmetic
progressions of length $k=4$.

Overview of the paper. In Section 2, we will show how Corollary 1.3 follows
from Theorem 1.1. The rest of the paper will be devoted to the proof of
Theorem 1.1.

Our proof is an adaptation of the work of Bloom and Sisask on three-term
arithmetic progressions [4]. We will use the same notation as in their paper.
We will recall some of it in Section 3, where we also restate some classical
lemmas that will be used throughout the proof. The more technical definitions,
such as those of additively non-smoothing sets or of additive frameworks, can
be found in [4].

The structure of the proof of Theorem 1.1 is shown in Fig. 1. Section 4 is
dedicated to the proof of Proposition 4.4, a result which by itself is
sufficient to prove a weaker version of Theorem 1.1, with the bound
${|G|}/{(\log|G|)^{1-o(1)}}$ instead. The proof of Proposition 4.4 is similar
to that of Theorem 1.1, but is considerably simpler. It uses a density
increment lemma from [1].

In Section 5, we prove Theorem 1.1 by adapting the work of Bloom and Sisask
[4] to our more general setting. Fortunately, large portions of their paper
can be used as a black box, without any modification. This is especially the
case for [4, Sections 9 and 10] (structure theorem for additively non-
smoothing sets), as well as [4, Section 11] (spectral boosting). We will
mostly need to adapt some results from [4, Sections 5, 8 and 12].

Proposition 4.4Proposition 4.3Proposition 5.3Lemma 3.8Lemma 4.2Proposition
5.2[4, Lemma 8.5][4, Proposition 11.8]Lemma 5.1Theorem 1.1[1, Corollary 3.7]

Figure 1: Dependency graph for the proof of Theorem 1.1 (only the main lemmas
and propositions are shown).

Comparison with the Bloom-Sisask proof. We strongly recommend the readers to
familiarize themselves with the article of Bloom and Sisask before reading
Sections 3, 4 and 5 of this paper. We have attempted to make as few changes to
their proof as possible, to make the comparison easier for the reader.

The proof of Bloom and Sisask can be immediately generalised to equations as
in Theorem 1.1 for automorphisms that are multiples of the identity. If
$T=n\,\mathrm{Id}_{G}$ and $B$ is a Bohr set, then the dilate $B_{\rho}$ is a
subset of both $B$ and $T^{-1}B$, provided that $\rho\leq 1/n$ (see Section 3
for the relevant definitions). This very useful property no longer holds for
general automorphisms.

Instead of considering a simple dilate $B_{\rho}$, we will need to work with
the intersection $B\cap T^{-1}B$. The dilate of a Bohr set is another Bohr set
of the same rank. By contrast, $B\cap T^{-1}B$ is still a Bohr set, but the
rank may have doubled! Controlling the rank of these repeated intersections is
the main additional difficulty. To overcome it, we need to keep track more
explicitly of the frequency sets of all the Bohr sets in the proof.

Carefully tracking the dependence on the coefficients allows us to show that
the rank of the successive Bohr sets in the density increment iteration grows
polynomially. To obtain this, we also need to assume that the automorphisms
$T_{i}$ commute (see Remark 3.4 for more details). Since three-term equations
always reduce to the case of commuting automorphisms (see Remark 4.1), there
is no commutativity assumption in Theorem 1.1.

Theorem 1.1 gives a bound to subsets of $\mathbf{Z}/N\mathbf{Z}$ without
solutions to $ax+by+cz=0$ for integers $a,b,c$ comprime to $N$ with $a+b+c=0$.
It is important to note that this bound is _uniform_ in $a,b,c$. Such
uniformity would not have been obtained through a ‘naive’ modification of the
Bloom-Sisask proof using dilates as above.

######  Remark 1.8.

Theorem 1.1 can be generalized to equations with more than three terms. More
precisely, a slight adaptation of the proof shows the following. If
$T_{1},T_{2},\dots,T_{k}$ are commuting automorphisms of an abelian group $G$
such that $T_{1}+T_{2}+\dots+T_{k}=0$, then any set $A\subset G$ without non-
trivial solutions to $T_{1}a_{1}+T_{2}a_{2}+\dots T_{k}a_{k}=0$ satisfies

$|A|\ll\frac{|G|}{(\log|G|)^{1+c}}$

where $c>0$ is an absolute constant. Corollary 1.3 can also be modified in a
similar way. However, for $k\geq 4$, considerably better bounds are available
using other methods (see [9]), which is why we restrict ourselves to the case
$k=3$.

##  2 Application to Matrix Coefficients

In this section, we show how Corollary 1.3 follows from Theorem 1.1. The proof
is standard and involves two steps: first truncating the set $A$, then
embedding this truncation of $A$ inside a finite abelian group.

###### Proof of Corollary 1.3.

Let $A$ be a subset of $\mathbf{Z}^{d}$ containing only finitely many non-
trivial solutions to the equation

$M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0.$ (2)

We want to prove that

$\sum_{a\in A\setminus\\{0\\}}\frac{1}{\left\|a\right\|^{d}}<+\infty.$ (3)

After removing a finite number of elements from $A$, we can assume that $A$
has no non-trivial solution to Eq. 2.

For $T\geq 1$, let $A_{T}$ be the truncated set

$A_{T}:=A\cap[-T,T]^{d}.$

It is sufficient to prove that, for all $T\geq 3$,

$|A_{T}|\ll\frac{T^{d}}{(\log{T})^{1+c}},$ (4)

where $c>0$ is the constant from Theorem 1.1.444In this proof, the implied
constants in the asymptotic notation $\ll$ and $\asymp$ depend only on the
dimension $d$ and the matrices $M_{1},M_{2},M_{3}$. Indeed, we have

$\sum_{\begin{subarray}{c}a\in A\setminus\\{0\\}\\\
\left\|a\right\|_{\infty}\leq
M\end{subarray}}\frac{1}{\left\|a\right\|^{d}}\asymp\sum_{N=1}^{M}\frac{1}{N^{d}}\cdot\\#\\{a\in
A:\left\|a\right\|_{\infty}=N\\},$

and by partial summation, together with Eq. 4, we get

$\sum_{\begin{subarray}{c}a\in A\setminus\\{0\\}\\\
\left\|a\right\|_{\infty}\leq
M\end{subarray}}\frac{1}{\left\|a\right\|^{d}}\asymp\frac{|A_{M}|}{M^{d}}+\int_{1}^{M}\frac{|A_{T}|}{T^{d+1}}\,\mathrm{d}T\ll
1+\int_{2}^{M}\frac{1}{T(\log{T})^{1+c}}\,\mathrm{d}T\ll 1.$

Taking $M\to+\infty$ proves Eq. 3.

Let $T\geq 3$. Let

$C=\max\big{(}\left\|M_{1}\right\|_{\mathrm{op}},\left\|M_{2}\right\|_{\mathrm{op}},\left\|M_{3}\right\|_{\mathrm{op}},\lvert\det
M_{1}\rvert,\lvert\det M_{2}\rvert,\lvert\det M_{3}\rvert\big{)},$

where $\left\|M_{i}\right\|_{\mathrm{op}}$ is the operator norm of the matrix
$M_{i}$, viewed as a map
$(\mathbf{R}^{d},\left\|\cdot\right\|_{\infty})\to(\mathbf{R}^{d},\left\|\cdot\right\|_{\infty})$.
Let $p$ be a prime number between $4CT$ and $8CT$, which exists by Bertrand’s
postulate.

We embed $A_{T}$ in the abelian group $(\mathbf{Z}/p\mathbf{Z})^{d}$. Let
$\overline{A_{T}}$, $\overline{M_{1}},\overline{M_{2}}$ and $\overline{M_{3}}$
be the reductions of $A_{T},M_{1},M_{2}$ and $M_{3}$ modulo $p$. Clearly, each
$\overline{M_{i}}$ is invertible as its determinant is not divisible by $p$.

We claim that the map

$\\{(a_{1},a_{2},a_{3})\in(A_{T})^{3}:M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0\\}\to\\{(x_{1},x_{2},x_{3})\in(\overline{A_{T}})^{3}:\overline{M_{1}}x_{1}+\overline{M_{2}}x_{2}+\overline{M_{3}}x_{3}=0\\}$

given by reduction modulo $p$ is surjective. Indeed, if
$(a_{1},a_{2},a_{3})\in(A_{T})^{3}$ is such that

$M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}\equiv 0\pmod{p},$

then $M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0$ in $\mathbf{R}^{d}$ since we also
have

$\left\|M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}\right\|_{\infty}\leq 3CT<p.$

It follows that $\overline{A_{T}}$ only has trivial solutions to the equation
$\overline{M_{1}}x_{1}+\overline{M_{2}}x_{2}+\overline{M_{3}}x_{3}=0$. By
Theorem 1.1, we obtain

$|A_{T}|=|\overline{A_{T}}|\ll\frac{p^{d}}{(\log
p^{d})^{1+c}}\asymp\frac{T^{d}}{(\log T)^{1+c}},$

which proves Eq. 4 and concludes the proof of Corollary 1.3. ∎

##  3 Notation and New Density Increments

We use the same notation as in the paper of Bloom and Sisask [4]. We recall
some of it below, but we encourage the readers to familiarize themselves with
their article before reading the rest of this paper.

