<EMAIL_ADDRESS>
# Densities probabilities of a Bose-Fermi Mixture
in 1D Double well potential

J. Nisperuza1, JP Rubio1 and R. Avella 1 [  1 Facultad de Ingenieria
Aeronautica, Fundación Universitaria los Libertadores, A. A. 75087 Bogotá,
Colombia.

###### Abstract

We use the two mode approximation for a interacting one-dimensional spinless
soft core bosons and one half spin fermions in a double-well potential with a
large central barrier. We include all the on-site boson-boson, fermion-fermion
and boson-fermion repulsive contact potential represented by delta-function
and considered bosonic and fermionic isotopes of ytterbium(Yb) ${}^{170}Yb$
and ${}^{171}Yb$ respectively. By means of the approximation, we find that in
the regime $U_{BF}>U_{BB}$ give rise to a immiscible phase and in the regime
$U_{BB}\geq U_{BF}$ give rise to a miscible phase, that is characterized by a
temporal overlap of the bosonic and fermionic probability densities. We also
report that due to the Bose-Fermi interaction, the system presents an apparent
destruction of the collapse-revival oscillation of boson density probability
at least in the ranges investigated.

††preprint: APS/123-QED

##  I INTRODUCTION

In the field of ultracold gases, there are perfectly controllable physical
parameters[1] and many interesting quasi-one-dimension experiments have been
realized in harmonic [2, 3], double-well[4, 5], periodic[6], and
bichromatic[7] optical-lattice traps. The atoms in these experiments can be
confined to different lattice sites and if two atomic species are considered,
both the effective intra and inter component interactions can be tuned with
great precision via the magnetic Feshbach resonance or confinement induced
resonance [8, 9, 10].

Advances in laser technology allow experimental realization in one, two and
three dimensions [11, 12, 13] of a single double-well and an array of many
copies of the double well system, know as dimers. In these systems the
tunneling between the local double-well potentials is negligible, compared to
tunneling inside the double well potential and each site have a well-defined
and almost identical quantum state [14, 11, 15]. These systems allow to study
a very important concepts in quantum information theory, the notion of state
as a linear superposition of ‘classical’ states[16], where the system can
reside in a superposition of two or more degenerate states[17, 18, 19] and
quantum tunneling, that have applications in solid-state devices, solar cells
and microscopes [20].

Particular interesting phenomena that is present in systems of interacting
atoms in double-well potentials are bosonic Josephson junction [21, 22, 4],
squeezing and entanglement of matter waves [23, 24], matter wave interference
[25, 26], and exact many-body quantum dynamic in one dimension. In quantum
information processing, this systems has been used as a way to make quantum
logic gates for ultracold neutral atoms confined in optical lattices [27].

One dimensional double well potential has been used to study both
theoretically [28, 29] and experimentally [30, 4] the simplest case of
Josephson effect using 4He and a superfluid Fermi gas with 3He [31, 32, 33].

The influence of fermions onto bosons has been investigated in different
mixture, for example in a mixture of ${}^{4}He-^{3}He$ has been reported the
observation of simultaneous quantum degeneracy [34, 35]. In mixture of
${}^{87}Rb-^{40}K$ [36, 37] has been found a pronounced asymmetry between
strong repulsion and strong attraction [38]. Phase separation was found in a
mixture of ${}^{41}K-^{6}Li$[39]. A localized phase of ultracold bosonic
quantum gases of ${}^{87}Rb$ induced by a small contribution of ${}^{40}K$
fermionic atoms was observed in [40]. Attractive and repulsive Bose-Fermi
interaction was study in a system composed by ${}^{170}Yb-^{173}Yb$
(attractive) and ${}^{174}Yb-^{173}Yb$ (Repulsive)[41]. Other Bose-Fermi
mixtures that have been experimentally reported are ${}^{7}Li-^{6}Li$ [42],
${}^{39}K-^{40}K$ and ${}^{41}K-^{40}K$ [43].

Theoretically, numerous studies have been carried out to analyze the Bose-
Fermi mixture (BFM). For example, the collapse in attractive mixtures has been
studied numerically [44, 45, 46, 47], semi-analytically [48, 49], or in the
Thomas–Fermi approximation[50]. The Bose-Fermi interaction induce the pairing
of fermions analogous to the formation of Cooper pairs in the BCS model [51,
52, 53] and the boson phase transition from the Mott insulator to super fluid
[54, 55, 56]. Other studies indicate an asymmetry between the attraction and
repulsion cases [57, 58], as well as phase separation, spatial modulation
[59], supersolid phase and charge density wave [60].

There is a fine line separating properties of bosons and fermions, for
example, in the Bose-Fermi mapping theorem for hard core particles [61]
realized experimentally for first time in [62], the ground-state is highly
degenerate in a limit of infinitely strong boson-fermion repulsions [63] due
to the freedom of fixing the sign of many-body wave function under the
exchange of a boson with a fermion [64, 65, 66]. To find a duality relation in
systems composed by spin-1/2 fermions interacting with two-component bosons
[67] was used the Cheon-Shigehara mapping.

The most general one-to-one mapping between bosons and fermions in one
dimension was found by M. Valiente. This general mapping not restricted to
pairwise forces, is valid for arbitrary single-particle dispersion, including
non-relativistic, relativistic, continuum limits of lattice hamiltonians and
can also be applied to any internal structure and spin [68].

This leaves open the possibility of studying the transmutation of bosons into
fermions and vice versa in a system of spinless bosons in the soft-core limit
and spin one-half fermions, that is studied in this paper. In this work we
focus on a repulsive interacting Bose-Fermi mixture of a few particles
confined in a 1D double well potential at zero temperature.

The paper is organized as follows. The model used to describe a mixture of
bosonic and fermionic atoms is introduced in Sec.II. In section III, we vary
the inter and intra species interactions and find two different regions and
finally in Sec. IV we make remarks.

##  II Physical Model

Due to the experimental possibility of confining quantum gases in an array of
many copies of the double well system, where the tunneling between the local
double-well potentials is negligible compared to tunneling inside the double
well potential, we considered the study in one of these potentials. In our
study we consider the experimental setup used in [69], where the radial
separation of the potential wells is $d=13\mu m$, a trap depth of $h\times
4.7Khz$ and we considered the width of each well of $a=6\mu m$. This potential
confines bosonic and fermionic isotopes of ytterbium(Yb) ${}^{170}Yb$ and
${}^{171}Yb$ respectively and we considered that the fermions isotopes have
two internal degrees of freedom, allowing a density of fermions per site of
$0\leq\rho_{F}\leq 2$; the maximum number of scalar bosons on the “soft-core”
boundary per site, is $n_{max}^{B}=2$. This value is due to the fact that in
several reports has been found that the qualitative physical properties
obtained for $n_{max}^{B}=2$ do not change when $n_{max}^{B}$ is increased
[70, 71].

The general wave function in the basis of the two states that is noncommittal
as to which particle is in which state and provid an accurate formulation [22,
72, 29] can be expressed as

$\Psi_{\pm}(x_{1},x_{2})=A[\psi_{s}^{1}(x_{1})\psi_{a}^{1}(x_{2})\pm\psi_{a}^{1}(x_{1})\psi_{s}^{1}(x_{2})],$
(1)

where $A$ is the normalization constant, plus sign is for symmetric (bosons) a
minus sign is for antisymmetric (fermions) functions, $\psi_{s}^{1}(x_{i})$
indicate the wave function of the ith particle in the ground state and
$\psi_{a}^{1}(x_{i})$ is the wave function of the ith particle in the first
excited level; these wave function satisfy the orthonormal condition

$\int\psi_{i}(x)\psi_{j}(x)dx=\delta_{ij}.$

The two particles wave function at time $t$ of these vectors are

$\begin{split}&\Psi_{R,L}(x_{1},x_{2},t)=\\\
&\frac{e^{-i\frac{E_{s}^{1}+E_{a}^{1}}{\hbar}}}{2}\Big{[}e^{i\Omega_{1}t}\psi_{s}^{1}(x_{1})\psi_{s}^{1}(x_{2})\pm\psi_{s}^{1}(x_{1})\psi_{a}^{1}(x_{2})\\\
&\pm\psi_{a}^{1}(x_{1})\psi_{s}^{1}(x_{2})+e^{i\Omega_{1}t}\psi_{a}^{1}(x_{1})\psi_{a}^{1}(x_{2})\Big{]},\end{split}$
(2)

where $\Omega_{1}=\frac{E_{a}^{1}-E_{s}^{1}}{\hbar}$ is the Bohr frequency and
$\Psi_{R,L}(x_{1},x_{2},t)$ describe the probability of find the two particles
on the right $\Psi_{R}(x_{1},x_{2},t)$ (plus sign) or on the left
$\Psi_{L}(x_{1},x_{2},t)$(minus sign) of the double well. From this we deduce
the probability density

$\begin{split}&|\Psi_{R,L}(x_{1},x_{2},t)|^{2}=\\\
&\Bigg{\\{}\left[\frac{1}{2}\left[\psi_{s}^{1}(x_{1})\right]^{2}\pm\frac{1}{2}\left[\psi_{a}^{1}(x_{1})\right]^{2}\pm\cos\left[\Omega_{1}t\psi_{s}^{1}(x_{1})\psi_{a}^{1}(x_{1})\right]\right]\\\
&\left[\frac{1}{2}\left[\psi_{s}^{1}(x_{2})\right]^{2}\pm\frac{1}{2}\left[\psi_{a}^{1}(x_{2})\right]^{2}\pm\cos\left[\Omega_{1}t\psi_{s}^{1}(x_{2})\psi_{a}^{1}(x_{2})\right]\right]\Bigg{\\}},\end{split}$
(3)

