# Diophantine approximation with a quaternary problem

Alessandro Gambini

###### Abstract.

Let $1<k<7/6$, $\lambda_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$ be non-
zero real numbers, not all of the same sign such that
$\lambda_{1}/\lambda_{2}$ is irrational and let $\omega$ be a real number. We
prove that the inequality
$|\lambda_{1}p_{1}^{2}+\lambda_{2}p_{2}^{2}+\lambda_{3}p_{3}^{2}+\lambda_{4}p_{4}^{k}-\omega|\leq(\max_{j}p_{j})^{-\frac{7-6k}{14k}+\varepsilon}$
has infinitely many solutions in prime variables $p_{1},p_{2},p_{3},p_{4}$ for
any $\varepsilon>0$.

##  1\. Introduction

Numerous recent papers have explored a Diophantine inequality involving prime
variables, each with a unique set of assumptions and conclusions. In their
work, Brüdern, Cook, and Perelli [1], focused on binary linear forms in prime
arguments. Cook and Fox [4], addressed a ternary form with primes squared, and
this was subsequently improved in terms of approximation by Harman in [12].
Cook [3], provided a more comprehensive description of the problem, which was
later refined by Cook and Harman [5].

There are several distinctions between the results mentioned above and the
scope of our research. Notably, in their papers, the assumption that all
coefficients $\lambda_{j}$ are positive is not a constraint. Additionally, the
values of $k_{j}$ are consistent positive integers for all $j$. However, the
pivotal aspect remains the requirement that $\lambda_{1}/\lambda_{2}$ must be
irrational. In our case, we will prove that there are infinitely many solution
to a the problem of the form

$\displaystyle|\lambda_{1}p_{1}^{k_{1}}+\cdots+\lambda_{r}p_{r}^{k_{r}}-\omega|\leq\eta$

when $\eta$ depends on the maximum of the $p_{j}$, whereas in the previously
cited papers, $\eta$ is a small negative power of $\omega$.

Vaughan [23] follows a similar approach to the one we employ in our article,
dealing with a ternary linear form in prime arguments and assuming more
suitable conditions on the $\lambda_{j}$. He proved that there are infinitely
many solutions to the problem:

$\displaystyle|\lambda_{1}p_{1}+\lambda_{2}p_{2}+\lambda_{3}p_{3}-\omega|\leq\eta$

when $\eta$ depends on the maximum of the $p_{j}$. In his case,
$\eta=(\max_{j}p_{j})^{-\frac{1}{10}}$. This result was enhanced by Baker and
Harman [2] with an exponent of $-\frac{1}{6}$, by Harman [11] with an exponent
of $-\frac{1}{5}$, and finally by Matomäki [19] with an exponent of
$-\frac{2}{9}$.

Languasco and Zaccagnini, in [16] and [15], examined a ternary problem with
varying powers $k_{j}$, one of which depended on a parameter $k$.
Additionally, Gambini, Languasco, and Zaccagnini [8], analyzed a ternary
problem involving two primes and a $k$-th power of a prime. In all these
cases, the value of $\eta$ still depends on the primes $p_{j}$ also contingent
on the parameter $k$. The concept in this scenario is to optimize the value of
$k$ to maximize the range in which the inequality holds.

Languasco and Zaccagnini also addressed a quaternary form [14] that involved a
prime and three squares of primes, resulting in
$\eta=(\max_{j}p_{j})^{-\frac{1}{18}}$. This was improved by Li and Wang [17]
and later by Liu and Sun in [18] with $\eta=(\max_{j}p_{j})^{-\frac{1}{16}}$
using the Harman technique. Mu [20] investigated a problem with five variables
comprising four squares of primes and a $k$-th power of a prime, optimizing
the value of $k$. Ge and Li [9], utilized a quaternary form with varying
integer powers $k_{j}$. Gambini [7], explored a quaternary problem featuring
one prime, two squares of primes, and a $k$-th power of a prime, while Gao and
Liu [10] and later Mu, Zhu, and Li [21] examined a problem with four squares
of a prime and a $k$-th power of a prime.

The case of this paper involves three squares of primes and one $k$-th power
of a prime. We prove the following theorem:

######  Theorem 1.

Assume that $1<k<7/6$, $\lambda_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$
be non-zero real numbers, not all of the same sign, that
$\lambda_{1}/\lambda_{2}$ is irrational and let $\omega$ be a real number. The
inequality

$\displaystyle\left|\lambda_{1}p_{1}^{2}+\lambda_{2}p_{2}^{2}+\lambda_{3}p_{3}^{2}+\lambda_{4}p_{4}^{k}-\omega\right|\leq\left(\max_{j}p_{j}\right)^{-\frac{7-6k}{14k}+\varepsilon}$
(1)

has infinitely many solutions in prime variables $p_{1},p_{2},p_{3},p_{4}$ for
any $\varepsilon>0$.

##  2\. Outline of the proof

We use a variant of the classical circle method that was introduced by
Davenport and Heilbronn in 1946 [6] substituting the integration over a
circle, or equivalently over the interval $[0,1]$, with integration across the
whole real line.