######  Notation 3.1.

Fix a finite abelian group $G$. If $A\subset B$, the _relative density_ of $A$
in $B$ is the ratio $|A|/|B|$. If $X\subset G$, the density of $X$ in $G$ is
denoted by $\mu(X):=|X|/|G|$. We write $\mu_{X}$ for the normalized indicator
function $\mu_{X}=\mu(X)^{-1}\mathbf{1}_{X}$.

For $f,g:G\to\mathbf{C}$, we use the normalizations

$\left\langle f,g\right\rangle:=\frac{1}{|G|}\sum_{x\in
G}f(x)\overline{g(x)}\quad\text{and}\quad f*g(x):=\frac{1}{|G|}\sum_{y\in
G}f(y){g(x-y)},$

while for $f,g:\widehat{G}\to\mathbf{C}$, we set

$\left\langle
f,g\right\rangle:=\sum_{\gamma\in\widehat{G}}f(\gamma)\overline{g(\gamma)}\quad\text{and}\quad
f*g(x):=\sum_{y\in\widehat{G}}f(y){g(x-y)}.$

In order to suppress logarithmic factors, we use the notation
$X\lesssim_{\alpha}Y$ or $X=\tilde{O}_{\alpha}(Y)$ to mean that $|X|\leq
C_{1}\log(2/\alpha)^{C_{2}}Y$ for some constants $C_{1},C_{2}>0$.

######  Notation 3.2 (Bohr sets).

For $\Gamma\subset\widehat{G}$ and $\nu:\Gamma\to[0,2]$, we define the _Bohr
set_ $B=\mathrm{Bohr}_{\nu}(\Gamma)$ to be the subset of $G$ defined by

$\mathrm{Bohr}_{\nu}(\Gamma)=\\{x\in G:|1-\gamma(x)|\leq\nu(\gamma)\text{ for
all }\gamma\in\Gamma\\}.$

The set $\Gamma$ is called the _frequency set_ of $B$ and $\nu$ its _width
function_. The _rank_ of $B$, denoted by $\mathrm{rk}(B)$, is defined to be
the size of $\Gamma$. Note that all Bohr sets are symmetric.

When we speak of a Bohr set, we implicitly refer to the triple
$(\mathrm{Bohr}_{\nu}(\Gamma),\Gamma,\nu)$, since the Bohr set
$\mathrm{Bohr}_{\nu}(\Gamma)$ alone does not uniquely determine the frequency
set nor the width.

The intersection of two Bohr sets is again a Bohr set. If
$B=\mathrm{Bohr}_{\nu}(\Gamma)$ and $\rho>0$, we denote by $B_{\rho}$ the
_dilate_ of $B$, i.e. the Bohr set given by
$B_{\rho}:=\mathrm{Bohr}_{\rho\nu}(\Gamma)$.

A Bohr set $B$ of rank $d$ is _regular_ if for all $|\kappa|\leq 1/(100d)$, we
have

$(1-100d|\kappa|)|B|\leq|B_{1+\kappa}|\leq(1+100d|\kappa|)|B|.$

An important property is that, for every Bohr set $B$, there is a dilate
$B_{\rho}$, for some $\rho\in[1/2,1]$, which is regular (see [4, Lemma 4.3]).

If $B$ is a Bohr set and $T$ is an automorphism, then $TB$ is a Bohr set and
$(TB)_{\rho}=TB_{\rho}$. If $B$ is regular, then so too is $TB$.

The sizes of Bohr sets can be controlled using the classical lemma [4, Lemma
4.4]. We restate it below as it will be used extensively throughout the
article.

######  Lemma 3.3.

Let $\Gamma\subset\widehat{G}$ and $\nu,\nu^{\prime}:\Gamma\to[0,2]$ be such
that $\nu^{\prime}(\gamma)\leq\nu(\gamma)$ for $\gamma\in\Gamma$. We have

$|\mathrm{Bohr}_{\nu^{\prime}}(\Gamma)|\geq\Bigg{(}\prod_{\gamma\in\Gamma}\frac{\nu^{\prime}(\gamma)}{4\nu(\gamma)}\Bigg{)}|\mathrm{Bohr}_{\nu}(\Gamma)|.$

In particular, if $\rho\in(0,1)$ and $B$ is a Bohr set of rank $d$, then
$|B_{\rho}|\geq(\rho/4)^{d}|B|$.

######  Remark 3.4.

One of the main difficulties that arise when working with general
automorphisms $T_{i}$ is that we often have to control intersections of Bohr
sets such as $B^{\prime}=T_{1}B\cap T_{2}B\cap T_{3}B$. If the Bohr set $B$
has frequency set $\Gamma=\\{\gamma_{i}\mid i\in I\\}$, then $B^{\prime}$ can
be viewed as a Bohr set with frequency set $\Gamma^{\prime}=\\{\gamma_{i}\circ
T_{j}^{-1}\mid i\in I,j\in\\{1,2,3\\}\\}$. If we don’t know anything about the
frequency set of $B$, then the best we can say about the rank of $B^{\prime}$
is that

$\mathrm{rk}(B^{\prime})\leq 3\,\mathrm{rk}(B).$

Suppose that $B_{0}$ is a Bohr set of rank $d$ and define, for $n\geq 0$,
$B_{n+1}=T_{1}B_{n}\cap T_{2}B_{n}\cap T_{3}B_{n}$. Using the above bound
would give an exponential growth for the ranks of these Bohr sets. Such a
naive bound would be completely insufficient to prove Theorem 1.1. However, we
can note that

$B_{n}=\bigcap_{T\in W_{n}}TB_{0},$

where $W_{n}$ is the set of all compositions of $n$ automorphisms from the set
$\\{T_{1},T_{2},T_{3}\\}$. If we know $T_{1}$, $T_{2}$ and $T_{3}$ commute,
then $|W_{n}|$ has polynomial growth and we can obtain an acceptable bound for
the rank of $B_{n}$, namely

$\mathrm{rk}(B_{n})\ll n^{2}\,\mathrm{rk}(B_{0}).$

In the density increment argument, we will need to be more explicit with the
definitions of the Bohr sets, in order to carefully keep track of their ranks
and frequency sets.

In the light of Remark 3.4, to make the proof of Bloom and Sisask work for
general coefficients, we need to change the definition of density increments
([4, Definition 5.1]).

######  Definition 3.5 (Increments).

Let $B$ be a regular Bohr set, and let $B^{\prime}\subset B$ be a regular Bohr
set of rank $d$. We say that $A\subset B$ of relative density $\alpha$ has a
_density increment_ of strength $[\delta,d^{\prime};C]$ relative to
$B^{\prime}$ if there is a regular Bohr set $B^{\prime\prime}$ of the form

$B^{\prime\prime}=B_{\rho}^{\prime}\cap\widetilde{B}$

such that

$\left\|\mathbf{1}_{A}*\mu_{B^{\prime\prime}}\right\|_{\infty}\geq(1+C^{-1}\delta)\alpha,$

where $\widetilde{B}=\mathrm{Bohr}(\widetilde{\Gamma},\widetilde{\nu})$ is a
Bohr set of rank $|\widetilde{\Gamma}|\leq Cd^{\prime}$, $\rho\in(0,1]$, and
$\rho,\widetilde{\nu}$ satisfy the inequality

$\left(\frac{\rho}{4}\right)^{d}\prod_{\gamma\in\widetilde{\Gamma}}\frac{\widetilde{\nu}(\gamma)}{8}\geq(2d(d^{\prime}+1))^{-C(d+d^{\prime})}.$
(5)

######  Remark 3.6.

If $A\subset B$ has a density increment of strength $[\delta,d^{\prime};C]$
with respect to $B^{\prime}$ in the sense of Definition 3.5, then $A$ has a
density increment of the same strength with respect to $B^{\prime}$ in the
sense of [4, Definition 5.1]. This is because Eq. 5 implies the bound

$|B^{\prime\prime}|\geq(2d(d^{\prime}+1))^{-C(d+d^{\prime})}|B^{\prime}|,$ (6)

by a direct application of Lemma 3.3.

The converse is not true in general, but it is true for all the density
increments present in [4]. That is, every density increment in [4] is also a
density increment in the sense of Definition 3.5, of the same strength. The
reason is that

  1. 1.

the Bohr set $B^{\prime\prime}$ in [4, Definition 5.1] is always chosen to be
of the form $B^{\prime\prime}=B_{\rho}^{\prime}\cap\widetilde{B}$ in [4],

  2. 2.

and every time the authors show that some set $A\subset B$ has a density
increment, they need to prove Eq. 6. To do this, the only tool they use is
Lemma 3.3, and thus they prove the stronger Eq. 5.