In this way the two-particle Hamiltonian for the system in one dimension is

$\begin{split}\hat{H}_{BF}=\hat{H}_{B}^{\lambda_{B}}(x_{1},x_{2})+\hat{H}_{F}^{\lambda_{F}}(x_{1},x_{2})+\lambda_{BF}\delta(x_{1},x_{2}),\end{split}$
(4)

where

$\begin{split}&\hat{H}_{B}^{\lambda_{B}}(x_{1},x_{2})=\\\
&-\frac{\hbar^{2}}{2m_{B}}\frac{d^{2}}{dx_{1}^{2}}+V(x_{1})-\frac{\hbar^{2}}{2m_{B}}\frac{d^{2}}{dx_{2}^{2}}+V(x_{2})+\lambda_{B}\delta(x_{1}-x_{2}),\end{split}$
(5)

and

$\begin{split}&\hat{H}_{F}^{\lambda_{F}}(x_{1},x_{2})=\\\
&-\frac{\hbar^{2}}{2m_{F}}\frac{d^{2}}{dx_{1}^{2}}+V(x_{1})-\frac{\hbar^{2}}{2m_{F}}\frac{d^{2}}{dx_{2}^{2}}+V(x_{2})+\lambda_{F}\delta(x_{1}-x_{2})\end{split}$
(6)

The repulsive contact potential between bosons (fermions) is represented by
delta-function potential $\lambda_{B}(_{F})\delta(x_{1}-x_{2})$, where
$\lambda_{B(F)}>0$ for a repulsive interaction, $m_{B}(_{F})$ is the mass of
${}^{170}Yb$ (${}^{171}Yb$), $V(x)$ is the external confinement potential that
is the same for both species and $\lambda_{BF}\delta(x_{1},x_{2})$ is the
repulsive interaction between two ultracold neutral atoms of different
statistic. We consider that the perturbation is sufficiently small, therefore
is considered that the amplitudes of other states do not mix [73], so we use
the projection of the wave function onto two states that is used in the study
of BEC in a double-well potential and in the investigation of Fermi super
fluid [74].

From the equations 2 and 3 we configure our system considering that the two
bosons and the two fermions are on the right side. The orbital wave function
of two fermions with opposite spin in a double-well potential is decomposed
into a singlet and three triplet states with respect to the pseudospin defined
by the double-well potential at $t=0$ and we adopt our units of length,
$l=1\mu$, energy $E_{a}=E/\xi$ with $\xi=10^{-31}$ and time
$\tau=\hbar/\xi$[75]. Henceforth, we will measure lengths, energies and time
in these units.

##  III Bose-Fermi probabilities

We considerer that the initial state of our system is configured, with two
spinless soft core bosons and two one half spin fermions in the right side of
the double well potential. The boson-boson $\lambda_{BB}$, fermion-fermion
$\lambda_{FF}$ and boson-fermion $\lambda_{BF}$ repulsive contact interaction
terms, were varied.

We found that although the initial state of the two species is considered on
the right side of the double potential well, the behavior of the probability
density of bosons and fermions is completely different in the regime of
$\lambda_{BF}>\lambda_{BB}$, where arise a immiscible phase characterized by
almost perfectly separated fermionic and bosonic probability densities. In the
regime of $\lambda_{BB}\geq\lambda_{BF}$, densities of bosons and fermions are
overlapping as a function of time, as see in 1 where $P_{RR}(\tau)$ represents
the probability density of finding the two bosonic (fermionic) particles on
the right side of the double well, as a function of the dimensionless
parameter of time $\tau$.

Time independent miscible phase, were found theoretically in [76], where was
studied the ground state and the nonequilibrium expansion dynamics of a Bose
Fermi mixture in an one-dimensional lattice with an imposed harmonic
oscillator and in [77], where was studied the ground state properties of a
Bose-Fermi mixture trapped in a one-dimensional optical lattice with a
superimposed harmonic oscillator.

In our research we found a complete miscibility of bosons (red line) and
fermions (blue line) probability densities until dimensionless time
$\tau\approx 1363.97$ for $\lambda_{BB}=\lambda_{FF}=1\times 10^{-5}$ and
$\lambda_{BF}=1\times 10^{-5}$ as show in figure 1(a). If fermion-fermion is
greater than boson-fermion interaction, we found a miscibility of
probabilities until $\tau\approx 905.90$ for
$\lambda_{BB}=1,\lambda_{FF}=3\times 10^{-5}$ and $\lambda_{BF}=1\times
10^{-5}$, which indicates that the interaction parameter $\lambda_{FF}$
induces a greater repulsive interaction in the bosons, forcing them to stay
together for a shorter time and giving rise to immiscible phase more quickly,
as illustrated in the figure 1(b).

(a) Bose (red line) and Fermi (blue line) density probabilities for:
$\lambda_{BB}=\lambda_{FF}=1\times 10^{-5}$ and $\lambda_{BF}=1\times
10^{-5}$. The two probability densities are the same until dimensionless time
$\tau\approx 1363.97$.

(b) Bose (red line) and Fermi (blue line) density probabilities for:
$\lambda_{BB}=1\times 10^{-5},\lambda_{FF}=3\times 10^{-5}$ and
$\lambda_{BF}=1\times 10^{-5}$. The two probability densities are the same
until dimensionless time $\tau\approx 905.90$.

(c) Bose (red line) and Fermi (blue line) density probabilities for:
$\lambda_{BB}=7\times 10^{-5},\lambda_{FF}=1\times 10^{-5}$ and
$\lambda_{BF}=5\times 10^{-5}$. The two probability densities are the same
until dimensionless time $\tau\approx 4090.97$.

Figure 1: Time evolution of Bose-Fermi density probabilities on the right side
of the double well, as a function of the dimensionless parameter of time
$\tau$.

Is interesting to note that by increasing the value of the interaction between
bosons to $\lambda_{BB}=7\times 10^{-5}$ and considering that
$\lambda_{BF}>\lambda_{FF}$ the two probabilities tend to be together until
$\tau\approx 4090.97$ and the separation between the two probabilities tend to
be closer than in the previous case. This is due to the fact that by
increasing the interaction parameter between bosons, they tend to separate
more quickly and this increase also affects the fermion-fermion interaction,
as can be seen in figure 1(c). For these values, our study revealed that due
to the small interaction between fermions, they tend to stay together on the
right side of the double well (blue lines) in relation to the bosons (red
lines), which separate more quickly for similar values, as seen in the figure
1.

By increasing the order of the interaction parameters, we found again that in
the regime of $\lambda_{BB}\geq\lambda_{BF}$, densities of bosons and fermions
are overlapping when the two species begin their tunneling process on the left
side of the double well in approximately the same time, again displaying a
miscible phase for $\lambda_{BB}=9\times 10^{-4},\lambda_{FF}=1\times 10^{-4}$
and $\lambda_{BF}=1\times 10^{-4}$ that is maintained until the two species
peak again on the right side of the well. The amplitud of bosons and fermions
probability densities is different after the first tunneling, which is
evidence of the time dependence of the miscible phase; before reaching the
second tunneling, the probability of the two species ceases to be the same,
giving rise to a immiscible phase as show in figure 2(a). We found a
transitive process from correlated Rabi oscillation of the boson particles to
the apparent uncorrelated collapse-revival oscillation of $P_{RR}(\tau)$ [78,
75] due to an increasing temporary decay of the bosons oscillation amplitude
(red line) as show in figure 2(a). However when we increase the adimensional
time, we found a apparent destruction of the uncorrelated collapse-revival
oscillation as seen in the figure 2(b), which indicates that the bosons tunnel
through the central barrier in correlated state at least in the ranges
investigated. This phenomenon indicates that due to the boson-fermion
interaction, an attractive boson-boson interaction is induced and that forces
the bosons to tunnel in a correlated state.

(a) Decrease in the boson probability amplitude (red line)

(b) Destruction of collapse-revival oscillation of bosons $P_{RR}(\tau)$ due
to Bose-Fermi interaction

Figure 2: Apparent destruction of collapse-revival oscillation of bosons
$P_{RR}(\tau)$.

##  IV SUMMARY AND CONCLUSION

We have investigated the ground state properties of a two-component model
where boson-boson, fermion-fermion and boson-fermion interaction is a weak
repulsive contact potential and the two species are confined in the same one
dimensional double well potential.

Tuning the ratio between the inter and intraspecies interaction strengths we
found that although in the initial state both species are considered to the
right side of the confinement potential, the probability densities of both
species are completely different, in the regime $\lambda_{BF}>\lambda_{BB}$,
giving rise to a immiscible phase. This phase occurs because fermions, despite
their repulsive interaction, tend to stay longer in the initial state, while
bosons tend to separate more quickly due to the fact that in addition to the
repulsive contact potential between bosons, a repulsive potential is induced
in them, for fermion-fermion and boson-fermion interaction

In the regime $\lambda_{BB}\geq\lambda_{BF}$ probability densities of bosons
and fermions are overlapping, giving rise to a miscibility phase that
disappears as a function of time. These results indicate that the immiscible
and miscible phases depend of the competition between the inter and intra
species interaction strength and are characterized by negligible and complete
overlap of the density probabilities, respectively.