In this paper, we denote prime numbers as $p$ and $p_{i}$, where $k\geq 1$ is
a real number, $\varepsilon$ represents a minute positive value whose
specifics might vary depending on occurrences, and $\omega$ is a fixed real
number. To establish the existence of infinitely many solutions for (1), it
suffices to create an increasing sequence $X_{n}$ that grows towards infinity,
ensuring that (1) has at least one solution with $\max p_{j}\in[\delta
X_{n},X_{n}]$, with $\delta>0$, a fixed value contingent upon the choice of
$\lambda_{j}$. Consider $q$ as the denominator of a convergent to
$\lambda_{1}/\lambda_{2}$, with $X_{n}=X$ (omitting the subscript $n$), and
traverses the sequence $X=q^{7/3}$. We set

$\displaystyle S_{k}(\alpha)=\sum_{\delta X\leq p^{k}\leq X}\log p\
e(p^{k}\alpha),$ (2) $\displaystyle U_{k}(\alpha)=\sum_{\delta X\leq n^{k}\leq
X}e(n^{k}\alpha),$ $\displaystyle T_{k}(\alpha)=\int_{(\delta
X)^{\frac{1}{k}}}^{X^{\frac{1}{k}}}e(\alpha t^{k})\,\mathrm{d}t,$ (3)

where $e(\alpha)=e^{2\pi i\alpha}$.

To obtain the most accurate estimate, we utilize the sieve function $\rho(m)$
as defined in (5.2) of [13] introduced by Harman and Kumchev and employed by
Wang and Yao in [25] for the case $k=2$. This function serves as a non-trivial
lower bound for the characteristic function of primes. It enables the
definition of an exponential function (4) with a distinct weight:

$\rho(m)=\psi(m,X^{5/42})-\sum_{X^{5/42}\leq p<X^{1/4}}\psi(m/p,z(p)),$

where

$\psi(m,z)=\left\\{\begin{array}[]{ll}1&\text{if }p|m\ \Rightarrow\ p\geq
z,\\\ 0&\text{otherwise}\end{array}\right.$

and

$z(p)=\left\\{\begin{array}[]{ll}X^{5/28}p^{-1/2}&\text{if }p<X^{1/7},\\\
p&\text{if }X^{1/7}\leq p\leq X^{3/14},\\\ X^{5/14}p^{-1}&\text{if
}p>X^{3/14}.\end{array}\right.$

The crucial property of $\rho(m)$ we focus on is the estimation (2.3) in [25]:

$\displaystyle\sum_{m\in I}\rho(m)=\ell|I|(\log X)^{-1}+O(X^{1/2}(\log
X)^{-2}),$

where $\ell>0$ is an absolute constant and $I$ is any subinterval of $[(\delta
X)^{1/2},X^{1/2}]$. Based on this, we define the following exponential
function:

$\displaystyle\widetilde{S}_{2}(\alpha)=\sum_{\delta X\leq m^{2}\leq
X}\rho(m)e(m^{2}\alpha).$ (4)

We will approximate $S_{k}$ with $T_{k}$ or $U_{k}$ and we will approximate
$\widetilde{S}_{2}$ with $T_{2}$.

By the Prime Number Theorem and first derivative estimates for trigonometric
integrals we establish

$\displaystyle S_{k}(\alpha)\ll
X^{\frac{1}{k}},\qquad\widetilde{S}_{2}(\alpha)\ll X^{\frac{1}{2}},\qquad
T_{k}(\alpha)\ll_{k,\delta}X^{\frac{1}{k}-1}\min(X,|\alpha|^{-1}),$ (5)

where $k\geq 1$ and $\delta>0$ are real numbers.

Moreover the Euler summation formula implies that, for $k\geq 1$,

$\displaystyle T_{k}(\alpha)-U_{k}(\alpha)\ll 1+|\alpha|X.$ (6)

We also require a continuous function to identify solutions of (1). Hence, we
introduce

$\displaystyle\widehat{K}_{\eta}(\alpha):=\max\\{0,\eta-|\alpha|\\}\quad\mbox{where}\quad\eta>0$

whose inverse Fourier transform is

$\displaystyle
K_{\eta}(\alpha)=\left(\frac{\sin(\pi\alpha\eta)}{\pi\alpha}\right)^{2}$

for $\alpha\neq 0$ and, by continuity, $K_{\eta}(0)=\eta^{2}$. It vanishes at
infinity like $|\alpha|^{-2}$ and in fact it is trivial to prove that

$\displaystyle K_{\eta}(\alpha)\ll\min(\eta^{2},|\alpha|^{-2}).$ (7)

The original works of Davenport-Heillbronn in [6] and later Vaughan in [23]
and [24] directly approximate the difference $|S_{k}(\alpha)-T_{k}(\alpha)|$,
estimating it as $O(1)$ using the Euler summation formula. Brüdern, Cook, and
Perelli in [1] enhanced these estimations by computing the $L^{2}$-norm of
$|S_{k}(\alpha)-T_{k}(\alpha)|$, leading to substantially improved conditions
and a broader major arc compared to the original approach. Introducing the
generalized version of the Selberg integral

$\displaystyle\mathcal{J}_{k}(X,h)=\int_{X}^{2X}\left(\theta((x+h)^{\frac{1}{k}})-\theta(x^{\frac{1}{k}})-((x+h)^{\frac{1}{k}}-x^{\frac{1}{k}})\right)^{2}\mathrm{d}x,$

where $\theta$ is the Chebyshev Theta function,

$\theta(x)=\sum_{p\leq x}\log p,$

we have the following lemmas.

######  Lemma 1 ([16], Lemma 1).

Let $k\geq 1$ be a real number. For $0<Y<\frac{1}{2}$, we have

$\displaystyle\int_{-Y}^{Y}|S_{k}(\alpha)-U_{k}(\alpha)|^{2}\mathrm{d}\alpha\ll_{k}\frac{{X^{\frac{2}{k}-2}}\log^{2}X}{Y}+Y^{2}X+Y^{2}\mathcal{J}_{k}\left(X,\frac{1}{2Y}\right).$

######  Lemma 2 ([16], Lemma 2).

Let $k\geq 1$ be a real number and $\varepsilon$ be an arbitrarily small
positive constant. There exists a positive constant $c_{1}(\varepsilon)$,
which does not depend on $k$, such that

$\displaystyle\mathcal{J}_{k}(X,h)\ll_{k}h^{2}X^{\frac{2}{k}-1}\exp\left(-c_{1}\left(\frac{\log
X}{\log\log X}\right)^{\frac{1}{3}}\right)$

uniformly for $X^{1-\frac{5}{6k}+\varepsilon}\leq h\leq X$.