We restate here [4, Lemma 5.2], which is an easy consequence of the definition
of density increment.

######  Lemma 3.7.

Let $B$ be a regular Bohr set and $B^{\prime}\subset B$ be a regular Bohr set
of rank $d$. Let $\rho\in(0,1)$. If $A\subset B$ has a density increment of
strength $[\delta,d^{\prime};C]$ relative to $B^{\prime}_{\rho/d}$, then $A$
has a density increment of strength
$[\delta,d^{\prime};C+\tilde{O}_{\rho}(1)]$ relative to $B^{\prime}$.

Finally, we reproduce the statement of [4, Lemma 12.1] for three smaller Bohr
sets instead of two. The proof of Lemma 3.8 is the same as that of [4, Lemma
12.1], so we shall not repeat it here.

######  Lemma 3.8.

There is a constant $c>0$ such that the following holds. Let $\mathcal{B}$ be
a regular Bohr set of rank $d$, let $\mathcal{A}\subset\mathcal{B}$ have
relative density $\alpha$, let $\varepsilon>0$ and suppose that
${B_{1},B_{2},B_{3}}\subset\mathcal{B}_{\rho}$ where $\rho\leq
c\alpha\varepsilon/d$. Then either

  1. 1.

($\mathcal{A}$ has almost full density on $B_{1}$, $B_{2}$ and $B_{3}$) there
is an $x\in\mathcal{B}$ such that

$\mathbf{1}_{\mathcal{A}}*\mu_{B_{i}}(x)\geq(1-\varepsilon)\alpha$

for $i=1,2,3$, or

  2. 2.

(density increment) $\mathcal{A}$ has an increment of strength
$[\varepsilon,0;O(1)]$ relative to one of the $B_{i}$’s.

##  4 Proof of a Weaker Bound

In this section, we prove Proposition 4.4, which can be regarded as a weaker
version of Theorem 1.1. On its own, Proposition 4.4 is sufficient to prove the
bound

$|A|\ll\frac{|G|}{(\log|G|)^{1-o(1)}},$

keeping the notation of Theorem 1.1. We will use Proposition 4.4 at the end of
the proof of Theorem 1.1 when, after a series of density increments, we arrive
at a subset $A^{\prime}$ of a Bohr set $B^{\prime}$ whose relative density is
substantially larger than the original density $|A|/|G|$.

######  Remark 4.1.

It suffices to prove Theorem 1.1 when the first automorphism $T_{1}$ is the
identity, something which we will assume from this point onward. To deduce the
case of a general automorphism $T_{1}$, simply apply Theorem 1.1 to the set
$T_{1}A$ and the automorphisms
$\mathrm{Id_{G}},T_{2}T_{1}^{-1},T_{3}T_{1}^{-1}$.

From now on, fix two automorphisms $T_{2}$, $T_{3}$ of $G$ such that
$\mathrm{Id}_{G}+T_{2}+T_{3}=0$. We count the number of solutions to the
equation $a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$ via the inner product

$T(A_{1},A_{2},A_{3}):=\left\langle\mathbf{1}_{A_{1}}*\mathbf{1}_{T_{2}A_{2}},\mathbf{1}_{-T_{3}A_{3}}\right\rangle,$

defined for $A_{1},A_{2},A_{3}\subset G$. Observe that

$T(A,A,A)=\frac{1}{|G|^{2}}\cdot\\#\\{(a_{1},a_{2},a_{3})\in
A^{3}:a_{1}+T_{2}a_{2}+T_{3}a_{3}=0\\}.$

We will obtain Proposition 4.4 by repeated applications of the following
lemma, which is a restatement of [1, Corollary 3.7] in the language of regular
Bohr sets.

######  Lemma 4.2.

There is a constant $c\in(0,\tfrac{1}{2})$ such that the following holds. Let
$0<\alpha\leq 1$. Let $B$ be a regular Bohr set of rank $d$ and $B^{\prime}$ a
regular Bohr set of rank $\leq 3d$ such that $B^{\prime}\subset B_{\rho}$,
where $\rho=c\alpha/d$. Suppose that $A_{1}\subset B$, $A_{2}\subset
T_{2}^{-1}B^{\prime}$ and $A_{3}\subset T_{3}^{-1}B$, each time with relative
density at least $\alpha$. Then

  1. 1.

either

$T(A_{1},A_{2},A_{3})\gg\alpha^{3}\mu(B)\mu(B^{\prime})$

  2. 2.

or there is a regular Bohr set $B^{\prime\prime}$ such that
$\left\|1_{A}*\mu_{B^{\prime\prime}}\right\|_{\infty}\geq(1+c)\alpha$, where

     * •

$A$ is either $A_{1}$ or $-T_{3}A_{3}$,

     * •

and $B^{\prime\prime}$ is of the form

$B^{\prime\prime}=B^{\prime}_{\eta}\cap\widetilde{B}$

for some $\eta\asymp\exp(-\tilde{O}_{\alpha}(1))/d$ and some
$\widetilde{B}=\mathrm{Bohr}(\widetilde{\Gamma},\widetilde{\nu})$ with
$|\widetilde{\Gamma}|\leq d^{\prime}/4$ and $\widetilde{\nu}\geq 1/d^{\prime}$
on $\widetilde{\Gamma}$, where $d^{\prime}=\tilde{O}_{\alpha}(\alpha^{-1})$.

###### Proof.

This follows directly from [1, Corollary 3.7], applied to the sets $A_{1}$,
$T_{2}A_{2}$ and $-T_{3}A_{3}$. Note that, since $B$ is a regular Bohr set,

$|(B+B^{\prime})\setminus B|\leq|B_{1+\rho}\setminus B|\leq(1+O(d\rho))|B|,$

so that $B^{\prime}$ is $(2^{-270}\alpha)$-sheltered by $B$, provided that $c$
is sufficiently small (see [1] for the definition of ‘sheltered’ in this
context). We see in a similar way that $B^{\prime\prime}$ has the required
amount of shelter. Finally, if $x\in B^{\prime\prime}$ and
$\gamma^{\prime}\in\langle\widetilde{\Gamma}\rangle$, say
$\gamma^{\prime}=\sum_{\gamma\in\Gamma}\varepsilon_{\gamma}\gamma$ with
$\varepsilon_{\gamma}\in\\{-1,0,1\\}$, we have

$\lvert 1-\gamma^{\prime}(x)\rvert\leq\sum_{\gamma\in\widetilde{\Gamma}}\lvert
1-\gamma(x)^{\varepsilon_{\gamma}}\rvert\leq\tfrac{1}{4}$

as required. ∎

######  Proposition 4.3.

Let $\mathcal{B}$ be a regular Bohr set of rank $d$, and let
$\mathcal{A}\subset\mathcal{B}$ of relative density $\alpha$. Let
$B^{\star}={\mathcal{B}\cap T_{2}\mathcal{B}\cap T_{3}\mathcal{B}}$. Then,
either

$T(\mathcal{A},\mathcal{A},\mathcal{A})\gg\exp\left(-\tilde{O}_{\alpha}(d\log(2d)\right)\mu(B^{\star})^{2}$

or $\mathcal{A}$ has a density increment of strength

$[1,\alpha^{-1};\tilde{O}_{\alpha}(1)]$

with respect to either $B^{\star}$, $T_{2}^{-1}B^{\star}$ or
$T_{3}^{-1}B^{\star}$.

###### Proof.

Let $\varepsilon=c/2$, where $c$ is the constant of Lemma 4.2. We apply Lemma
3.8 with $B_{1}=(B^{\star})_{\rho}$,
$B_{2}=T_{2}^{-1}(B^{\star})_{\rho\rho^{\prime}}$,
$B_{3}=T_{3}^{-1}(B^{\star})_{\rho}$, where $\rho=c_{1}\alpha\varepsilon/d$
and $\rho^{\prime}=c_{2}\alpha/d$, with $c_{1},c_{2}>0$ being two small
constants, chosen in particular such that $B_{1}$, $B_{2}$ and $B_{3}$ are
regular.

If the second case of Lemma 3.8 holds, then $\mathcal{A}$ has a density
increment of strength $[1,0;O(1)]$ relative to one of the $B_{i}$’s. By Lemma
3.7, this implies that $\mathcal{A}$ has a density increment of strength
$[1,0;\tilde{O}_{\alpha}(1)]$ relative to $B^{\star}$, $T_{2}^{-1}B^{\star}$
or $T_{3}^{-1}B^{\star}$.