As a final attempt we increase the order of the interaction parameters, and
found that the miscible phase is a time function again and as a result of the
interaction between bosons and fermions the system presents an apparent
destruction of the uncorrelated collapse-revival oscillation of boson density
probability at least in the ranges investigated.

Our findings can help interpret experimental results in bosonic Josephson
junction, squeezing, entanglement of matter waves, matter wave interference
and in quantum information processing.

##  V ACKNOWLEDGMENTS

R.A. is thankful for the support of Departamento Administrativo de Ciencia,
Tecnología e Innovación (COLCIENCIAS) (Grant No. FP44842-135-2017).

We gratefully acknowledge support from ”Fundación Universitaria los
libertadores”. We are especially indebted to Mistakidis, for sending the
arcticle [76] before this publication.

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# A polylogarithmic bound in the nonlinear Roth theorem

Sarah Peluse  Mathematical Institute
University of Oxford
UK<EMAIL_ADDRESS>and  Sean Prendiville  Department of
Mathematics and Statistics
Lancaster University
UK<EMAIL_ADDRESS>
###### Abstract.

We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^{2}$
have at most polylogarithmic density.

###### Contents

  1. 1 Introduction
  2. 2 Iterating the density increment
  3. 3 The cut norm inverse theorem
  4. 4 A weak regularity lemma
  5. 5 The density increment lemma
  6. 6 Global control by major arc Fourier coefficients
  7. 7 Longer progressions

##  1\. Introduction

###  1.1. Density bound

In [9] the authors obtained, for the first time, an effective bound for
subsets of $\left\\{1,\dots,N\right\\}$ lacking the nonlinear Roth
configuration $x$, $x+y$, $x+y^{2}$. There it was established that such sets
have cardinality at most $O(N/(\log\log N)^{c})$, where $c>0$ is an absolute
constant. The key breakthrough of [9] was a “local $U^{1}$-control” result,
from which a bound for sets lacking the nonlinear Roth configuration follows
via standard methods. Here, we combine this local $U^{1}$-control result with
a more sophisticated argument to remove a logarithm from the bound of [9].

######  Theorem 1.1 (Density bound).

There exists an absolute constant $c>0$ such that the following holds. Suppose
that $A\subset\left\\{1,\dots,N\right\\}$ lacks configurations of the form

$x,\ x+y,\ x+y^{2}\qquad(y\neq 0).$ (1.1)

Then

$|A|=O\left(N/(\log N)^{c}\right).$

A careful analysis shows that the exponent $c=2^{-150}$ is permissible, where
150 represents the combined number of times we utilise the Cauchy–Schwarz
inequality in [9] and this paper

###  1.2. Major arc correlation

The techniques which yield Theorem 1.1 also allow us to show, in a
quantitatively effective manner, that the major arc Fourier coefficients of a
set determine how many nonlinear Roth configurations (1.1) the set contains.

######  Theorem 1.2 (Major-arc control).

Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with
support in $\left\\{1,\dots,N\right\\}$. Suppose that

$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta
N^{3/2}.$

Then either $N\ll\delta^{-O(1)}$, or there is a frequency
$\alpha\in\mathbb{R}$ and a positive integer $q\ll\delta^{-O(1)}$ such
that111Here $\left\|\cdot\right\|$ denotes the distance to the nearest
integer, and $e(\alpha):=e^{2\pi i\alpha}$. For our conventions regarding
asymptotic notation see §1.5. $\left\|q\alpha\right\|\ll\delta^{-O(1)}/N$ and

$\left|\sum_{x\in\mathbb{Z}}h(x)e(\alpha x)\right|\gg\delta^{O(1)}N.$

In the nomenclature of [14], the major arc linear phases are the only
obstructions to uniformity for the nonlinear Roth configuration. We emphasise
that Theorem 1.2 is not used in the proof of Theorem 1.1.

The major arc Fourier coefficients of a subset of $\\{1,\dots,N\\}$
essentially measure its distribution in arithmetic progressions of common
difference $\ll 1$ and length $\gg N$. To illustrate this, the following
definition is useful.

######  Definition 1.3 (Local function).

We call a function $\phi:\mathbb{Z}\to\mathbb{C}$ a _local function of
resolution $M$ and modulus $q$_ if there exists a partition of $\mathbb{Z}$
into intervals of length $M$ such that $\phi$ is constant on the intersection
of every such interval with every congruence class mod $q$.

######  Corollary 1.4 (Local control of the nonlinear term).

Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with
support in $\left\\{1,\dots,N\right\\}$. Suppose that

$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta
N^{3/2}.$

Then either $N\ll\delta^{-O(1)}$, or there is a 1-bounded local function
$\phi$ of resolution $M\gg\delta^{O(1)}N$ and modulus $q\ll\delta^{-O(1)}$
such that

$\left|\sum_{x\in\mathbb{Z}}h(x)\phi(x)\right|\gg\delta^{O(1)}N.$

One cannot hope to prove that the functions $f$ and $g$ above also correlate
globally with local functions, as the following example illustrates. For any
positive integers $x_{1},x_{2}\leqslant N^{1/2}$, set

$f\left(x_{1}+(x_{2}-1)\left\lfloor
N^{1/2}\right\rfloor\right)=\begin{cases}1&\text{ if }x_{2}\equiv
0\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 1\pmod{4},\\\ -1&\text{ if }x_{2}\equiv
2\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 3\pmod{4};\end{cases}$

and set $f(x)=0$ everywhere else. Taking $g:=f$ and $h:=1_{\\{1,\dots,N\\}}$,
one can check that either $N\ll 1$ or

$\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\gg N^{3/2}.$

However, for any arithmetic progression $P\subset\\{1,\dots,N\\}$, we have

$\left|\sum_{x\in P}f(x)\right|\ll N^{1/2}.$

Hence, for any 1-bounded local function $\phi$ of resolution $\geqslant\delta
N$ and modulus $\leqslant\delta^{-1}$, the triangle inequality gives the
discorrelation

$\left|\sum_{x\in\mathbb{Z}}f(x)\phi(x)\right|\ll\delta^{-2}N^{1/2}.$

This example is a local obstruction coming from the real numbers: the nature
of our counting operator means that we cannot disentangle possible
correlations between the $f$ and $g$ functions on subintervals of length
$N^{1/2}$. We can, however, show that these are the only other possible
obstructions to uniformity.

######  Theorem 1.5 (Local control of all terms).

Let $\delta>0$ and $f_{1},f_{2},f_{3}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded
functions with support in $\left\\{1,\dots,N\right\\}$. Suppose that

$\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f_{1}(x)f_{2}(x+y)f_{3}(x+y^{2})\right|\geqslant\delta
N^{3/2}.$

Then either $N\ll\delta^{-O(1)}$, or for each $i=1,2,3$ there is a 1-bounded
local function $\phi_{i}$ of resolution $\gg\delta^{O(1)}N^{1/2}$ and modulus
$q_{i}\ll\delta^{-O(1)}$ such that

$\left|\sum_{x\in\mathbb{Z}}f_{i}(x)\phi_{i}(x)\right|\gg\delta^{O(1)}N.$

###### Proof.

This is an immediate consequence of Corollary 1.4 and Lemma 3.2. ∎

###  1.3. Longer polynomial progressions

In analogy with the first author’s generalisation [8] of [9], it is natural to
ask whether the methods of this paper yield polylogarithmic bounds for sets of
integers lacking longer progressions

$x,\ x+P_{1}(y),\ \dots,\ x+P_{m}(y),$ (1.2)

where the $P_{i}\in\mathbb{Z}[y]$ have zero constant term and $\deg
P_{1}<\dots<\deg P_{m}$.

As was mentioned above, the key input to this paper is the local
$U^{1}$-control result [9, Theorem 7.1]. Replacing this with [8, Theorem 3.3],
our argument generalises in a straightforward manner to yield polylogarithmic
bounds for subsets of $\\{1,\dots,N\\}$ lacking (1.2) when $m=2$, that is, for
all three-term polynomial progressions with distinct degrees and zero constant
term.

Obtaining polylogarithmic bounds for longer polynomial progressions requires
an additional idea. We sketch a strategy in §7, which relies on obtaining an
appropriate generalisation of [8, Theorem 3.3], a generalisation that would
require re-running the majority of the arguments therein.

### Acknowledgements

S. Peluse is supported by the NSF Mathematical Sciences Postdoctoral Research
Fellowship Program under Grant No. DMS-1903038

###  1.4. An outline of our argument

Effective Szemerédi-type theorems are commonly proved via a density increment
strategy, the prototypical example being the proof of Roth’s theorem [11] on
three-term arithmetic progressions. This strategy begins with a set
$A\subset\\{1,\dots,N\\}$ of density $\delta:=|A|/N$ that lacks the
configuration in question. It then proceeds to show that there is a
substructure $S\subset\\{1,\dots,N\\}$ on which $A$ has increased density
$\delta+\Omega_{\delta}(1)$. One then hopes to iterate the argument with
$A\cap S$ in place of $A$ and $S$ in place of $\\{1,\dots,N\\}$.