###  2.1. Setting the problem

Let

$\displaystyle\mathcal{P}(X)=\\{(p_{1},p_{2},p_{3},p_{4}):\delta
X<p_{1}^{2},p_{2}^{2},p_{3}^{2},p_{4}^{k}<X\\}$

and let us define

$\displaystyle\mathcal{I}(\eta,\omega,\mathfrak{X})=\int_{\mathfrak{X}}\widetilde{S}_{2}(\lambda_{1}\alpha)S_{2}(\lambda_{2}\alpha)S_{2}(\lambda_{3}\alpha)S_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$

where $\mathfrak{X}$ is a measurable subset of $\mathbb{R}$.

It follows from the construction of $\rho(m)$ that, if $\omega(m)$ denotes the
characteristic function of the set of primes,

$\rho(m)\leq\omega(m).$

Then, from the definitions of $\widetilde{S}_{2}(\lambda_{1}\alpha)$ and
$S_{j}(\lambda_{i}\alpha)$, and performing the Fourier transform for
$K_{\eta}(\alpha)$, we obtain

$\displaystyle\mathcal{I}(\eta,\omega,\mathbb{R})$
$\displaystyle=\sum_{p_{i}\in\mathcal{P}(X)}\rho(m_{1})\log p_{2}\log
p_{3}\log p_{4}\cdot$
$\displaystyle\quad\left(\max(0,\eta-\left|\lambda_{1}m_{1}^{2}+\lambda_{2}p_{2}^{2}+\lambda_{3}p_{3}^{2}+\lambda_{4}p_{4}^{k}-\omega\right|)\right)$
$\displaystyle\leq\eta(\log X)^{3}\mathcal{N}(X),$

where $\mathcal{N}(X)$ denotes the number of solutions of the inequality (1)
with $(p_{1},p_{2},p_{3},p_{4})\in\mathcal{P}(X)$. In other words
$\mathcal{I}(\eta,\omega,\mathbb{R})$ provides a lower bound for the quantity
we are interested in; therefore it is sufficient to prove that
$\mathcal{I}(\eta,\omega,\mathbb{R})>0$.

Next, we partition $\mathbb{R}$ into subsets $\mathfrak{M}$, $\mathfrak{m}$,
and $t$, where $\mathbb{R}=\mathfrak{M}\cup\mathfrak{m}\cup t$, with
$\mathfrak{M}$ as the major arc, $\mathfrak{m}$ as the minor arc (or
intermediate arc), and $t$ as the trivial arc, defined as follows:

$\displaystyle\mathfrak{M}=\left[-\frac{P}{X},\frac{P}{X}\right]\qquad\mathfrak{m}=\left[-R,-\frac{P}{X}\right]\cup\left[\frac{P}{X},R\right]\qquad
t=\mathbb{R}\backslash(\mathfrak{M}\cup\mathfrak{m}),$

so that
$\mathcal{I}(\eta,\omega,\mathbb{R})=\mathcal{I}(\eta,\omega,\mathfrak{M})+\mathcal{I}(\eta,\omega,\mathfrak{m})+\mathcal{I}(\eta,\omega,t)$.

The parameters $P=P(X)\gg\log(X)$, and $R=R(X)>1/\eta$ are chosen later (see
(11) and (14)), along with $\eta=\eta(X)$, which, as previously mentioned, we
desire to be a small negative power of $\max p_{j}$ and therefore of $X$ as in
(19).

We anticipate having the main term with the correct order of magnitude on
$\mathfrak{M}$ without any special hypotheses on the coefficients
$\lambda_{j}$. It is crucial to prove that $\mathcal{I}(\eta,\omega,m)$ and
$\mathcal{I}(\eta,\omega,t)$ are both
$o(\mathcal{I}(\eta,\omega,\mathfrak{M}))$: the contribution from the trivial
arc is significantly smaller in comparison to the main term. The main
challenge lies within the minor arc, where we will require the complete power
of the assumptions on the $\lambda_{j}$ and the theory of continued fractions.

Remark: From this point forward, whenever we use the symbols $\ll$ or $\gg$,
we omit the dependence of the approximation on the constants
$\lambda_{j},\delta$, and $k$.

###  2.2. Lemmas

In this paper we will also use Lemmas 5 of [8] and (2.5) of [25] that allow us
to have an estimation of mean value of $|S_{k}(\alpha)|^{4}$ and
$|\widetilde{S}_{2}(\alpha)|^{4}$:

######  Lemma 3 ([8], Lemma 5).

Let $k>1$, $\tau>0$. We have

$\displaystyle\int_{-\tau}^{\tau}|S_{k}(\alpha)|^{2}\mathrm{d}\alpha\ll\left(\tau
X^{1/k}+X^{2/k-1}\right)(\log
X)^{3}\qquad\int_{n}^{n+1}|S_{k}(\alpha)|^{2}\mathrm{d}\alpha\ll X^{1/k}(\log
X)^{3}.$

Finally, we will use the following Lemma.