We may thus suppose that the first case of Lemma 3.8 holds. That is, there is
some $x\in G$ such that, if we let

$A_{1}=(\mathcal{A}-x)\cap B_{1},\quad A_{2}=(\mathcal{A}-x)\cap
B_{2}\quad\text{and}\quad A_{3}=(\mathcal{A}-x)\cap B_{3},$

then each $A_{i}$ has relative density at least $(1-\varepsilon)\alpha$ in the
corresponding $B_{i}$. We now use Lemma 4.2 with $B=(B^{\star})_{\rho}$ and
$B^{\prime}=(B^{\star})_{\rho\rho^{\prime}}$.

  1. 1.

In the first case, we get

$T(A_{1},A_{2},A_{3})\gg(1-\varepsilon)^{3}\alpha^{3}\mu(B)\mu(B^{\prime})\gg\exp\left(-\tilde{O}_{\alpha}(d\log(2d)\right)\mu(B^{\star})^{2},$

where the last inequality follows from Lemma 3.3. Since the $A_{i}$’s are
subsets of the same translate of $\mathcal{A}$ and the equation
$a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$ is translation-invariant, we have

${T(\mathcal{A},\mathcal{A},\mathcal{A})\geq T(A_{1},A_{2},A_{3})},$

which gives the claimed bound.

  2. 2.

In the second case, there is a regular Bohr set $B^{\prime\prime}$ as in the
statement of the lemma such that

$\left\|1_{A}*\mu_{B^{\prime\prime}}\right\|_{\infty}\geq(1+c)(1-\varepsilon)\alpha\geq(1+c/4)\alpha,$
(7)

where $A$ is either $A_{1}$ or $-T_{3}A_{3}$. We therefore deduce that
$\mathcal{A}$ has a density increment of strength
$[1,\alpha^{-1};\tilde{O}_{\alpha}(1)]$ relative to
$(B^{\star})_{\rho\rho^{\prime}}$ or
$T_{3}^{-1}(B^{\star})_{\rho\rho^{\prime}}$. By Lemma 3.7, this means that
$\mathcal{A}$ has a density increment of the same strength relative to
$B^{\star}$ or $T_{3}^{-1}B^{\star}$.∎

We now iteratively apply Proposition 4.3 to obtain Proposition 4.4, which
plays the same role as [4, Theorem 5.4] in the proof of Bloom and Sisask.

######  Proposition 4.4.

Let $B=\mathrm{Bohr}(\Gamma,\nu)$ be a regular Bohr set of rank $d$ and
suppose that $A\subset B$ has density $\alpha$. Then

$T(A,A,A)\geq\exp\left(-\tilde{O}_{\alpha}(d+\alpha^{-1})\log{2d}\right)\Bigg{(}\prod_{\gamma\in\Gamma}\frac{\nu(\gamma)}{8}\Bigg{)}^{\tilde{O}_{\alpha}(1)}.$

###### Proof.

Let $C=\tilde{O}_{\alpha}(1)$ be the constant in the density increment case of
Proposition 4.3 ($C$ is fixed as $\alpha$ is given). Recall that
$\tilde{O}_{\alpha^{\prime}}(1)$ is short for
$C_{1}\log(2/\alpha^{\prime})^{C_{2}}$, which is a decreasing function of
$\alpha^{\prime}$. Thus, if we use Proposition 4.3 with some pair
$\mathcal{A}^{\prime}\subset\mathcal{B}^{\prime}$ having relative density
$\alpha^{\prime}\geq\alpha$ and the second case applies, we will have a
density increment of strength $[1,(\alpha^{\prime})^{-1};C]$.

We inductively construct two sequences $(A_{n})$ and $(B_{n})$, where, for
each $n$, $A_{n}$ is a subset of $B_{n}$ with relative density $\alpha_{n}$.
Let $A_{0}=A$ and $B_{0}=B$. Assume that $A_{i}$ and $B_{i}$ have been
constructed for $i<n$. We use Proposition 4.3 with $\mathcal{A}=A_{n-1}$ and
$\mathcal{B}=B_{n-1}$. If the first case of the proposition holds, we stop the
construction. Otherwise, we are in the density increment case and there are
sets $A_{n}\subset B_{n}$ such that

  * •

$A_{n}$ is a subset of a translate of $A_{n-1}$;

  * •

$A_{n}$ is a subset of $B_{n}$ of relative density
$\alpha_{n}\geq(1+C^{-1})\alpha_{n-1}$;

  * •

$B_{n}$ is a regular Bohr set of the form

$B_{n}=(S_{n}B_{n-1}^{\star})_{\rho_{n}}\cap\widetilde{B}_{n},$ (8)

where

    * –

$B_{n-1}^{\star}$ is the Bohr set

$B_{n-1}^{\star}:=B_{n-1}\cap T_{2}B_{n-1}\cap T_{3}B_{n-1},$ (9)

whose rank we denote by $d_{n-1}^{\star}$,

    * –

$S_{n}$ is either $\mathrm{Id}_{G}$, $T_{2}^{-1}$ or $T_{3}^{-1}$,

    * –

$\widetilde{B}_{n}=\mathrm{Bohr}(\widetilde{\Gamma}_{n},\widetilde{\nu}_{n})$,
where $|\widetilde{\Gamma}_{n}|\leq C\alpha^{-1}$ and

$\bigg{(}\frac{\rho_{n}}{4}\bigg{)}^{d_{n-1}^{\star}}\prod_{\gamma\in\widetilde{\Gamma}_{n}}\frac{\widetilde{\nu}_{n}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(d_{n-1}^{\star}+\alpha^{-1}\right)\log\left(d_{n-1}^{\star}\right)\right).$
(10)

Note that, since $1\geq\alpha_{n}\geq(1+C^{-1})^{n}\alpha$, this construction
must terminate in $l=\tilde{O}_{\alpha}(1)$ steps. We then arrive at
$A_{l}\subset B_{l}$, for which

$T(A,A,A)\geq
T(A_{l},A_{l},A_{l})\gg\exp\left(-\tilde{O}_{\alpha}\left(d_{l}\log(2d_{l})\right)\right)\mu(B_{l}^{\star})^{2},$
(11)

where $d_{l}$ is the rank of $B_{l}$ and, as usual, $B_{l}^{\star}=B_{l}\cap
T_{2}B_{l}\cap T_{3}B_{l}$.

Let $W_{i}$ be the set of all automorphisms obtained by composing $i$ elements
of $\\{\mathrm{Id}_{G},T_{2},T_{3},T_{2}^{-1},T_{3}^{-1}\\}$. Since
$T_{3}=-\mathrm{Id}_{G}-T_{2}$, these automorphisms commute, which implies
that $|W_{i}|\leq(2i+1)^{2}$.

An immediate induction using Eqs. 8 and 9 shows that

$B_{i}^{\star}\supset\Bigg{(}\bigcap_{T\in
W_{2i+1}}(TB)_{\rho_{1}\cdots\rho_{i}}\Bigg{)}\cap\Bigg{(}\bigcap_{j=1}^{i}\bigcap_{T\in
W_{2i+1}}(T\widetilde{B}_{j})_{\rho_{j+1}\cdots\rho_{i}}\Bigg{)}$ (12)

for $0\leq i\leq l$. The same reasoning shows that the frequency set of
$B_{i}$ is contained in

$\Bigg{(}\bigcup_{T\in
W_{2i}}T\Gamma\Bigg{)}\cup\Bigg{(}\bigcup_{j=1}^{i}\bigcup_{T\in
W_{2i}}T\widetilde{\Gamma}_{j}\Bigg{)}.$

This shows that

$d_{l}\ll l^{2}d+l^{3}C\alpha^{-1}=\tilde{O}_{\alpha}(d+\alpha^{-1}).$

In particular, Eq. 10 becomes

$\bigg{(}\frac{\rho_{n}}{4}\bigg{)}^{d_{n-1}^{\star}}\prod_{\gamma\in\widetilde{\Gamma}_{n}}\frac{\widetilde{\nu}_{n}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(d+\alpha^{-1}\right)\log(2d)\right),$
(13)

for $1\leq n\leq l$.