One avenue to obtaining polylogarithmic bounds in a Szemerédi-type theorem is
to obtain a constant proportion density increment $\delta+\Omega(\delta)$ on a
substructure $S$ of polynomial size $|S|\approx N^{\Omega(1)}$. This was
accomplished for three-term arithmetic progressions by Heath–Brown [7] and
Szemerédi [13] (in fact, they were able to handle a smaller lower bound on
$|S|$).

An alternative strategy for obtaining polylogarithmic bounds is to obtain the
weaker polynomial increment $\delta+\Omega(\delta^{O(1)})$, yet on a _dense_
or _global_ substructure $S$, that is, a substructure of size
$|S|\geqslant\exp(-O(\delta^{-O(1)}))N$. This was accomplished by Sárközy [12]
for the configuration $x,x+y^{2}$ and for three-term arithmetic progressions
by Bourgain [2].

Both of these strategies are achievable for the nonlinear Roth configuration.
The global structure strategy is perhaps the most natural, and may be
accomplished by utilising a generalisation of Theorem 1.2. In this note we do
not pursue this, and instead give details for a constant-proportion density
increment, as our argument is somewhat cleaner in this form.

More specifically, we show that if $A\subset\left\\{1,\dots,N\right\\}$ has
density $\delta$ and lacks nontrivial configurations of the form
$x,x+y,x+y^{2}$, then there exists an arithmetic progression $P$ of length
$|P|\gg\delta^{O(1)}N^{1/2}$ and common difference $q\ll\delta^{-O(1)}$ such
that we have the density increment

$\frac{|A\cap P|}{|P|}\geqslant(1+\Omega(1))\frac{|A|}{N}.$ (1.3)

As outlined in [9], the ‘almost bounded’ size of $q$ allows us to iterate this
procedure. (In [9], we obtain the weaker density increment
$(1+\Omega(\delta^{O(1)}))|A|/N$, which leads to the extra logarithm appearing
in the bound there.)

We obtain the constant-proportion increment (1.3) by combining the local
$U^{1}$-control result of [9] with a strategy of Heath–Brown [7] and Szemerédi
[13], which has a very robust formulation due to Green and Tao [6]. To
accomplish this, we first give a structural characterisation of sets lacking
the nonlinear Roth configuration (this is Lemma 3.3, whose essence is captured
in the weaker Theorem 1.5). These sets resemble the level sets of the product
of a function that is constant on intervals of length $N^{1/2}$ and a function
that is constant on congruence classes modulo a bounded $q$.

Having obtained such a structural characterisation, an energy increment
procedure closely following [6] allows us to approximate an arbitrary set of
integers by these level sets, up to an error that does not contribute
substantially to the count of nonlinear Roth configurations. A combinatorial
argument then allows us to deduce that our set must have a substantial density
increment on one of these level sets, of the form $\delta+\Omega(\delta)$. As
a result, our density increment procedure requires only
$\log(\delta^{-1})+O(1)$ iterations, compared with the $O(\delta^{-O(1)})$
required in [9], and this yields the polylogarithmic improvement over our
previous density increment iteration.

The remainder of this paper is organized as follows. We derive Theorem 1.1 in
§2 via a density increment iteration. Our deduction uses a density increment
lemma that is established in §§3–5. We prove Theorem 1.2 and Corollary 1.4 in
§6.

###  1.5. Notation

####  1.5.1. Standard conventions

We use $\mathbb{N}$ to denote the positive integers. For a real number
$X\geqslant 1$, write $[X]=\\{1,2,\ldots,\left\lfloor X\right\rfloor\\}$. A
complex-valued function is said to be _1-bounded_ if the modulus of the
function does not exceed 1.

We use counting measure on $\mathbb{Z}$, so that for
$f,g:\mathbb{Z}\to\mathbb{C}$, we have

$\left\|f\right\|_{\ell^{p}}:=\biggl{(}\sum_{x}|f(x)|^{p}\biggr{)}^{\frac{1}{p}},\
\left\langle f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)},\ \text{and}\
(f*g)(x)=\sum_{y}f(y)g(x-y).$

Any sum of the form $\sum_{x}$ is to be interpreted as a sum over
$\mathbb{Z}$. The _support_ of $f$ is the set
$\mathrm{supp}(f):=\left\\{x\in\mathbb{Z}:f(x)\neq 0\right\\}$. We write
$\left\|f\right\|_{\infty}$ for $\sup_{x\in\mathbb{Z}}|f(x)|$.

We use Haar probability measure on $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, so
that for measurable $F:\mathbb{T}\to\mathbb{C}$, we have

$\left\|F\right\|_{L^{p}}:=\biggl{(}\int_{\mathbb{T}}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}=\biggl{(}\int_{0}^{1}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}.$

We write $\left\|\alpha\right\|_{\mathbb{T}}$ for the distance from
$\alpha\in\mathbb{R}$ to the nearest integer
$\min_{n\in\mathbb{Z}}|\alpha-n|.$ This remains well-defined on $\mathbb{T}$.

We define the Fourier transform of $f:\mathbb{Z}\to\mathbb{C}$ by

$\hat{f}(\alpha):=\sum_{x}f(x)e(\alpha x)\qquad(\alpha\in\mathbb{T}),$ (1.4)

when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i\alpha}$.

For a finite set $S$ and function $f:S\to\mathbb{C}$, denote the average of
$f$ over $S$ by

$\mathbb{E}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$

For a complex-valued function $f$ and positive-valued function $g$, write
$f\ll g$ or $f=O(g)$ if there exists a constant $C$ such that $|f(x)|\leq
Cg(x)$ for all $x$. We write $f=\Omega(g)$ if $f\gg g$. We subscript this
notation when the implicit constant may depend on the subscripted parameters.

####  1.5.2. Local conventions

Up to normalisation, all of the above are widely used in the literature. Next,
we list notation specific to our paper. We have tried to minimise this in
order to aid the casual reader.

The quantity $(N/q)^{1/2}$ appears repeatedly, where $N$ and $q$ are integers
fixed throughout the majority of our paper. We therefore adopt the convention
that

$M:=\left\lfloor\sqrt{N/q}\right\rfloor.$ (1.5)

Assuming this, define the _counting operator_ on the functions
$f_{i}:\mathbb{Z}\to\mathbb{C}$ by

$\Lambda_{q,N}(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+y)f_{2}(x+qy^{2}).$
(1.6)

When $f_{0}=f_{1}=f_{2}=f$, we simply write $\Lambda_{q,N}(f)$ for
$\Lambda_{q,N}(f_{0},f_{1},f_{2})$.

For a real parameter $H\geqslant 1$, we use $\mu_{H}:\mathbb{Z}\to[0,1]$ to
represent the following normalised Fejér kernel

$\mu_{H}(h):=\frac{1}{\left\lfloor
H\right\rfloor}\left(1-\frac{|h|}{\left\lfloor
H\right\rfloor}\right)_{+}=\frac{(1_{[H]}*1_{-[H]})(h)}{\left\lfloor
H\right\rfloor^{2}}.$ (1.7)

This is a probability measure on $\mathbb{Z}$ with support in the interval
$(-H,H)$.

##  2\. Iterating the density increment

In this section we prove Theorem 1.1 using the following lemma, which we will
devote §§3–5 to proving.

######  Lemma 2.1 (Density increment lemma).

Let $q\leqslant N$ be positive integers and $\delta>0$. Suppose that
$A\subset[N]$ satisfies $|A|\geqslant\delta N$ and lacks the configuration

$x,\ x+y,\ x+qy^{2}\qquad(y\neq 0).$ (2.1)

Then either $N\ll(q/\delta)^{O(1)}$ or there exists
$q^{\prime}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$ and
$N^{\prime}\geqslant
q^{-O(1)}\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{1/2}$ such that, for
some $a\in\mathbb{Z}$, we have

$|A\cap(a+qq^{\prime}\cdot[N^{\prime}])|\geqslant(1+\Omega(1))\delta
N^{\prime}.$ (2.2)

###### Proof of Theorem 1.1 given Lemma 2.1.

This is the same as the proof of [9, Theorem 1.1], but using the improved
density increment lemma above in place of the density increment lemma of [9].
Note first that if $A$ lacks the configuration (2.1), then the set

$\\{x:a+qq^{\prime}x\in A\\},$

lacks configurations of the form

$x,\ x+y,\ x+q^{2}q^{\prime}y^{2}\qquad(y\neq 0).$

Let $A\subset[N]$ have size $\delta N$, and suppose that it has no non-linear
Roth configurations (1.1). Setting $A_{0}:=A$, $N_{0}:=N$ and $q_{0}=1$, let
us suppose we have a sequence of tuples $(A_{i},N_{i},q_{i})$ for
$i=0,1,\dots,n$ that each satisfy the following:

  1. (i)

$A_{i}$ lacks configurations of the form

$x,\ x+y,\ x+q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm
q_{i-1}^{2}q_{i}y^{2}\qquad(y\neq 0).$

  2. (ii)

$q_{i}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$;

  3. (iii)

$A_{i}\subset[N_{i}]$ and for $i\geqslant 1$ we have

$\frac{|A_{i}|}{N_{i}}\geqslant(1+c)\frac{|A_{i-1}|}{N_{i-1}},$

where $c=\Omega(1)$ is a positive absolute constant;

  4. (iv)

for $i\geqslant 1$ we have the lower bound

$N_{i}\geqslant\frac{N_{i-1}^{1/2}}{\left(q_{0}^{2^{i-1}}\dotsm
q_{i-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}.$

Applying Lemma 2.1 with $q=q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm
q_{i-1}^{2}q_{i}$, either

$N_{n}\ll\left(q_{0}^{2^{n}}q_{1}^{2^{n-1}}\dotsm
q_{n-1}^{2}q_{n}/\delta\right)^{O(1)},$ (2.3)

or we may obtain $(A_{n+1},N_{n+1},q_{n+1})$ satisfying conditions (i)–(iv).
If (2.3) holds, then our iterative process terminates at stage $n$.