######  Lemma 4.

$\displaystyle\int_{0}^{1}|S_{2}(\alpha)|^{4}\mathrm{d}\alpha\ll
X\log^{2}X\qquad\int_{\mathbb{R}}|S_{2}(\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha\ll\eta
X\log^{2}X.$
$\displaystyle\int_{0}^{1}|\widetilde{S}_{2}(\alpha)|^{4}\mathrm{d}\alpha\ll
X(\log
X)^{c}\qquad\int_{\mathbb{R}}|\widetilde{S}_{2}(\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha\ll\eta
X(\log X)^{c}.$

###### Proof.

The first two statements are based on Satz 3 of [22], p. 94 and the last two
derive directly from (2.5) of [25]. ∎

##  3\. The major arc

We begin with the major arc and the computation of the main term. Substituting
all $S_{2}$, $S_{k}$, and $\widetilde{S}_{2}$ defined in (2) and (4) with
their respective $T_{i}$ defined in (3) brings forth some discrepancies that
require estimation using Lemma 1, the Cauchy-Schwarz inequality, and the
Hölder inequality. We proceed to calculate

$\displaystyle\mathcal{I}(\eta,\omega,$
$\displaystyle\mathfrak{M})=\int_{\mathfrak{M}}S_{2}(\lambda_{1}\alpha)S_{2}(\lambda_{2}\alpha)S_{2}(\lambda_{3}\alpha)S_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle=$ $\displaystyle\ell(\log
X)^{-1}\int_{\mathfrak{M}}T_{2}(\lambda_{1}\alpha)T_{2}(\lambda_{2}\alpha)T_{2}(\lambda_{3}\alpha)T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle+\int_{\mathfrak{M}}(\widetilde{S}_{2}(\lambda_{1}\alpha)-\ell(\log
X)^{-1}T_{2}(\lambda_{1}\alpha))S_{2}(\lambda_{2}\alpha)T_{2}(\lambda_{3}\alpha)T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle+\int_{\mathfrak{M}}\widetilde{S}_{2}(\lambda_{1}\alpha)(S_{2}(\lambda_{2}\alpha)-T_{2}(\lambda_{2}\alpha))T_{2}(\lambda_{3}\alpha)T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle+\int_{\mathfrak{M}}\widetilde{S}_{2}(\lambda_{1}\alpha))S_{2}(\lambda_{2}\alpha)(S_{2}(\lambda_{3}\alpha)-T_{2}(\lambda_{3}\alpha))T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle+\int_{\mathfrak{M}}\widetilde{S}_{2}(\lambda_{1}\alpha)S_{2}(\lambda_{2}\alpha)S_{2}(\lambda_{3}\alpha)(S_{k}(\lambda_{4}\alpha)-T_{k}(\lambda_{4}\alpha))K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle=$ $\displaystyle J_{1}+J_{2}+J_{3}+J_{4}+J_{5},$

say. Since the computations for $J_{3}$ is similar to, but simpler than, the
corresponding ones for $J_{2}$, $J_{4}$ and $J_{5}$, we will leave it to the
reader.

###  3.1. Main Term: lower bound for $J_{1}$

As the reader might expect the main term is given by the summand $J_{1}$.

Let
$H(\alpha)=T_{1}(\lambda_{1}\alpha)T_{2}(\lambda_{2}\alpha)T_{2}(\lambda_{3}\alpha)T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)$
so that

$\displaystyle J_{1}=\ell(\log
X)^{-1}\int_{\mathbb{R}}H(\alpha)\mathrm{d}\alpha+\mathcal{O}\left(\int_{P/X}^{+\infty}|H(\alpha)|\mathrm{d}\alpha\right).$

Using inequalities (5) and (7) ,

$\displaystyle\int_{P/X}^{+\infty}|H(\alpha)|\mathrm{d}\alpha\ll$
$\displaystyle
X^{-\frac{1}{2}}X^{-\frac{1}{2}}X^{-\frac{1}{2}}X^{\frac{1}{k}-1}\eta^{2}\int_{P/X}^{+\infty}\frac{\mathrm{d}\alpha}{\alpha^{4}}$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{k}-\frac{5}{2}}\eta^{2}\frac{X^{3}}{P^{3}}=X^{\frac{1}{k}+\frac{1}{2}}\eta^{2}P^{-3}=o\left(X^{\frac{1}{k}+\frac{1}{2}}\eta^{2}\right)$

provided that $P\rightarrow+\infty$. Let $D=[(\delta
X)^{\frac{1}{2}},X^{\frac{1}{2}}]^{3}\times[(\delta
X)^{\frac{1}{k}},X^{\frac{1}{k}}]$ we have

$\displaystyle\int_{\mathbb{R}}H(\alpha)\mathrm{d}\alpha$
$\displaystyle=\idotsint_{D}\int_{\mathbb{R}}e((\lambda_{1}t_{1}^{2}+\lambda_{2}t_{2}^{2}+\lambda_{3}t_{3}^{2}+\lambda_{4}t_{4}^{k}-\omega)\alpha)K_{\eta}(\alpha)\mathrm{d}\alpha\,\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}$
$\displaystyle=\idotsint_{D}\max(0,\eta-|\lambda_{1}t_{1}^{2}+\lambda_{2}t_{2}^{2}+\lambda_{3}t_{3}^{2}+\lambda_{4}t_{4}^{k}-\omega)|)\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\mathrm{d}t_{4}.$

Apart from very slight changes in the computation, we proceed with a change of
variables as in [7] and we obtain

$\displaystyle J_{1}\gg(\log X)^{-1}\eta^{2}X^{\frac{1}{k}+\frac{1}{2}},$

which is the expected lower bound.

###  3.2. Bound for $J_{2}$

We expect the main term to have the dominant asymptotic behavior, then we
shall prove that all the remaining terms of the sum are $o\left((\log
X)^{-1}\eta^{2}X^{\frac{1}{k}+\frac{1}{2}}\right)$.