We now use Lemma 3.3 to give a lower bound for $\mu(B_{l}^{\star})$. By Eq.
12, we have

$\mu(B_{l}^{\star})\geq\left(\Bigg{(}\frac{1}{4}\prod_{j=1}^{l}\rho_{j}\Bigg{)}^{d}\prod_{\gamma\in\Gamma}\frac{\nu(\gamma)}{8}\right)^{|W_{2l+1}|}\cdot\prod_{j=1}^{l}\left(\Bigg{(}\frac{1}{4}\prod_{n=j+1}^{l}\rho_{n}\Bigg{)}^{\mathrm{rk}(\widetilde{B}_{j})}\prod_{\gamma\in\widetilde{\Gamma}_{j}}\frac{\widetilde{\nu}_{j}(\gamma)}{8}\right)^{|W_{2l+1}|}.$

Using Eq. 13 and the simple inequalities $d\leq d_{j-1}^{\star}$ and
$\mathrm{rk}(\widetilde{B}_{j})\leq d_{n-1}^{\star}$ for $j\geq 1$ and
$j+1\leq n\leq l$, this yields

$\mu(B_{l}^{\star})\geq\exp\left(-\tilde{O}_{\alpha}\left(d+\alpha^{-1}\right)\log(2d)\right)\Bigg{(}\prod_{\gamma\in\Gamma}\frac{\nu(\gamma)}{8}\Bigg{)}^{\tilde{O}_{\alpha}(1)}.$

Together with Eq. 11, this concludes the proof of Proposition 4.4. ∎

##  5 Proof of the Main Theorem

This section is dedicated to the proof of Theorem 1.1. Each statement in this
section is an adaptation of a corresponding statement in [4]. To help the
reader, we will highlight the changes made to the original statements of [4]
in blue.

We start by proving an analogue of [4, Lemma 8.2] in our setting. When
$A\subset B$, the notation $\mu_{A/B}$ stands for the _balanced function_
$\mu_{A/B}:=\mu_{A}-\mu_{B}$.

######  Lemma 5.1.

There is a constant $c>0$ such that the following holds. Let $0<\alpha\leq 1$.
Let $B$ be a regular Bohr set of rank $d$, and $B^{\prime}$ another regular
Bohr set such that $T_{3}B^{\prime}\subset B_{\rho}$, with $\rho\leq
c\alpha/d$. Let $A_{1}\subset B$, $A_{2}\subset T_{2}^{-1}B$ and $A_{3}\subset
B^{\prime}$, each time with relative density in $[\alpha/2,2\alpha]$. Then
either

  1. 1.

(many solutions)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T(A_{1},A_{2},A_{3})}\geq\tfrac{1}{16}\alpha^{3}\mu(B)\mu(B^{\prime})$,
or

  2. 2.

(large $L^{2}$ mass on a spectrum) there is some $\eta\gg\alpha$ such that

$\sum_{\gamma\in\Delta_{\eta}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-T_{3}A_{3}})}|\widehat{\mu_{A/B}}(\gamma)|^{2}\gtrsim_{\alpha}\eta^{-1}\mu(B)^{-1},$

where $A$ is either $A_{1}$ or $T_{2}A_{2}$.

###### Proof.

We have

$T(A_{1},A_{2},A_{3})=\left\langle\mathbf{1}_{A_{1}}*\mathbf{1}_{T_{2}A_{2}},\mathbf{1}_{-T_{3}A_{3}}\right\rangle\geq\tfrac{1}{8}\alpha^{3}\mu(B)^{2}\mu(B^{\prime})\left\langle\mu_{A_{1}}*\mu_{T_{2}A_{2}},\mu_{-T_{3}A_{3}}\right\rangle.$

Replacing $\mu_{A_{1}}$ and $\mu_{T_{2}A_{2}}$ with their balanced functions
$\mu_{A_{1}/B}$ and $\mu_{T_{2}A_{2}/B}$, we have

$\displaystyle\left\langle\mu_{A_{1}}*\mu_{T_{2}A_{2}},\mu_{-T_{3}A_{3}}\right\rangle$
$\displaystyle=\left\langle\mu_{A_{1}/B}*\mu_{T_{2}A_{2}/B},\mu_{-T_{3}A_{3}}\right\rangle+E,$

where

$E=\left\langle\mu_{A_{1}}*\mu_{B},\mu_{-T_{3}A_{3}}\right\rangle+\left\langle\mu_{B}*\mu_{T_{2}A_{2}},\mu_{-T_{3}A_{3}}\right\rangle-\left\langle\mu_{B}*\mu_{B},\mu_{-T_{3}A_{3}}\right\rangle.$

We can estimate $E$ using regularity. Since $-T_{3}A_{3}\subset B_{\rho}$, we
have $\left\|\mu_{-T_{3}A_{3}}*\mu_{B}-\mu_{B}\right\|_{1}=O(\rho d)$ by [4,
Lemma 4.5]. Moreover,
$\left\|\mu_{A_{1}}\right\|_{\infty},\left\|\mu_{T_{2}A_{2}}\right\|_{\infty},\left\|\mu_{B}\right\|_{\infty}\leq
2\alpha^{-1}\mu(B)^{-1}$. Therefore

$\displaystyle E$
$\displaystyle=\left\langle\mu_{A_{1}},\mu_{-T_{3}A_{3}}*\mu_{B}\right\rangle+\left\langle\mu_{T_{2}A_{2}},\mu_{-T_{3}A_{3}}*\mu_{B}\right\rangle-\left\langle\mu_{B},\mu_{-T_{3}A_{3}}*\mu_{B}\right\rangle$
$\displaystyle=\left\langle\mu_{A_{1}},\mu_{B}\right\rangle+\left\langle\mu_{T_{2}A_{2}},\mu_{B}\right\rangle-\left\langle\mu_{B},\mu_{B}\right\rangle+O(\rho
d\alpha^{-1}\mu(B)^{-1})$
$\displaystyle=\mu(B)^{-1}+\mu(B)^{-1}-\mu(B)^{-1}+O\big{(}\rho
d\alpha^{-1}\mu(B)^{-1}\big{)}$ $\displaystyle=\mu(B)^{-1}+O\left(\rho
d\alpha^{-1}\mu(B)^{-1}\right).$

In particular, $E\geq\frac{3}{4}\mu(B)^{-1}$, provided $\rho$ is small enough.
Thus

$T(A_{1},A_{2},A_{3})\geq\tfrac{1}{8}\alpha^{3}\mu(B)^{2}\mu(B^{\prime})\Big{(}\left\langle\mu_{A_{1}/B}*\mu_{T_{2}A_{2}/B},\mu_{-T_{3}A_{3}}\right\rangle+\tfrac{3}{4}\mu(B)^{-1}\Big{)}$

If the first case of the conclusion doesn’t hold, then

$\left\langle\mu_{A_{1}/B}*\mu_{T_{2}A_{2}/B},\mu_{-T_{3}A_{3}}\right\rangle\leq-\tfrac{1}{4}\mu(B)^{-1}.$

By Parseval’s identity, followed by the triangle inequality, we deduce that

$\left\langle|\widehat{\mu_{A_{1}/B}}||\widehat{\mu_{T_{2}A_{2}/B}}|,|\widehat{\mu_{-T_{3}A_{3}}}|\right\rangle\geq\tfrac{1}{4}\mu(B)^{-1}.$

Using $xy\leq\tfrac{1}{2}(x^{2}+y^{2})$, we find that

$\sum_{\gamma\in\widehat{G}}|\widehat{\mu_{A/B}}(\gamma)|^{2}|\widehat{\mu_{-T_{3}A_{3}}}(\gamma)|=\left\langle|\widehat{\mu_{A/B}}|^{2},|\widehat{\mu_{-T_{3}A_{3}}}|\right\rangle\geq\tfrac{1}{4}\mu(B)^{-1},$

where $A$ is either $A_{1}$ or $T_{2}A_{2}$. Since
$\left\|\mu_{A/B}\right\|_{2}^{2}\leq\alpha^{-1}\mu(B)^{-1}$, we can discard
the terms of the above sum with
$|\widehat{\mu_{-T_{3}A_{3}}}|\leq\tfrac{1}{8}\alpha$ to obtain

$\sum_{\gamma\in\Delta_{\alpha/8}(-T_{3}A_{3})}|\widehat{\mu_{A/B}}(\gamma)|^{2}|\widehat{\mu_{-T_{3}A_{3}}}(\gamma)|\geq\tfrac{1}{8}\mu(B)^{-1}.$

By the dyadic pigeonhole principle, we conclude that there is some
$1\geq\eta\gg\alpha$ such that

$\sum_{\gamma\in\Delta_{\eta}(-T_{3}A_{3})\setminus\Delta_{2\eta}(-T_{3}A_{3})}|\widehat{\mu_{A/B}}(\gamma)|^{2}|\widehat{\mu_{-T_{3}A_{3}}}(\gamma)|\gtrsim_{\alpha}\mu(B)^{-1}.$

This concludes the proof since
$|\widehat{\mu_{-T_{3}A_{3}}}(\gamma)|\asymp\eta$ on the set
$\Delta_{\eta}(-T_{3}A_{3})\setminus\Delta_{2\eta}(-T_{3}A_{3})$. ∎

Next, we modify the statement of [4, Proposition 8.1] as follows.

######  Proposition 5.2.

There is a constant $c>0$ such that the following holds. Let $k,h,t\geq 20$ be
some parameters.