If the number of iterations $n$ is at least $c^{-1}$, then the density of
$A_{n}$ on $[N_{n}]$ is at least $2\delta$. After an additional
$\tfrac{1}{2}c^{-1}$ iterations, the density is at least $4\delta$. Hence if
the number of iterations is at least

$\left\lceil
c^{-1}\right\rceil+\left\lceil\tfrac{1}{2}c^{-1}\right\rceil+\left\lceil\tfrac{1}{4}c^{-1}\right\rceil+\dots+\left\lceil\tfrac{1}{2^{m-1}}c^{-1}\right\rceil,$

then the density is at least $2^{m}\delta$. The density therefore exceeds one
if the number of iterations exceeds $2c^{-1}+\log_{2}(\delta^{-1})$. Since
this cannot happen, it follows that there exists
$n\leqslant\log_{2}(\delta^{-1})+O(1)$ such that the procedure terminates at
stage $n$.

At the point of termination, the smallness assumption (2.3) must hold, so that

$N_{n}\leqslant\exp\left(O\Bigl{(}\delta^{-O(1)}\Bigr{)}\right).$

On the other hand, iteratively applying the lower bound (iv), we have

$\begin{split}N_{n}&\geqslant\frac{N_{n-1}^{1/2}}{\left(q_{0}^{2^{n-1}}\dotsm
q_{n-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}\\\ &\geqslant
N^{1/2^{n}}\left[q_{0}^{2^{n-1}}\dotsm
q_{n-1}\exp\left(\delta^{-O(1)}\right)\right]^{-O(1+\frac{1}{2}+\frac{1}{4}+\dots+2^{1-n})}\\\
&\gg\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{\Omega(\delta)},\end{split}$

where we use the upper bound (ii) on the $q_{i}$’s, together with
$n\leqslant\log_{2}(\delta^{-1})+O(1)$. Taking a logarithm and comparing upper
and lower bounds for $N_{n}$ gives $\log N\ll\delta^{-O(1)},$ which yields the
bound claimed in Theorem 1.1. ∎

##  3\. The cut norm inverse theorem

The first step of the proof of Lemma 2.1 is to use the main technical result
of [9] to prove an inverse theorem for the cut norm associated to
$\Lambda_{q,N}$, which we now define.

######  Definition 3.1 (Cut norm).

For positive integers $q\leqslant N$, we define the _cut norm_ of
$f:\mathbb{Z}\to\mathbb{C}$ by

$\left\|f\right\|_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\
|\Lambda_{q,N}(g_{1},f,g_{2})|,\ |\Lambda_{q,N}(g_{1},g_{2},f)|\\},$ (3.1)

where the supremum is taken over all 1-bounded functions
$g_{i}:[N]\to\mathbb{C}$. We note that, in spite of our nomenclature, this is
not a norm, but a seminorm. One could remedy this by summing over $y\geqslant
0$ in the counting operator (1.6).

Initially, the cut norm is too restrictive for us, so we begin by working with
the weaker quantity

$\left\|f\right\|^{\flat}_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,|\Lambda_{q,N}(g_{1},f,g_{2})|:|g_{i}|\leqslant
1\text{ and }\mathrm{supp}(g_{i})\subset[N]\\},$ (3.2)

which we refer to as the _partial cut norm_.

The following lemma is simply a rephrasing of [9, Theorem 7.1], which is the
technical heart of that paper. See Definition 1.3 for the meaning of ‘local
function’.

######  Lemma 3.2 (Partial cut norm inverse theorem).

Let $q\leqslant N$ be positive integers, $\delta>0$, and
$f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$.
Suppose that

$\left\|f\right\|^{\flat}_{q,N}\geqslant\delta.$

Then either $N\ll(q/\delta)^{O(1)}$ or there exists a 1-bounded local function
$\phi$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, modulus $qq^{\prime}$ for
some $q^{\prime}\ll\delta^{-O(1)}$, and such that

$\sum_{x\in[N]}f(x)\phi(x)\gg\delta^{O(1)}N.$

###### Proof.

By compactness, there exist 1-bounded functions $g_{1},g_{2}:[N]\to\mathbb{C}$
such that either $|\Lambda_{q,N}(f,g_{1},g_{2})|\geqslant\delta$ or
$|\Lambda_{q,N}(g_{1},f,g_{2})|\geqslant\delta.$ In the latter case, we may
apply [9, Theorem 7.1] to deduce that there exist positive integers
$q^{\prime}\ll\delta^{-O(1)}$ and $N^{\prime}\gg(\delta/q)^{O(1)}N^{1/2}$ such
that

$\sum_{x}\left|\sum_{y\in[N^{\prime}]}f(x+qq^{\prime}y)\right|\gg\delta^{O(1)}NN^{\prime}.$

In the former case, the reader may check that the argument of [9, Theorem 7.1]
delivers the same conclusion222For details see the second author’s exposition
[10]..

To ease notation, write $Q:=qq^{\prime}$. Partitioning the integers into
arithmetic progressions of length $N^{\prime}$ and common difference $Q$ gives

$\delta^{O(1)}NN^{\prime}\ll\sum_{z\in[N^{\prime}]}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|\\\
\leqslant
N^{\prime}\max_{z}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|.$

Defining $\psi_{z}(u,x)$ to be the conjugate phase of the inner sum, we deduce
the existence of $z$ for which

$\displaystyle\delta^{O(1)}N\ll\sum_{u\in[Q]}\sum_{x}\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\psi_{z}(u,x).$

The result follows on noting that every integer has a unique representation of
the form $QN^{\prime}x+u+Qy$ with $u\in[Q]$, $x\in\mathbb{Z}$ and
$y\in[N^{\prime}]$. Hence the map

$Qz+QN^{\prime}x+u+Qy\mapsto\psi_{z}(u,x)$

is a local function of resolution $QN^{\prime}$ and modulus $Q$. ∎

Now we can prove an inverse theorem for the cut norm itself.

######  Lemma 3.3 (Full cut norm inverse theorem).

Let $q\leqslant N$ be positive integers, $\delta>0$, and
$f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$.
Suppose that

$\left\|f\right\|_{q,N}\geqslant\delta.$

Then either $N\ll(q/\delta)^{O(1)}$ or there exist 1-bounded local functions
$\phi_{1}$ and $\phi_{2}$, of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and
moduli $qq_{1}$ and $qq_{2}$, respectively, for some
$q_{1},q_{2}\ll\delta^{-O(1)}$ such that

$\left|\sum_{x\in[N]}f(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$
(3.3)

###### Proof.

By the definition of the cut norm (3.1) and Lemma 3.2, we may assume that
there are 1-bounded functions $g,h:[N]\to\mathbb{C}$ such that

$|\Lambda_{q,N}(g,h,f)|\geqslant\delta.$ (3.4)

Recalling that $M:=\lfloor\sqrt{N/q}\rfloor$, define the dual function

$F(x):=\mathbb{E}_{y\in[M]}h(x+y)f(x+qy^{2}).$

Re-parametrising (3.4) and applying the Cauchy–Schwarz inequality, we have
that

$\delta^{2}\leqslant\mathbb{E}_{x\in[N]}F(x)^{2}=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}F(x)h(x+y)f(x+qy^{2}).$

Recalling the definition of the partial cut norm (3.2), we deduce that

$\left\|F\right\|_{q,N}^{\flat}\geqslant\delta^{2}.$

Applying the partial cut norm inverse theorem (Lemma 3.2), there exists a
1-bounded local function $\phi_{1}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{1}$ for some
$q_{1}\ll\delta^{-O(1)}$ such that

$\left|\sum_{x\in[N]}F(x)\phi_{1}(x)\right|\gg\delta^{O(1)}N.$

Thus

$|\Lambda_{q,N}(\phi_{1},h,f)|\gg\delta^{O(1)}.$

We now re-run our argument on $h$ instead of $f$, deducing the existence of a
1-bounded local function $\phi_{2}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{2}$ for some
$q_{2}\ll\delta^{-O(1)}$ such that

$|\Lambda_{q,N}(\phi_{1},\phi_{2},f)|\gg\delta^{O(1)}.$

Expanding the counting operator and taking a maximum over $y\in[M]$ gives

$\displaystyle\delta^{O(1)}NM$
$\displaystyle\ll\left|\sum_{y\in[M]}\sum_{x}f(x)\phi_{1}(x-qy^{2})\phi_{2}(x-qy^{2}+y)\right|$
$\displaystyle\leqslant
M\left|\sum_{x}f(x)\tilde{\phi}_{1}(x)\tilde{\phi}_{2}(x)\right|,$

where both $\tilde{\phi}_{i}$ are 1-bounded local functions of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and moduli $qq_{i}$ for some
$q_{i}\ll\delta^{-O(1)}$. ∎

##  4\. A weak regularity lemma

Much of the material is this section is standard, and closely follows the
expositions in Green [4] and Green–Tao [6]. To simplify the exposition of
later arguments, while the factors in [4] and [6] are $\sigma$-algebras, our
factors will be the set of atoms of certain $\sigma$-algebras (which can
obviously be recovered by taking the $\sigma$-algebra generated by the set of
atoms).