From partial summation on (4) we get

$\widetilde{S}_{2}(\lambda_{1}\alpha)=\int_{(\delta
X)^{\frac{1}{2}}}^{X^{\frac{1}{2}}}e(\lambda
t^{2}\alpha)\,\mathrm{d}\bigg{(}\sum_{\begin{subarray}{c}m_{2}\leq t\\\
m_{2}\in[(\delta X)^{1/2},X^{1/2}]\end{subarray}}\rho(m_{2})\bigg{)},$

then

$\widetilde{S}_{2}(\lambda_{1}\alpha)-\ell(\log
X)^{-1}T_{2}(\lambda_{1}\alpha)\ll X^{\frac{1}{2}}(\log X)^{-2}(1+|\alpha|X).$

Retrieving (7) and using the triangle inequality,

$\displaystyle J_{2}\ll$
$\displaystyle\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)-\ell(\log
X)^{-1}T_{2}(\lambda_{1}\alpha)||T_{2}(\lambda_{2}\alpha)||T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$ $\displaystyle\eta^{2}X^{\frac{1}{2}}(\log
X)^{-2}(1+|\alpha|X)\int_{\mathfrak{M}}|T_{2}(\lambda_{2}\alpha)||T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$ $\displaystyle\eta^{2}X^{\frac{1}{2}}(\log
X)^{-2}\int_{0}^{1/X}|T_{2}(\lambda_{1}\alpha)||T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle+\eta^{2}X^{\frac{3}{2}}(\log
X)^{-2}\int_{1/X}^{P/X}\alpha|T_{2}(\lambda_{1}\alpha)||T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle=$ $\displaystyle
o\left(\eta^{2}X^{\frac{1}{k}+\frac{1}{2}}(\log X)^{-1}\right).$

###  3.3. Bound for $J_{4}$

Using the triangle inequality and (7),

$\displaystyle J_{4}=$
$\displaystyle\int_{\mathfrak{M}}\widetilde{S}_{2}(\lambda_{1}\alpha)S_{2}(\lambda_{2}\alpha)(S_{2}(\lambda_{3}\alpha)-T_{2}(\lambda_{3}\alpha))T_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)e(-\omega\alpha)\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||S_{2}(\lambda_{3}\alpha)-T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle\leq$
$\displaystyle\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||S_{2}(\lambda_{3}\alpha)-U_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle+\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||U_{2}(\lambda_{3}\alpha)-T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle=$ $\displaystyle\eta^{2}(A_{4}+B_{4}),$

say. Using Theorem 5 and the Hölder inequality,

$\displaystyle A_{4}\ll$ $\displaystyle
X^{\frac{1}{k}}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||S_{2}(\lambda_{3}\alpha)-U_{2}(\lambda_{3}\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{k}}\left(\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{4}}\left(\int_{\mathfrak{M}}|S_{2}(\lambda_{2}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{4}}\left(\int_{\mathfrak{M}}|S_{2}(\lambda_{3}\alpha)-U_{2}(\lambda_{3}\alpha)|^{2}\mathrm{d}\alpha\right)^{\frac{1}{2}}.$

Using Lemmas 1-2-4, for any fixed $A$,

$\displaystyle A_{4}\ll X^{\frac{1}{k}}(X\log^{2}X)^{\frac{1}{2}}(\log
X)^{-\frac{A}{2}}=X^{\frac{1}{2}+\frac{1}{k}}(\log
X)^{1-\frac{A}{2}}=o\left((\log X)^{-1}X^{\frac{1}{k}+\frac{1}{2}}\right)$

as long as $A>4$.

As for $A_{2}$ we used in the estimation above Lemma 1 that has two more
terms, but also in this case these terms are negligible if we want to meet the
hypothesis of Lemma 2: in fact it requires that

$X^{1-\frac{5}{12}+\varepsilon}\leq\frac{X}{P}\leq X$

and this is consistent with the choice we will make in (8).

Again using Theorem 6,

$\displaystyle B_{4}=$
$\displaystyle\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||U_{2}(\lambda_{3}\alpha)-T_{2}(\lambda_{3}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\int_{0}^{1/X}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle+X\int_{1/X}^{P/X}\alpha|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha.$

Remembering that $|\alpha|\leq\frac{P}{X}$ on $\mathfrak{M}$ and using the
Hölder inequality, trivial bounds and Lemma 4 we have

$\displaystyle B_{4}\ll$ $\displaystyle
X^{\frac{1}{2}}\,X^{\frac{1}{2}}X^{\frac{1}{k}}\frac{1}{X}+X\,X^{\frac{1}{k}}\left(\int_{1/X}^{P/X}\alpha^{2}\right)^{\frac{1}{2}}\left(\int_{1/X}^{P/X}|\widetilde{S}_{2}(\lambda_{1}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{4}}\left(\int_{1/X}^{P/X}|S_{2}(\lambda_{2}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{4}}$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{k}}+X^{1+\frac{1}{k}}\left(X\log^{2}X\right)^{\frac{1}{2}}\left(\int_{1/X}^{P/X}\alpha^{2}\mathrm{d}\alpha\right)^{\frac{1}{2}}$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{k}}+X^{\frac{3}{2}+\frac{1}{k}}\log
X\left(\frac{P}{X}\right)^{\frac{3}{2}}=X^{\frac{1}{k}}P^{\frac{3}{2}}\log X.$

We assume

$\displaystyle P\leq X^{\frac{1}{3}-\varepsilon},$ (8)

so that $P^{\frac{3}{2}}=o(X^{\frac{1}{2}}/\log^{2}X)$ which, with the upper
bound for $B_{4}$ here above, ensures that

$B_{4}=o((\log X)^{-1}X^{1/2+1/k}).$

###  3.4. Bound for $J_{5}$

In order to provide an estimation for $J_{5}$, we use (7),

$\displaystyle J_{5}\ll$
$\displaystyle\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||S_{2}(\lambda_{3}\alpha)||S_{k}(\lambda_{4}\alpha)-T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$

and then the arithmetic-geometric inequality:

$\displaystyle J_{5}\ll$
$\displaystyle\eta^{2}\sum_{j=2}^{3}\left(\int_{\mathfrak{M}}|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{j}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)-T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha\right).$

The three terms may be estimated in the same way and produce the same upper
bound. We show the details of the bound only for the case $j=2$:

$\displaystyle\eta^{2}\int_{\mathfrak{M}}$
$\displaystyle|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)-T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)-U_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle+\eta^{2}\int_{\mathfrak{M}}|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)|^{2}|U_{k}(\lambda_{4}\alpha)-T_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$
$\displaystyle=$ $\displaystyle\eta^{2}(A_{5}+B_{5}),$

say. Using trivial estimates,

$\displaystyle A_{5}\ll
X^{\frac{1}{2}}\int_{\mathfrak{M}}|S_{2}(\lambda_{2}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)-U_{k}(\lambda_{4}\alpha)|\mathrm{d}\alpha$

then using the Cauchy-Schwartz inequality, for any fixed $A>4$, by Lemmas 4, 1
and 2 we have

$\displaystyle A_{5}\ll$ $\displaystyle
X^{\frac{1}{2}}\left(\int_{\mathfrak{M}}|S_{2}(\lambda_{1}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{2}}\left(\int_{\mathfrak{M}}|S_{k}(\lambda_{4}\alpha)-U_{k}(\lambda_{4}\alpha)|^{2}\mathrm{d}\alpha\right)^{\frac{1}{2}}$
$\displaystyle\ll$ $\displaystyle X^{\frac{1}{2}}\,X^{\frac{1}{2}}\log
X\frac{P}{X}\mathcal{J}_{k}\left(X,\frac{X}{P}\right)^{\frac{1}{2}}\ll_{A}X^{\frac{1}{2}+\frac{1}{k}}(\log
X)^{1-\frac{A}{2}}=o\left((\log X)^{-1}X^{\frac{1}{2}+\frac{1}{k}}\right)$

provided that $\frac{X}{P}\geq X^{1-\frac{5}{6k}+\varepsilon}$ (condition of
Lemma 2), that is,

$\displaystyle(\log X)^{A}\ll_{A}P\leq X^{\frac{5}{6k}-\varepsilon}.$ (9)

Now we turn to $B_{5}$, using Theorem 6:

$\displaystyle
B_{5}\ll\int_{0}^{1/X}|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)|^{2}\mathrm{d}\alpha+X\int_{1/X}^{P/X}\alpha|\widetilde{S}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)|^{2}\mathrm{d}\alpha.$

Using trivial estimates and Lemma 4,

$\displaystyle B_{5}\ll$ $\displaystyle X^{\frac{3}{2}}\frac{1}{X}+X\cdot
X^{\frac{1}{2}}\left(\int_{1/X}^{P/X}\alpha^{2}\mathrm{d}\alpha\cdot\int_{1/X}^{P/X}|S_{2}(\lambda_{1}\alpha)|^{4}\mathrm{d}\alpha\right)^{\frac{1}{2}}$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{2}}+X^{\frac{3}{2}}(P/X)^{\frac{3}{2}}X^{\frac{1}{2}}\log
X=X^{\frac{1}{2}}+P^{\frac{3}{2}}X^{\frac{1}{2}}\log X.$

The case $j=3$ can be estimated in the same way. We need

$\displaystyle P=o\left(X^{\frac{2}{3k}-\varepsilon}\right).$

Summing up with (9),

$\displaystyle P\leq X^{\frac{2}{3k}-\varepsilon}.$ (10)

Collecting all the bounds for $P$, that is, (8), (9), (10) we can take

$\displaystyle P\leq X^{\frac{1}{3}-\varepsilon}.$ (11)

In fact, if we consider (8), (9) and (10) we should choose the most
restrictive condition among the three but as we expect that the value of $k$
is smaller than 2, (8) is the most restrictive: $\frac{2}{3k}\leq\frac{5}{6k}$
and $\frac{1}{3}\geq\frac{2}{3k}$ only if $k\geq 2$.

##  4\. Trivial arc

By the arithmetic-geometric mean inequality and the trivial bound for
$\widetilde{S}_{2}(\lambda_{1}\alpha)$, we see that

$\displaystyle|\mathcal{I}(\eta,\omega,\mathfrak{t})|\ll$
$\displaystyle\int_{R}^{+\infty}|\widetilde{S}_{2}(\lambda_{1}\alpha)S_{2}(\lambda_{2}\alpha)S_{2}(\lambda_{3}\alpha)S_{k}(\lambda_{4}\alpha)K_{\eta}(\alpha)|\mathrm{d}\alpha$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{2}}\sum_{j=2}^{3}\int_{R}^{+\infty}|S_{2}(\lambda_{j}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)|K_{\eta}(\alpha)\mathrm{d}\alpha.$

The three terms may be estimated in the same way and produce the same upper
bound. We show the details of the bound only for the case $j=1$:

$\displaystyle X^{\frac{1}{2}}\int_{R}^{+\infty}$
$\displaystyle|S_{2}(\lambda_{j}\alpha)|^{2}|S_{k}(\lambda_{4}\alpha)|K_{\eta}(\alpha)\mathrm{d}\alpha$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{2}}\left(\int_{R}^{+\infty}|S_{2}(\lambda_{1}\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha\right)^{\frac{1}{2}}\left(\int_{R}^{+\infty}|S_{k}(\lambda_{4}\alpha)|^{2}K_{\eta}(\alpha)\mathrm{d}\alpha\right)^{\frac{1}{2}}$
$\displaystyle\ll$ $\displaystyle
X^{\frac{1}{2}}\left(\int_{R}^{+\infty}\frac{|S_{2}(\lambda_{1}\alpha)|^{4}}{\alpha^{2}}\mathrm{d}\alpha\right)^{\frac{1}{2}}\left(\int_{R}^{+\infty}\frac{|S_{k}(\lambda_{4}\alpha)|^{2}}{\alpha^{2}}\mathrm{d}\alpha\right)^{\frac{1}{2}}=X^{\frac{1}{2}}C_{1}^{\frac{1}{2}}C_{2}^{\frac{1}{2}},$

say. Using Lemma 4, we have

$\displaystyle C_{1}=$
$\displaystyle\int_{R}^{+\infty}\frac{|S_{2}(\lambda_{2}\alpha)|^{4}}{\alpha^{2}}\mathrm{d}\alpha\ll\int_{\lambda_{1}R}^{+\infty}\frac{|S_{2}(\alpha)|^{4}}{\alpha^{2}}\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\sum_{n\geq|\lambda_{1}|R}\frac{1}{(n-1)^{2}}\int_{n-1}^{n}|S_{2}(\alpha)|^{4}\mathrm{d}\alpha\ll\frac{X\log^{2}X}{R}.$
(12)

Now using Lemma 3,

$\displaystyle C_{2}=$
$\displaystyle\int_{R}^{+\infty}\frac{|S_{k}(\lambda_{4}\alpha)|^{4}}{\alpha^{2}}\mathrm{d}\alpha\ll\int_{\lambda_{4}R}^{+\infty}\frac{|S_{k}(\alpha)|^{2}}{\alpha^{2}}\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\sum_{n\geq|\lambda_{4}|R}\frac{1}{(n-1)^{2}}\int_{n-1}^{n}|S_{k}(\alpha)|^{2}\mathrm{d}\alpha\ll\frac{X^{\frac{1}{k}}\log^{3}X}{R}.$
(13)

Collecting (4) and (4),

$\displaystyle|\mathcal{I}(\eta,\omega,\mathfrak{t})|\ll
X^{\frac{1}{2}}\left(\frac{X\log^{2}X}{R}\right)^{\frac{1}{2}}\left(\frac{X^{\frac{1}{k}}\log^{3}X}{R}\right)^{\frac{1}{2}}\ll\frac{X^{1+\frac{1}{2k}}(\log
X)^{\frac{5}{2}}}{R}.$

Hence, remembering that $|\mathcal{I}(\eta,\omega,t)|$ must be $o\left((\log
X)^{-1}\eta^{2}X^{\frac{1}{k}+1}\right)$, i.e. little-o of the main term, the
choice

$\displaystyle R=\frac{X^{\frac{1}{2}-\frac{1}{2k}}\log^{4}X}{\eta^{2}}$ (14)

is admissible.

##  5\. The minor arc

In [25] section 4 it is proven that the measure of the set where
$|\widetilde{S}_{2}(\lambda_{1}\alpha)|^{\frac{1}{2}}$ and
$|\widetilde{S}_{2}(\lambda_{2}\alpha)|$ are both large for $\alpha\in m$ is
small, exploiting the fact that the ratio $\lambda_{1}/\lambda_{2}$ is
irrational.

######  Lemma 5 (Wang-Yao [25], Lemma 1).

Suppose that $X^{\frac{1}{2}}\geq Z\geq
X^{\frac{1}{2}-\frac{1}{14}+\varepsilon}$ and
$|\widetilde{S}_{2}(\lambda\alpha)|>Z$. Then there are coprime integers
$(a,q)=1$ satisfying

$\displaystyle 1\leq
q\leq\left(\frac{X^{\frac{1}{2}+\varepsilon}}{Z}\right)^{4},\qquad|q\lambda\alpha-a|\ll
X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon}}{Z}\right)^{4}.$

In this case we need only Lemma 5. Let us split $\mathfrak{m}$ into subsets
$\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$ and
$\mathfrak{m}^{*}=\mathfrak{m}\backslash(\mathfrak{m}_{1}\cup\mathfrak{m}_{2})$
where

$\displaystyle\mathfrak{m}_{i}=\\{\alpha\in\mathfrak{m}\,:\
|\widetilde{S}_{2}(\lambda_{i}\alpha)|\leq X^{\frac{1}{2}-u+\varepsilon}\\}$

remembering that Lemma 5 holds for $0\leq u\leq\frac{1}{14}$. In this case we
leave only the parameter $u$ free. Using the Hölder inequalities and the
definition of $m_{i}$ we obtain

$\displaystyle\left|\mathcal{I}(\eta,\omega,\mathfrak{m}_{i})\right|\ll$
$\displaystyle\int_{\mathfrak{m}_{i}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||S_{2}(\lambda_{2}\alpha)||S_{2}(\lambda_{3}\alpha)||S_{k}(\lambda_{4}\alpha)|K_{\eta}(\alpha)\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\max{|\widetilde{S}_{2}(\lambda_{1}\alpha)|}\left(\int_{\mathfrak{m}_{i}}|S_{2}(\lambda_{2}\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha)\right)^{1/4}$
$\displaystyle\left(\int_{\mathfrak{m}_{i}}|S_{2}(\lambda_{3}\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha)\right)^{1/4}\left(\int_{\mathfrak{m}_{i}}|S_{k}(\lambda_{4}\alpha)|^{2}K_{\eta}(\alpha)\mathrm{d}\alpha)\right)^{1/2}$
$\displaystyle\ll$ $\displaystyle X^{\frac{1}{2}-u+\varepsilon}(\eta
X\log^{2}X)^{\frac{1}{4}}(\eta X\log^{2}X)^{\frac{1}{4}}\left(\eta
X^{\frac{1}{k}}\log^{3}X\right)^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\eta X^{1-u+\frac{1}{2k}+\varepsilon}\log^{\frac{5}{2}}X.$ (15)

by Lemma 5. The bound (5) must be
$o\left(\eta^{2}X^{\frac{1}{2}+\frac{1}{k}}\right)$, consequently we have the
following condition:

$\displaystyle\eta=\infty\left(X^{{\frac{1}{2}}-\frac{1}{2k}-u+\varepsilon}\right),$

where we used the notation $f=\infty(g)$ for $g=o(f)$.