Let $0<\alpha\leq 1$. Let $B$ be a regular Bohr set of rank $d$, and
$B^{\prime}$ another regular Bohr set, of rank at most $3d$, such that
$B^{\prime}\subset T_{3}^{-1}B_{\rho}$, where $\rho\leq c\alpha^{2}/d$. Let
$A_{1}\subset B$, $A_{2}\subset T_{2}^{-1}B$ and $A_{3}\subset B^{\prime}$,
each time with relative density in $[\alpha/2,2\alpha]$. Then for either
$A=A_{1}$ or $A=T_{2}A_{2}$, one of the following holds

  1. 1.

(large density) $\alpha\gg 1/k^{2}$, or

  2. 2.

(many solutions)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T(A_{1},A_{2},A_{3})}\gg\alpha^{3}\mu(B)\mu(B^{\prime})$,
or

  3. 3.

$A$ has a density increment of strength either

    1. (a)

(small increment) $[1,\alpha^{-1/k};\tilde{O}_{\alpha}(h\log t)]$ or

    2. (b)

(large increment) $[\alpha^{-1/k},\alpha^{-1+1/k};\tilde{O}_{\alpha}(h\log
t)]$

relative to
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T_{3}B^{\prime}}$,
or

  4. 4.

(non-smoothing large spectrum) there is a set $\Delta$ and three quantities
${\rho_{\mathrm{top}},\rho_{\mathrm{bottom}},\rho^{\prime}\in(0,1)}$
satisfying

$\rho_{\mathrm{top}}\gg\alpha^{O(1)}(c/dt)^{O(h)},\quad\rho_{\mathrm{bottom}}\gg(\alpha/d)^{O(1)},\quad\text{and}\quad\rho^{\prime}\gg(\alpha/d)^{O(1)},$

such that

    1. (a)

$\alpha^{-3+O(1/k)}\ll|\Delta|\lesssim_{\alpha}\alpha^{-3}$,

    2. (b)

there exists an additive framework $\tilde{\Gamma}$ of height $h$ and
tolerance $t$ between

$\Gamma_{\mathrm{top}}:=\Delta_{1/2}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T_{3}B_{\rho_{\mathrm{top}}}^{\prime}})\quad\text{and}\quad\Gamma_{\mathrm{bottom}}:=\Delta_{1/2}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T_{3}B_{\rho_{\mathrm{bottom}}}^{\prime}}),$

    3. (c)

$\Delta$ is $\tfrac{1}{4}$-robustly $(\tau,k^{\prime})$-additively non-
smoothing relative to $\tilde{\Gamma}$ for some
$\alpha^{2-O(1/k)}\gg\tau\gg\alpha^{2+O(1/k)}$ and $k\geq k^{\prime}\gg k$,
and

    4. (d)

if we let
$B^{\prime\prime}=({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}T_{3}B_{\rho_{\mathrm{top}}}^{\prime}})_{\rho^{\prime}}$
then for all $\gamma\in\Delta+\Gamma_{\mathrm{top}}$

$|\widehat{\mu_{A/B}}|^{2}\circ|\widehat{\mu_{B^{\prime\prime}}}|^{2}(\gamma)\gg\alpha^{2+O(1/k)}\mu(B)^{-1},$

and

    5. (e)

$\Big{\|}\mathbf{1}_{\Delta}\circ|\widehat{\mu_{B^{\prime\prime}}}|^{2}\Big{\|}_{\infty}\leq
2.$

Few changes have to be made to the proof of [4, Proposition 8.1], so we only
give an overview of the modified proof.

###### Proof sketch.

We keep the notation $B^{(0)}:=B^{\prime}$ and $B^{(i+1)}=B^{(i)}_{\rho_{i}}$
for some $\rho_{i}$ that are the same as those in the original proof.

By Lemma 5.1, either we are in the second case or there is some
$\eta\gg\alpha$ such that

$\sum_{\gamma\in\Delta_{\eta}(-T_{3}A_{3})}|\widehat{\mu_{A/B}}(\gamma)|^{2}\gtrsim_{\alpha}\eta^{-1}\mu(B)^{-1},$

where $A$ is either $A_{1}$ or $T_{2}A_{2}$.

Suppose first that this is true for some $\eta\geq\tfrac{1}{2}K^{-1}$. In this
case we apply [4, Corollary 7.11] with $T_{3}B^{\prime}$ instead of $B$, with
$T_{3}B^{(1)}$ in place of $B^{\prime}$, the function $f$ chosen to be
$\mathbf{1}_{-T_{3}A_{3}}$ and and the weight function $\omega$ given by
$\omega=|\widehat{\mu_{A/B}}|^{2}$, restricted to
$\Delta_{\eta}(-T_{3}A_{3})$. We apply [4, Lemma 7.8] and [4, Lemma 5.7] in
the same way as in the original proof, except that we obtain a small density
increment for $A$ relative to $T_{3}B^{\prime}$ instead of $B^{\prime}$.

The case $\tfrac{1}{2}K^{-1}\geq\eta\geq K^{2}\alpha$ is similar. After using
[4, Corollary 7.12], [4, Lemma 7.8] and [4, Lemma 5.7], we conclude that $A$
has a large increment relative to $T_{3}B^{\prime}$.

Finally, in the case $\alpha\ll\eta\ll K^{2}\alpha$, we have

$\sum_{\gamma\in\tilde{\Delta}}|\widehat{\mu_{A/B}}(\gamma)|^{2}\gtrsim_{\alpha}K^{2}\alpha^{-1}\mu(B)^{-1},$

where $\tilde{\Delta}=\Delta_{c\alpha}(-T_{3}A_{3})$ for some absolute
constant $c>0$. We use [4, Lemma 6.2] to construct an additive framework
between $\Gamma_{\mathrm{top}}=\Delta_{1/2}(T_{3}B^{(2)})$ and
$\Gamma_{\mathrm{bottom}}=\Delta_{1/2}(T_{3}B^{(1)})$. Next, we use [4, Lemma
8.5] with $A^{\prime}$ being replaced by $-T_{3}A_{3}$, $B^{\prime}$ being
replaced by $T_{3}B^{\prime}$, $B^{(1)}$ being replaced by $T_{3}B^{(2)}$ and
$B^{(2)}$ being replaced by $T_{3}B^{(3)}$. This either gives a density
increment for $A$ with respect to $T_{3}B^{\prime}$, or else produces a set
$\Delta$ satisfying most of the conditions of the final case of Proposition
5.2. The rest of the proof is the same, after replacing every occurrence of
$2\cdot A^{\prime}$ by $-T_{3}A_{3}$ and every occurrence of $2\cdot B^{(i)}$
by $T_{3}B^{(i)}$. ∎

Proposition 5.3 is the adaptation of [4, Proposition 5.5] to general
coefficients.

######  Proposition 5.3.

There is a constant $C>0$ such that, for all $k\geq C$, the following holds.
Let $\mathcal{B}$ be a regular Bohr set of rank $d$ and suppose that
$\mathcal{A}\subset\mathcal{B}$ has density $\alpha$. Let
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{B}^{\star}:=\mathcal{B}\cap
T_{2}\mathcal{B}\cap T_{3}\mathcal{B}}$. Either

  1. 1.

$\alpha\geq 2^{-O(k^{2})}$,

  2. 2.

$T(\mathcal{A},\mathcal{A},\mathcal{A})\gg\exp\left(-\tilde{O}_{\alpha}(d\log{2d})\right)\mu({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{B}^{\star}})^{2},$

or

  3. 3.

$\mathcal{A}$ has a density increment of one of the following strengths
relative to $\mathcal{B}_{1}:=\mathcal{B}^{\star}$,
$\mathcal{B}_{2}:=T_{2}^{-1}\mathcal{B}^{\star}$ or
$\mathcal{B}_{3}:=T_{3}^{-1}\mathcal{B}^{\star}$:

    1. (a)

(small increment)
$[\alpha^{O(\varepsilon(k))},\alpha^{-O(\varepsilon(k))};\tilde{O}_{\alpha}(1)]$,
or

    2. (b)

(large increment) $[\alpha^{-1/k},\alpha^{-1+1/k};\tilde{O}_{\alpha}(1)]$,

where $\varepsilon(k)=\frac{\log\log\log k}{\log\log k}$.

###### Proof.

Let $\varepsilon=c_{0}\alpha^{C_{0}\frac{\log\log\log k}{\log\log k}}$, for
some small constant $0<c_{0}\leq\frac{1}{3}$ and some large constant
$C_{0}>0$. We apply Lemma 3.8 with

$B_{1}=(\mathcal{B}^{\star})_{\rho},\quad
B_{2}=T_{2}^{-1}(\mathcal{B}^{\star})_{\rho},\quad\text{and}\quad
B_{3}=T_{3}^{-1}(\mathcal{B}^{\star})_{\rho\rho^{\prime}},$

where $\rho=c\alpha\varepsilon/d$ and $\rho^{\prime}=c^{\prime}\alpha^{2}/d$
($c$ and $c^{\prime}$ being small constants, chosen in particular such that
$B_{1}$ and $B_{3}$ are regular.555Note that the regularity of $B_{2}$ follows
immediately from that of $B_{1}$. If we are in the second case of Lemma 3.8,
then $\mathcal{A}$ has a small increment with respect to $B_{1}$, $B_{2}$ or
$B_{3}$. By Lemma 3.7, this translates into a density increment of the same
strength with respect to $\mathcal{B}_{1}$, $\mathcal{B}_{2}$ or
$\mathcal{B}_{3}$, as required.