######  Definition 4.1 (Factor).

We define a _factor_ $\mathcal{B}$ of $[N]$ to be a partition of $[N]$, so
that $[N]=\sqcup_{B\in\mathcal{B}}B$. We say that a factor
$\mathcal{B}^{\prime}$ _refines_ $\mathcal{B}$ if every element of
$\mathcal{B}$ is a union of elements of $\mathcal{B}^{\prime}$. The _join_
$\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}$ of factors
$\mathcal{B}_{1},\dots,\mathcal{B}_{d}$ is the factor formed by taking the
$d$-fold intersections of the elements of $\mathcal{B}_{1}$, …,
$\mathcal{B}_{d}$, that is,

$\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}:=\\{B_{1}\cap\dots\cap
B_{d}:B_{i}\in\mathcal{B}_{i}\text{ for }i=1,\dots,d\\}.$

######  Definition 4.2 (Measurability, projection).

Given a factor $\mathcal{B}$, we say that a function $f:[N]\to\mathbb{C}$ is
_$\mathcal{B}$ -measurable_ if it is constant on the elements of
$\mathcal{B}$.

Define the _projection_ of any function $f:[N]\to\mathbb{C}$ onto
$\mathcal{B}$ by

$\Pi_{\mathcal{B}}f(x)=\mathbb{E}_{y\in B_{x}}f(y),$ (4.1)

where $B_{x}$ is the element of $\mathcal{B}$ that contains $x$. Notice that
$\Pi_{\mathcal{B}}f$ is $\mathcal{B}$-measurable, and is just the conditional
expectation of $f$ with respect to the $\sigma$-algebra generated by the
elements of $\mathcal{B}$.

We record some well-known properties of the projection operator
$\Pi_{\mathcal{B}}$ (that is, properties of conditional expectation) in the
next lemma.

######  Lemma 4.3 (Properties of the projection operator).

  1. (i)

The operator $\Pi_{\mathcal{B}}$ linearly projects onto the space of
$\mathcal{B}$-measurable functions.

  2. (ii)

$\Pi_{\mathcal{B}}$ is self-adjoint with respect to the inner product

$\left\langle
f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)}\qquad(f,g:[N]\to\mathbb{C}),$

so that $\left\langle
f,\Pi_{\mathcal{B}}g\right\rangle=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle$.

  3. (iii)

If $\mathcal{B}^{\prime}$ is a refinement of $\mathcal{B}$ then

$\Pi_{\mathcal{B}^{\prime}}\Pi_{\mathcal{B}}f=\Pi_{\mathcal{B}}f.$

  4. (iv)

If $\mathcal{B}^{\prime}$ refines $\mathcal{B}$ then $\Pi_{\mathcal{B}}f$ is
orthogonal to $\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$.

###### Proof.

Inspecting the formula (4.1) reveals that $\Pi_{\mathcal{B}}$ is linear, that
$\Pi_{\mathcal{B}}f$ is constant on elements of $\mathcal{B}$, and that if $f$
itself is constant on elements of $\mathcal{B}$, then $\Pi_{\mathcal{B}}f=f$.
This establishes (i).

Interchanging the order of summation gives

$\begin{split}\left\langle
f,\Pi_{\mathcal{B}}g\right\rangle=\sum_{B\in\mathcal{B}}|B|^{-1}\sum_{x,y\in
B}f(x)\overline{g(y)}=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle.\end{split}$

This proves that $\Pi_{\mathcal{B}}$ is self-adjoint.

The first refinement property follows from the fact that $\Pi_{\mathcal{B}}f$
is $\mathcal{B}^{\prime}$-measurable.

We utilise self-adjointness of $\Pi_{\mathcal{B}}$ and the first refinement
property to conclude that

$\begin{split}\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}^{\prime}}f\right\rangle&=\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-f\right\rangle=\left\langle
f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}}f\right\rangle=0.\end{split}$

∎

Now we describe the particular type of factors that will be relevant to us.

######  Definition 4.4 (Local factor).

A _simple real factor_ of resolution $M$ is a factor of $[N]$ obtained by
partitioning $\mathbb{R}$ into intervals all of length $M$.

A _simple congruence factor_ of modulus $q$ is the factor of $[N]$ obtained by
partitioning into congruence classes mod $q$.

We say that $\mathcal{B}$ is a _simple local factor_ of resolution $M$ and
modulus $q$ if it is the join of a simple real factor of resolution $M$ and a
simple congruence factor of modulus $q$. Notice that $\mathcal{B}$ is a simple
local factor if and only if it consists of the level sets of a local function
(Definition 1.3) of resolution $M$ and modulus $q$.

A _local factor_ of dimension $d$, resolution $M$ and modulus $q$ is the join
of $d$ simple local factors $\mathcal{B}_{i}$, each of resolution $M_{i}$ and
modulus $q_{i}$, where $M_{i}\geqslant M$ and
$q=\mathrm{lcm}[q_{1},\dots,q_{d}]$.

Local factors of large resolution and small modulus and dimension necessarily
contain few sets. This fact will be useful later in the proof of Lemma 2.1.

######  Lemma 4.5 (Size of a local factor).

If $\mathcal{B}$ is a local factor of dimension $d$, resolution $M$, and
modulus $q$, then

$|\mathcal{B}|\leqslant qd\left(\frac{N}{M}+2\right).$

###### Proof.

By the definition of a local factor, it suffices to bound the size of the join
of $d$ simple real factors, and then bound the size of the join of $d$ simple
congruence factors. The product of these two numbers gives us our final bound.

Joining $d$ congruence simple factors with moduli $q_{1},\dots,q_{d}$ results
in another congruence simple factor of modulus
$q=\mathrm{lcm}[q_{1},\dots,q_{d}]$. The number of parts in such a partition
is $q$.

The join of $d$ simple real factors partitions $[N]$ into intervals. The upper
endpoint of each of these intervals is either equal to $N$ or is equal to an
endpoint of an interval in one of the original simple real factors. For a
simple real factor of resolution $M$, at most $1+N/M$ upper endpoints lie in
$[1,N)$. Hence the number of intervals in the join of $d$ simple real factors
of resolutions $M_{1}$, …, $M_{d}$ is at most
$2d+N(M_{1}^{-1}+\dots+M_{d}^{-1})$.∎

We now prove a weak regularity lemma for the cut norm via an energy increment
argument.

######  Lemma 4.6 (Weak regularity).

Let $q\leqslant N$ be positive integers and $\delta>0$. Either
$N\ll(q/\delta)^{O(1)}$, or for any function $f:[N]\to[0,1]$ there exists a
local factor $\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some
$q^{\prime}\leqslant O\left(1/\delta\right)^{O(d)}$ such that

$\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.2)

###### Proof.

We run an energy increment argument, initialising at stage $0$ with the
trivial factor $\mathcal{B}_{0}:=\left\\{[N]\right\\}$. Suppose that at stage
$d$ of this iteration we have a local factor $\mathcal{B}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, dimension at most $2d$, and modulus
$qq^{\prime}$ for some $q^{\prime}\leqslant O(1/\delta)^{O(d)}$. In addition,
suppose that we have the energy lower bound

$\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}\gg d\delta^{O(1)}N.$ (4.3)

With these assumptions in place, we query if the following holds

$\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.4)

If so, then the process terminates. If not, we show how our iteration may
proceed to stage $d+1$.

Applying the cut norm inverse theorem (Lemma 3.3), we conclude that there
exist 1-bounded local functions $\phi_{i}$ of resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{i}$ for some
$q_{i}\leqslant\delta^{-O(1)}$ such that

$\left|\left\langle
f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|=\left|\sum_{x\in[N]}(f-\Pi_{\mathcal{B}}f)(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$

Let $\mathcal{B}^{\prime}$ denote the join of $\mathcal{B}$ and the simple
local factors generated by $\phi_{1}$ and $\phi_{2}$, so that
$\mathcal{B}^{\prime}$ is a local factor of dimension at most $2(d+1)$,
resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq^{\prime\prime}$ for
some $q^{\prime\prime}\leqslant q^{\prime}q_{1}q_{2}\leqslant
O(1/\delta)^{O(d+1)}$. Since $\phi_{1}\phi_{2}$ is
$\mathcal{B}^{\prime}$-measurable, we can use the properties listed in Lemma
4.3 together with the Cauchy–Schwarz inequality to deduce that

$\begin{split}\left|\left\langle
f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|&=\left|\left\langle
f-\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}^{\prime}}(\phi_{1}\phi_{2})\right\rangle\right|=\left|\left\langle\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|\\\
&\leqslant
N^{1/2}\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}.\end{split}$

It follows that

$\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}\gg\delta^{O(1)}N^{1/2}.$

Lemma 4.3 (iv) tells us that $\Pi_{\mathcal{B}}f$ is orthogonal to
$\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$, hence by Pythagoras’s
theorem

$\left\|\Pi_{\mathcal{B}^{\prime}}f\right\|_{\ell^{2}}^{2}=\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}+\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}.$

The energy bound (4.3) follows for $\mathcal{B}^{\prime}$, allowing us to
proceed to the next stage of our iteration.