Let us assume henceforth that we are in the first case of Lemma 3.8. Let

$A_{1}=(\mathcal{A}-x)\cap B_{1},\quad A_{2}=(\mathcal{A}-x)\cap
B_{2}\quad\text{and}\quad A_{3}=(\mathcal{A}-x)\cap B_{3}.$

If $\alpha_{i}$ is the density of $A_{i}$ relative to $B_{i}$, for $1\leq
i\leq 3$, then Lemma 3.8 ensures that

$\alpha_{i}\in[(1-\varepsilon)\alpha,(1+\varepsilon)\alpha].$

We now apply Proposition 5.2 with $B=B_{1}$, $B^{\prime}=B_{3}$, $h=\lceil
c_{1}\log\log k/\log\log\log k\rceil$ and $t=\lceil C_{2}\log k\rceil$, for
some suitable constants $c_{1},C_{2}>0$.

  1. 1.

In the first case of the conclusion of Proposition 5.2, $\alpha\gg 1/k^{2}\geq
2^{-O(k^{2})}$.

  2. 2.

In the second case,

$T(A_{1},A_{2},A_{3})\gg\alpha^{3}\mu(B_{1})\mu(B_{3})\gg\exp\left(-\tilde{O}_{\alpha}(d\log
2d)\right)\mu(\mathcal{B}^{\star})^{2}$

by Lemma 3.3. Since the $A_{i}$’s are subsets of the same translate of
$\mathcal{A}$ and the equation $a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$ is translation-
invariant, we have $T(\mathcal{A},\mathcal{A},\mathcal{A})\geq
T(A_{1},A_{2},A_{3})$ and we are done.

  3. 3.

In the third case, either $A_{1}\subset B_{1}$ or $T_{2}A_{2}\subset B_{1}$
has a density increment of strength

$[1,\alpha^{-1/k};\tilde{O}_{\alpha}(h\log
t)]\quad\text{or}\quad[\alpha^{-1/k},\alpha^{-1+1/k};\tilde{O}_{\alpha}(h\log
t)]$

with respect to $T_{3}B_{3}=(B_{1})_{\rho^{\prime}}$. Note that $h\log
t=\tilde{O}_{\alpha}(1)$, or else we have the first case of the conclusion.
Therefore, $\mathcal{A}$ has a density increment of strength

$[1,\alpha^{-1/k};\tilde{O}_{\alpha}(1)]\quad\text{or}\quad[\alpha^{-1/k},\alpha^{-1+1/k};\tilde{O}_{\alpha}(1)]$

relative to either $(B_{1})_{\rho^{\prime}}$ or
$T_{2}^{-1}(B_{1})_{\rho^{\prime}}=(B_{2})_{\rho^{\prime}}$ (here we use the
fact that $\varepsilon$ is sufficiently small, similarly as in Eq. 7). By
Lemma 3.7, this implies that $\mathcal{A}$ has an increment of the same
strength relative to $\mathcal{B}_{1}$ or $\mathcal{B}_{2}$.

  4. 4.

Finally, suppose that the last case of the conclusion of Proposition 5.2
holds. Then we may apply [4, Proposition 11.8] with $B=B_{1}$,
$B^{\prime}=(T_{3}B_{3})_{\rho_{\mathrm{top}}}$ and
$B^{\prime\prime}=(T_{3}B_{3})_{\rho_{\mathrm{top}}\rho^{\prime}}$. The
hypotheses of [4, Proposition 11.8] exactly match the last case of Proposition
5.2 for some $K=\alpha^{-O(1/k)}$.

The number $M$ in [4, Proposition 11.8] satisfies
$M=\alpha^{-O(\varepsilon(k))}$, or else $\alpha\geq 2^{-O(k^{2})}$ and we are
in the first case of our conclusion. Taking $C$ large enough in the statement
of Proposition 5.3, we see that the first case of [4, Proposition 11.8] cannot
hold. In the other two cases, either $A_{1}\subset B_{1}$ or
$T_{2}A_{2}\subset B_{1}$ has a density increment of strength

$[\alpha^{O(\varepsilon(k))},\alpha^{-O(\varepsilon(k))};\tilde{O}_{\alpha}(1)]\quad\text{or}\quad[\alpha^{-1/k},\alpha^{-1+1/k};\tilde{O}_{\alpha}(1)]$

with respect to $B^{\prime\prime}$. As in the previous case, we conclude that
$\mathcal{A}$ has a density increment of the same strength with respect to
$\mathcal{B}_{1}$ or $\mathcal{B}_{2}$.∎

We are now ready to prove Theorem 1.1. The strategy is to iterate Proposition
5.3 as long as we are in the small increment case, and then apply Proposition
4.4 when one of the other cases applies.

###### Proof of Theorem 1.1.

Let $A\subset G$ of density $\alpha$. In this proof, $\alpha$ will always
denote the density of this initial $A$.

Let $1\leq C_{1}=O(1)$ be some absolute constant, chosen in particular larger
than the implied constants in the exponents of the small increment case of
Proposition 5.3. Let $k$ be some constant large enough such that Proposition
5.3 holds and such that $8C_{1}\varepsilon(k)\leq 1/2$. Let $1\leq
C_{2}=\tilde{O}_{\alpha}(1)$ be some fixed quantity (depending only on
$\alpha$), chosen in particular larger than the implicit constants of
Proposition 5.3 hidden in the $\gg$, $O(\cdot)$ and $\tilde{O}_{\alpha}(1)$
notation. By definition of $\tilde{O}_{\alpha}(\cdot)$, these implicit
constants are still bounded by $C_{2}$ if we use Proposition 5.3 with some
different relative density $\alpha^{\prime}$, as long as
$\alpha^{\prime}\geq\alpha$. Note that we may assume that
$2^{C_{2}k^{2}}\leq\alpha^{-1/2}$, or else we are done by an application of
Proposition 4.4 with $B=G$.

Iterative construction. We inductively construct two sequences $(A_{n})$ and
$(B_{n})$, where, for each $n$, $A_{n}$ is a subset of $B_{n}$ relative
density $\alpha_{n}$. Let $A_{0}=A$ and $B_{0}=G$. Assume that $A_{i}$ and
$B_{i}$ have been constructed for $i<n$. We use Proposition 5.3 with
$\mathcal{A}=A_{n-1}$ and $\mathcal{B}=B_{n-1}$. If we are not in case (3)(a),
then we stop the construction of the sequences. Otherwise, case (3)(a) occurs,
and we have a small increment for $A_{n}$. Hence, there are sets $A_{n}\subset
B_{n}$ such that

  * •

$A_{n}$ is a subset of a translate of $A_{n-1}$;

  * •

$A_{n}$ is a subset of $B_{n}$ of relative density
$\alpha_{n}\geq\left(1+C_{2}^{-1}\alpha^{C_{1}\varepsilon(k)}\right)\alpha_{n-1}$;

  * •

$B_{n}$ is a regular Bohr set of the form

$B_{n}=(S_{n}B_{n-1}^{\star})_{\rho_{n}}\cap\widetilde{B}_{n},$ (14)

where

    * –

$B_{n-1}^{\star}$ is the Bohr set

$B_{n-1}^{\star}:=B_{n-1}\cap T_{2}B_{n-1}\cap T_{3}B_{n-1},$ (15)

whose rank we denote by $d_{n-1}^{\star}$,

    * –

$S_{n}$ is either $\mathrm{Id}_{G}$, $T_{2}^{-1}$ or $T_{3}^{-1}$,

    * –

$\widetilde{B}_{n}=\mathrm{Bohr}(\widetilde{\Gamma}_{n},\widetilde{\nu}_{n})$,
where $|\widetilde{\Gamma}_{n}|\leq C_{2}\alpha^{-C_{1}\varepsilon(k)}$ and

$\bigg{(}\frac{\rho_{n}}{4}\bigg{)}^{d_{n-1}^{\star}}\prod_{\gamma\in\widetilde{\Gamma}_{n}}\frac{\widetilde{\nu}_{n}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(d_{n-1}^{\star}+\alpha^{-C_{1}\varepsilon(k)}\right)\log\left(d_{n-1}^{\star}\right)\right).$
(16)

Analysis of the algorithm. Note that, since
$1\geq\alpha_{n}\geq\left(1+C_{2}^{-1}\alpha^{C_{1}\varepsilon(k)}\right)^{n}\alpha$,
this construction must terminate in

$l=\tilde{O}_{\alpha}\left(\alpha^{-C_{1}\varepsilon(k)}\right)$

steps. We then arrive at $A_{l}\subset B_{l}$ for which one of the cases (1),
(2) and (3)(b) of Proposition 5.3 applies.