Since the function $f$ is $1$-bounded, the projection $\Pi_{\mathcal{B}}f$ is
also 1-bounded, hence the energy (4.3) is always bounded above by $N$. It
follows that this energy increment must terminate at stage $d$ for some
$d\ll\delta^{-O(1)}$, yielding the lemma.∎

##  5\. The density increment lemma

In this section we prove Lemma 2.1, modelling our argument on that given by
Green and Tao [6, Corollary 5.8]. We first record, for the sake of
convenience, the following immediate consequence of the triangle inequality.

######  Lemma 5.1 ($\ell^{1}$-control).

Suppose that $N\geqslant q$. Then for any $f_{0},f_{1},f_{2}:[N]\to\mathbb{C}$
we have

$|\Lambda_{q,N}(f_{0},f_{1},f_{2})|\leqslant
N^{-1}\left\|f_{i}\right\|_{\ell^{1}}\prod_{j\neq
i}\left\|f_{j}\right\|_{\infty}.$

###### Proof.

We prove the result for $i=1$, the other cases being similar. A
reparametrisation gives

$\displaystyle\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|$
$\displaystyle=\left|\mathbb{E}_{x\in[N]}f_{1}(x)\mathbb{E}_{y\in[M]}f_{0}(x-y)f_{2}(x+qy^{2}-y)\right|$
$\displaystyle\leqslant\mathbb{E}_{x\in[N]}|f_{1}(x)|\mathbb{E}_{y\in[M]}|f_{0}(x-y)||f_{2}(x+qy^{2}-y)|.$

∎

We are now in a position to prove Lemma 2.1, and thereby complete our proof of
Theorem 1.1.

###### Proof of Lemma 2.1.

Let $A$ satisfy the assumptions of Lemma 2.1. Increasing $\delta$ only
strengthens our conclusion, so we may assume that $|A|=\delta N$. Since
$\Lambda_{q,N}(1_{A})=0$, we have that
$\left|\Lambda_{q,N}(1_{A})-\Lambda_{q,N}(\delta
1_{[N]})\right|=\delta^{3}\Lambda_{q,N}(1_{[N]})\gg\delta^{3}$.

Applying the weak regularity lemma (Lemma 4.6), there exists a local factor
$\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution
$\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some
$q^{\prime}\leqslant O(1/\delta)^{O(d)}$ such that

$\left\|1_{A}-\Pi_{\mathcal{B}}1_{A}\right\|_{q,N}\leqslant\tfrac{1}{6}\delta^{3}\Lambda_{q,N}({1_{[N]}}).$

Setting $f:=\Pi_{\mathcal{B}}1_{A}$, a telescoping identity thus yields

$\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(\delta
1_{[N]})\right|\geqslant\tfrac{1}{2}\delta^{3}\Lambda_{q,N}({1_{[N]}})\gg\delta^{3}.$

Define the $\mathcal{B}$-measurable set

$S:=\left\\{x\in[N]:f(x)\geqslant(1+c)\delta\right\\},$

where $c>0$ is a sufficiently small absolute constant that will be chosen to
make the following argument valid. By Lemma 5.1 and a telescoping identity, we
have $\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(f1_{S^{c}})\right|\leqslant
3|S|/N$, so that

$\tfrac{|S|}{N}+\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta
1_{[N]})\right|\gg\delta^{3}.$

Yet another telescoping identity, in conjunction with Lemma 5.1, gives

$\displaystyle\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta
1_{[N]})\right|$
$\displaystyle\ll\tfrac{\delta^{2}}{N}\left\|f1_{S^{c}}-\delta
1_{[N]}\right\|_{\ell^{1}}\leqslant\tfrac{\delta^{2}}{N}\left\|f-\delta
1_{[N]}\right\|_{\ell^{1}}+\tfrac{|S|}{N},$

so that

$|S|+\delta^{2}\left\|f-\delta 1_{[N]}\right\|_{\ell^{1}}\gg\delta^{3}N.$

Since $f-\delta 1_{[N]}$ has mean zero, its $\ell^{1}$-norm is equal to twice
the $\ell^{1}$-norm of its positive part. The function $\left(f-\delta
1_{[N]}\right)_{+}$ can only exceed $c\delta$ on $S$, so taking $c$ small
enough gives $|S|\gg\delta^{3}N$. Letting $B$ denote the largest element of
$\mathcal{B}$ for which $B\subset S$, the bound in Lemma 4.5 yields

$|B|\gg q^{-O(1)}\delta^{O(d)}2^{-O(d)}N^{1/2}.$

By construction (see Definition 4.4), the set $B$ is an arithmetic progression
of common difference $qq^{\prime}$ with $q^{\prime}\leqslant
O(1/\delta)^{O(d)}$. Moreover, the density of $A$ on $B$ is equal to the value
of $f(x)$ for any $x\in B$, and this is at least $(1+c)\delta$ by the
definition of $S$. ∎

##  6\. Global control by major arc Fourier coefficients

The purpose of this section is to prove Theorem 1.2 and Corollary 1.4. We
begin with an alternative version of Lemma 3.2, replacing the rigid local
function found therein with something more continuous.

######  Definition 6.1 ($C$-Lipschitz).

We say that $\phi:\mathbb{Z}\to\mathbb{C}$ is _$C$ -Lipschitz along
$q\cdot\mathbb{Z}$_ if for any $x,y\in\mathbb{Z}$ we have

$|\phi(x+qy)-\phi(x)|\leqslant C|y|.$

Recalling our definition for the Fejér kernel (1.7), we observe that a
function of the form

$x\mapsto\sum_{h}\mu_{H}(h)f(x+qh)$ (6.1)

is Lipschitz along $q\cdot\mathbb{Z}$.

######  Lemma 6.2.

Let $q,H$ be positive integers and $f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded.
If $\phi$ is defined as in (6.1), then $\phi$ is $O(H^{-1})$-Lipschitz along
$q\cdot\mathbb{Z}$.

###### Proof.

Recalling (1.7), the triangle inequality for $|\cdot|$ and $\max\\{\cdot,0\\}$
show that $|\mu_{H}(h+y)-\mu_{H}(h)|\leqslant|y|/\left\lfloor
H\right\rfloor^{2}$ for all $h,y\in\mathbb{Z}$. Hence a change of variables
gives

$|\phi(x+qy)-\phi(x)|\leqslant\sum_{h}|\mu_{H}(h-y)-\mu_{H}(h)|\ll\frac{|y|}{H^{2}}\sum_{h\in(-H,H)\cup(y-H,y+H)}1.$

∎

Now we prove another partial cut norm inverse theorem, this time getting
correlation with functions that are Lipschitz along progressions with small
common difference.

######  Lemma 6.3 (Partial cut norm inverse theorem II).

Let $N$ be a positive integer, $\delta>0$, and $f,g,h:\mathbb{Z}\to\mathbb{C}$
be $1$-bounded functions with support in $[N]$. Suppose that

$\left|\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/2}]}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta.$

Then either $N\ll\delta^{-O(1)}$, or there exists $q\ll\delta^{-O(1)}$ and a
1-bounded function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along
$q\cdot\mathbb{Z}$ such that

$\sum_{x\in[N]}g(x)\phi(x)\gg\delta^{O(1)}N.$

###### Proof.

Applying [9, Theorem 7.1], we obtain positive integers $q\ll\delta^{-O(1)}$
and $N^{1/2}\geqslant M\gg\delta^{O(1)}N^{1/2}$ such that

$\sum_{x}\left|\sum_{y\in[M]}g(x+qy)\right|\gg\delta^{O(1)}NM.$

By the Cauchy–Schwarz inequality and a change of variables, we have

$\sum_{x}g(x)\sum_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))}\gg\delta^{O(1)}NM^{2}.$

Setting

$\phi(x):=\mathbb{E}_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))},$

Lemma 6.2 shows this function has the required properties. ∎

Before proving Theorem 1.2, we record two standard facts.

######  Lemma 6.4.

There are at most $O(N^{4})$ solutions $x\in[N]^{6}$ to the equation

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=x_{4}^{2}+x_{5}^{2}+x_{6}^{2}.$

###### Proof.

There are a number of ways to prove this. Perhaps the most robust is via the
circle method, see [3]. The result can be read out of [1, Proposition 1.10]. ∎

######  Lemma 6.5 (Weyl’s inequality).

Let $P\subset\mathbb{Z}$ be an arithmetic progression with common difference
$q$ and let $0<\delta\leqslant 1$. Suppose that

$\left|\sum_{x\in P}e(\alpha x^{2})\right|\geqslant\delta|P|.$

Then either $|P|\ll\delta^{-O(1)}$ or there exists a positive integer
$q^{\prime}\ll\delta^{-O(1)}$ such that

$\|q^{\prime}q^{2}\alpha\|\ll\delta^{-O(1)}|P|^{-2}.$

###### Proof.

Let $P=x_{0}+q\cdot[N]$, so that our exponential sum becomes

$\sum_{x\in P}e(\alpha x^{2})=\sum_{y\in[N]}e(\alpha q^{2}y^{2}+2\alpha
qx_{0}y+\alpha x_{0}^{2}).$

Applying [5, Lemma A.11], either $N\ll\delta^{-O(1)}$ or the conclusion of our
lemma follows. ∎

###### Proof of Theorem 1.2.

Write $\Lambda_{N}$ for the counting operator $\Lambda_{1,N}$ (that is, the
average (1.6) with $q=1$). Let $f,g,h:[N]\to\mathbb{C}$ be 1-bounded functions
satisfying

$|\Lambda_{N}(f,g,h)|\geqslant\delta.$

Define the seminorm

$\left\|g\right\|:=\sup\left\\{|\Lambda_{N}(g_{1},g,g_{2})|:|g_{i}|\leqslant
1\text{ and }\mathrm{supp}(g_{i})\subset[N]\right\\}.$

and the dual function

$F(x):=\mathbb{E}_{y\in[N^{1/2}]}f(x-y)h(x+y^{2}-y).$

We follow the argument in the proof of Lemma 3.3 to deduce that

$\left\|F\right\|\geqslant\delta^{2}.$

Hence, by Lemma 6.3, there exists $q\ll\delta^{-O(1)}$ and a 1-bounded
function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along
$q\cdot\mathbb{Z}$ and satisfies

$\sum_{x\in[N]}F(x)\phi(x)\gg\delta^{O(1)}N.$

Expanding the definition of the dual function, we have

$\sum_{x\in[N]}\sum_{y\in[N^{1/2}]}f(x)\phi(x+y)h(x+y^{2})\gg\delta^{O(1)}N^{3/2}.$

Let us partition $\mathbb{Z}$ into arithmetic progressions $P$ each of common
difference $q$ and length $M$, where $M$ will be chosen shortly. For each such
arithmetic progression $P$, fix an element $y_{P}\in P$. Using the Lipschitz
property of $\phi$, for any $x\in\mathbb{Z}$ and $y\in P$ we have

$|\phi(x+y_{P})-\phi(x+y)|\ll\delta^{-O(1)}MN^{-1/2}.$

Hence,

$\left|\sum_{P}\sum_{x\in[N]}\sum_{y\in
P\cap[N^{1/2}]}f(x)[\phi(x+y)-\phi(x+y_{P})]h(x+y^{2})\right|\ll\delta^{-O(1)}MN.$

We can therefore take $M$ sufficiently small to satisfy both
$M\gg\delta^{O(1)}N^{1/2}$ and

$\left|\sum_{P}\sum_{x}\sum_{y\in
P\cap[N^{1/2}]}f(x)\phi(x+y_{P})h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$

Set $f_{P}(x):=f(x)\phi(x+y_{P})$. The number of progressions $P$ that
intersect $[N^{1/2}]$ is at most $O(N^{1/2}M^{-1}+q)=O(\delta^{-O(1)})$.
Therefore, the pigeon-hole principle gives a progression $P$ for which

$\left|\sum_{x}\sum_{y\in
P\cap[N^{1/2}]}f_{P}(x)h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$ (6.2)

In particular, $|P\cap[N^{1/2}]|\gg\delta^{O(1)}N^{1/2}$.

Writing $S_{P}(\alpha)$ for $\sum_{y\in P\cap[N^{1/2}]}e\left(\alpha
y^{2}\right)$, the orthogonality relations allow us to reformulate (6.2) as

$\displaystyle\left|\int_{\mathbb{T}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$

Let $\eta>0$ be a parameter to be determined shortly, and define the major
arcs

$\mathfrak{M}:=\left\\{\alpha\in\mathbb{T}:|S_{P}(\alpha)|\geqslant\eta
N^{1/2}\right\\}.$

Parseval’s identity then gives

$\left|\int_{\mathbb{T}\setminus\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\eta
N^{1/2}\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}\leqslant\eta
N^{3/2}.$

Hence we may take $\eta\gg\delta^{O(1)}$ and ensure that

$\displaystyle\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$

By Lemma 6.4 and orthogonality, we have $\left\|S_{P}\right\|_{6}\ll N^{1/3}$.
Thus, by Hölder’s inequality, we get that

$\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}^{2/3}\big{\|}S_{P}\big{\|}_{6}\sup_{\alpha\in\mathfrak{M}}\bigl{|}\hat{h}(-\alpha)\bigr{|}^{1/3}.$

We therefore deduce that there exists $\alpha\in\mathfrak{M}$ such that

$\bigl{|}\hat{h}(-\alpha)\bigr{|}\gg\delta^{O(1)}N.$

Finally, an application of Weyl’s inequality (Lemma 6.5) shows that if
$-\alpha\in\mathfrak{M}$ then $\alpha$ has the required Diophantine
approximation property. ∎

###### Proof of Corollary 1.4.

Let $\alpha\in\mathbb{R}$ be the frequency and $q$ the positive integer
provided by Theorem 1.2. For any integer $a$ and positive integer $M$, if
$x,y\in a+q\cdot[M]$, then

$\left|e(\alpha x)-e(\alpha y)\right|\leqslant
2\pi\left\|\alpha(x-y)\right\|\ll\delta^{-O(1)}MN^{-1}.$

Partitioning $\mathbb{Z}$ into arithmetic progressions of common difference
$q$ and length $M$ then gives

$\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}+\delta^{-O(1)}M.$

We thus take $M\gg\delta^{O(1)}N$ sufficiently small to ensure that

$\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}.$

Write $\theta_{P}$ for the conjugate phase of the inner sum. Then the map
$x\mapsto\sum_{P}\theta_{P}1_{P}(x)$ is a local function of resolution
$\gg\delta^{O(1)}N$ and modulus $\ll\delta^{-O(1)}$, yielding the corollary. ∎

##  7\. Longer progressions

As mentioned in §1.3, the main obstacle to generalising our polylogarithmic
bound to longer configurations such as (1.2) is in obtaining an appropriate
generalisation of Lemma 3.3; in particular, showing that if the relevant
counting operator is large, then _all_ functions must correlate with a product
of a bounded number of local functions.

Let us demonstrate where the argument breaks down for $m>2$. Given polynomials
as in (1.2) and 1-bounded functions
$f_{0},f_{1},\dots,f_{m}:[N]\to\mathbb{C}$, define the counting operator

$\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m}):=\\\
\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/\deg
P_{m}}]}f_{0}(x)f_{1}(x+P_{1}(y))\dotsm f_{m}(x+P_{m}(y)).$

Using the main technical result of [8], [8, Theorem 3.3], one can show that if

$\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})\right|\geqslant\delta,$

then both $f_{0}$ and $f_{1}$ correlate with local functions $\phi_{0}$ and
$\phi_{1}$. Combining this with a dual function argument, as in our proofs of
Theorem 1.2 and Lemma 3.3, one may conclude that

$\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(\phi_{0},\phi_{1},f_{2},\dots,f_{m})\right|\gg\delta^{O(1)},$

If $m=2$, one can then pigeon-hole in the smaller $y$ variable appearing in
the counting operator (as we do in the proof of Lemma 3.3) to conclude that
$f_{2}$ correlates with a product of two local functions. It is this simple
pigeon-holing argument that fails when $m>2$.

###  7.1. An alternative strategy for longer progressions

A more productive strategy is to follow our proof of Theorem 1.2 instead of
Theorem 1.1. In proving Theorem 1.2 we replace the counting operator
$\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$ with
$\Lambda_{y,y^{2}}^{N}(f_{0},\phi,f_{2})$, where $\phi$ is a local function
that is constant on progressions of length $\approx N^{1/2}$ with common
difference of size $\approx O(1)$. Provided that we pass to appropriate
subprogressions in all of the variables appearing in our counting operator, we
can exploit the properties of this local function and ‘remove’ it from our
count. In effect (after passing to subprogressions of bounded common
difference), we replace the count $\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$
with one of the form $\Lambda_{Q}^{N^{\prime}}(f_{0},f_{2})$, where $Q$ is a
quadratic polynomial and $N^{\prime}$ is slightly smaller than $N$.

Generalising this approach, one can use [8, Theorem 3.3] to replace the
counting operator $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})$
with $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$, where
$\phi$ is a local function. Provided that this local function has resolution
$\gg N^{\deg P_{1}/\deg P_{m}}$ and common difference $q\ll 1$, we have

$\phi(x+P_{1}(y))\approx\phi(x)$

for any $x\in\mathbb{Z}$ and any $y$ constrained to a subprogression of common
difference $q$ and length $\approx N^{\deg P_{1}/\deg P_{m}}$. Passing to
subprogressions in $x$ and $y$, one should then be able to replace the
operator

$\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$

by one of the form

$\Lambda_{Q_{2},\dots,Q_{m}}^{N^{\prime}}(f_{0},f_{2},\dots,f_{m}).$

Applying induction on $m$ may then allow one to show that every function in
the original counting operator correlates with a local function.

The main impediment to carrying out this strategy is that the polynomials
$Q_{2}$, …, $Q_{m}$, which arise on passing to a subprogression, may not
satisfy the hypotheses required to reapply [8, Theorem 3.3]. It is likely that
the polynomials are sufficiently well-behaved for the arguments of [8] to
remain valid, but we leave this verification to the energetic reader.

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