Let $W_{i}$ be the set of all automorphisms obtained by composing $i$ elements
of $\\{\mathrm{Id}_{G},T_{2},T_{3},T_{2}^{-1},T_{3}^{-1}\\}$. Since
$T_{3}=-\mathrm{Id}_{G}-T_{2}$, these automorphisms commute, which implies
that $|W_{i}|\leq(2i+1)^{2}$.

An immediate induction using Eqs. 14 and 15 shows that

$B_{i}^{\star}\supset\bigcap_{j=1}^{i}\bigcap_{T\in
W_{2i+1}}(T\widetilde{B}_{j})_{\rho_{j+1}\cdots\rho_{i}}$ (17)

for $0\leq i\leq l$. Similarly, we see that the frequency set of
$B_{i}^{\star}$ is contained in

$\bigcup_{j=1}^{i}\bigcup_{T\in W_{2i+1}}T\widetilde{\Gamma}_{j}.$

This implies that

$d_{l}^{\star}\ll
l^{3}C_{2}\alpha^{-C_{1}\varepsilon(k)}=\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}\right).$

In particular, Eq. 16 becomes

$\bigg{(}\frac{\rho_{n}}{4}\bigg{)}^{d_{n-1}^{\star}}\prod_{\gamma\in\widetilde{\Gamma}_{n}}\frac{\widetilde{\nu}_{n}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}\right)\right),$
(18)

for $1\leq n\leq l$.

We now use Lemma 3.3 to give a lower bound for $\mu(B_{l}^{\star})$. By Eq.
17, we have

$\mu(B_{l}^{\star})\geq\prod_{j=1}^{l}\Bigg{(}\bigg{(}\frac{1}{4}\prod_{n=j+1}^{l}\rho_{n}\bigg{)}^{\mathrm{rk}(\widetilde{B}_{j})}\prod_{\gamma\in\widetilde{\Gamma}_{j}}\frac{\widetilde{\nu}_{j}(\gamma)}{8}\Bigg{)}^{|W_{2l+1}|}.$

Using Eq. 18 and the fact that $\mathrm{rk}(\widetilde{B}_{j})\leq
d_{n-1}^{\star}$ for $j+1\leq n\leq l$, this yields

$\mu(B_{l}^{\star})\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}\right)\right)^{\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}\right)}\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}\right)\right).$
(19)

If $B_{l}^{\star}=\mathrm{Bohr}(\Gamma_{l}^{\star},\nu_{l}^{\star})$, this
reasoning actually shows the more precise bound

$\prod_{\gamma\in\Gamma_{l}^{\star}}\frac{\nu_{l}^{\star}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}\right)\right).$
(20)

Concluding the proof. We now apply Proposition 5.3 to $\mathcal{A}=A_{l}$ and
$\mathcal{B}=B_{l}$. The small increment case cannot occur, by construction of
the sequences $(A_{l})$ and $(B_{l})$.

  * •

If we are in the case (1) of Proposition 5.3, then $\alpha_{l}\geq
2^{-C_{2}k^{2}}$. In this case we apply Proposition 4.4 and obtain the bound

$T(A_{l},A_{l},A_{l})\gg\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}+2^{C_{2}k^{2}}\right)\right)\bigg{(}\prod_{\gamma\in\Gamma_{l}}\frac{\nu_{l}(\gamma)}{8}\bigg{)}^{\tilde{O}_{\alpha}(1)},$

where $B_{l}=\mathrm{Bohr}(\Gamma_{l},\nu_{l})$. Using Eq. 20, we deduce that

$T(A,A,A)\geq
T(A_{l},A_{l},A_{l})\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}+2^{C_{2}k^{2}}\right)\right).$

  * •

In the second case, we directly obtain

$T(A_{l},A_{l},A_{l})\gg\exp\left(-\widetilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}\right)\right)\mu(B_{l})^{2},$

and thus

$T(A,A,A)\geq
T(A_{l},A_{l},A_{l})\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}\right)\right)$

by Eq. 19.

  * •

Finally, in the large increment case, there are some $\rho>0$,
$T\in\\{\mathrm{Id}_{G},T_{2}^{-1},T_{3}^{-1}\\}$ and
$\widetilde{B}=\mathrm{Bohr}(\widetilde{\Gamma},\widetilde{\nu})$ such that
${|\widetilde{\Gamma}|\leq C_{2}\alpha^{-1+1/k}}$ and, if

$B^{\prime\prime}:=(TB_{l}^{\star})_{\rho}\cap\widetilde{B},$

then
$\left\|\mathbf{1}_{A}*\mu_{B^{\prime\prime}}\right\|\gtrsim_{\alpha}\alpha^{1-1/k}$
and

$\left(\frac{\rho}{4}\right)^{\mathrm{rk}(TB_{l}^{\star})}\prod_{\gamma\in\widetilde{\Gamma}}\frac{\widetilde{\nu}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}+\alpha^{-1+1/k}\right)\right).$
(21)

Write
$B^{\prime\prime}=\mathrm{Bohr}(\Gamma^{\prime\prime},\nu^{\prime\prime})$.
Then

$\displaystyle\prod_{\gamma\in\Gamma^{\prime\prime}}\frac{\nu^{\prime\prime}(\gamma)}{8}$
$\displaystyle\geq\left(\left(\frac{\rho}{4}\right)^{\mathrm{rk}(TB_{l}^{\star})}\prod_{\gamma\in\widetilde{\Gamma}}\frac{\widetilde{\nu}(\gamma)}{8}\right)\left(\prod_{\gamma\in\Gamma_{l}^{\star}}\frac{\nu_{l}^{\star}(\gamma)}{8}\right)$
$\displaystyle\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}+\alpha^{-1+1/k}\right)\right)$

by Eqs. 21 and 20. We now apply Proposition 4.4 to a suitable subset
$A^{\prime\prime}$ of a translate of $A$ and the Bohr set $B^{\prime\prime}$
to find that

$\displaystyle T(A^{\prime\prime},A^{\prime\prime},A^{\prime\prime})$
$\displaystyle\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-4C_{1}\varepsilon(k)}+\alpha^{-1+1/k}\right)\right)\bigg{(}\prod_{\gamma\in\Gamma^{\prime\prime}}\frac{\nu^{\prime\prime}(\gamma)}{8}\bigg{)}^{\tilde{O}_{\alpha}(1)}$
$\displaystyle\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}+\alpha^{-1+1/k}\right)\right).$

Therefore, we obtain, in all three cases, the lower bound

$T(A,A,A)\geq\exp\left(-\tilde{O}_{\alpha}\left(\alpha^{-8C_{1}\varepsilon(k)}+\alpha^{-1+1/k}+2^{C_{2}k^{2}}\right)\right).$

Choosing $c=1/(2k)$, say, we obtain

$T(A,A,A)\geq\exp\left(-{O}(\alpha^{-1+c})\right).$

On the other hand, since $A$ contains only trivial solutions to
$a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$, we have

$T(A,A,A)=\frac{\alpha}{|G|}\leq\frac{1}{|G|}.$

Therefore, $|G|\leq\exp\left({O}(\alpha^{-1+c})\right)$, which can be
rewritten as

$|A|\ll\frac{|G|}{(\log{|G|})^{1+c^{\prime}}},$

where $c^{\prime}=\frac{1}{1-c}-1$. This finishes the proof of Theorem 1.1. ∎

## Acknowledgments

I am deeply grateful to Thomas Bloom and Olof Sisask for bringing this problem
to my attention and for the many fruitful conversations. I would also like to
thank Timothy Gowers for his guidance and the very helpful discussions
throughout my research internship for the École Normale Supérieure.

## References

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  * [4] , _Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions_ , arXiv preprint 2007.03528v2 (v1:2020, v2:2021).
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  * [7] Tom Sanders, _On Roth’s theorem on progressions_ , Annals of Mathematics 174 (2011), no. 1, 619–636.
  * [8] Tomasz Schoen, _Improved bound in Roth’s theorem on arithmetic progressions_ , Advances in Mathematics 386 (2021), 107801.
  * [9] Tomasz Schoen and Olof Sisask, _Roth’s theorem for four variables and additive structures in sums of sparse sets_ , Forum of Mathematics. Sigma 4 (2016), e5, 28.
  * [10] Ilya Shkredov and Jozsef Solymosi, _Tilted corners in integer grids_ , INTEGERS Journal, Ron Graham Memorial Volume 21A (2021), A20.

[pgom] Cédric Pilatte
École Normale Supérieure
Paris, France
cedric.pilatteensfr