It remains to discuss the set $\mathfrak{m}^{*}$ in which the following bounds
hold simultaneously

$\displaystyle|\widetilde{S}_{2}(\lambda_{i}\alpha)|>X^{\frac{1}{2}-u+\varepsilon},\qquad\frac{P}{X}=X^{-\frac{2}{3}}<|\alpha|\leq\frac{\log^{2}X}{\eta^{2}}.$

Following the dyadic dissection argument shown by Harman in [12] we divide
$\mathfrak{m}^{*}$ into disjoint sets $E(Z_{1},Z_{2},y)$ in which, for
$\alpha\in E(Z_{1},Z_{2},y)$, we have

$\displaystyle Z_{1}<|\widetilde{S}_{2}(\lambda_{1}\alpha)|\leq 2Z_{1},\qquad
Z_{2}<|\widetilde{S}_{2}(\lambda_{2}\alpha)|\leq 2Z_{2},\qquad y<|\alpha|\leq
2y$

where $Z_{i}=2^{k_{i}}X^{\frac{1}{2}-u+\varepsilon}$ for $i=1,2$, and
$y=2^{k_{3}}X^{-\frac{2}{3}-\varepsilon}$ for some non-negative integers
$k_{1},k_{2},k_{3}$.

It follows that the disjoint sets are, at most, $\ll\log^{3}X$. Let us define
$\mathcal{A}$ a shorthand for the sets $E(Z_{1},Z_{2},y)$; we have the
following result about the Lebesgue measure of $\mathcal{A}$ following the
same lines of Lemma 6 in [20].

In the subsequent Lemma, it is essential for both integers $a_{1}$ and $a_{2}$
involved in (16) below not to equal zero: specifically, if $a_{1}=0$, for
instance, then $q_{1}=1$, and $|\alpha|$ becomes so small that it cannot
belong to $\mathfrak{m}^{*}$. Consequently, to apply the Harman technique, we
are compelled to move away from the major arc, where $a_{1}a_{2}=0$. As we
shall later observe, upon defining the parameter $u$, we won’t encounter a gap
between the major and minor arcs, obviating the necessity to introduce an
intermediate arc.

######  Lemma 6.

We have

$\displaystyle\mu(\mathcal{A})\ll yX^{2+8u+3\varepsilon}Z_{1}^{-4}Z_{2}^{-4}$

where $\mu(\cdot)$ denotes the Lebesgue measure.

###### Proof.

If $\alpha\in\mathcal{A}$, by Lemma 5 there are coprime integers
$(a_{1},q_{1})$ and $(a_{2},q_{2})$ such that

$\displaystyle 1\leq
q_{2}\ll\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4},\qquad|q_{2}\lambda_{2}\alpha-
a_{2}|\ll X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4}$
(16)

We remark that $a_{1}a_{2}\neq 0$ otherwise we would have
$\alpha\in\mathfrak{M}$. In fact, if $a_{i}=0$ recalling the definitions of
$Z_{1}$ and (16), we get

$\displaystyle|\alpha|\ll
q_{2}^{-1}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4}\ll\frac{X}{X^{2-4u+3\varepsilon}}=X^{-1+4u-3\varepsilon}.$

It means that, on the minor arc

$|\alpha|\gg X^{-1+4u-3\varepsilon}.$

We wonder now if there is a gap between the end of the major arc and the
beginning of the minor arc: from Lemma 5 we are sure that $u\leq\frac{1}{16}$;
furthermore, from the previous lower bound for $\alpha$, we need to check
whether $\frac{P}{X}$ is greater than it:

$\displaystyle
X^{-1+4u-3\varepsilon}<\frac{P}{X}=X^{-\frac{2}{3}}\quad\Rightarrow\quad
u<\frac{1}{12}.$

It is clear that we can choose any parameter $u$ with the condition given by
Lemma 5 without leaving any gap from the two arcs.

Now, we can further split $\mathfrak{m}^{*}$ into sets
$I(Z_{1},Z_{2},y,Q_{1},Q_{2})$ where $Q_{j}\leq q_{j}\leq 2Q_{j}$ on each set.
Note that $a_{i}$ and $q_{i}$ are uniquely determined by $\alpha$. In the
opposite direction, for a given quadruple $a_{1}$, $q_{1}$, $a_{2}$, $q_{2}$
the inequalities (16) define an interval of $\alpha$ of length

$\displaystyle\mu(I)\ll\min\left(Q_{1}^{-1}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{1}}\right)^{4},Q_{2}^{-1}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4}\right).$

Taking the geometric mean ($\min(a,b)\leq\sqrt{a}\sqrt{b}$) we can write

$\displaystyle\mu(I)\ll
Q_{1}^{-\frac{1}{2}}Q_{2}^{-\frac{1}{2}}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{1}}\right)^{2}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{2}\ll\frac{X^{1+\varepsilon}}{Q_{1}^{\frac{1}{2}}Q_{2}^{\frac{1}{2}}Z_{1}^{2}Z_{2}^{2}}.$
(17)

Now we need a lower bound for $Q_{1}^{\frac{1}{2}}Q_{2}^{\frac{1}{2}}$: by
(16)

$\displaystyle\left|a_{2}q_{1}\frac{\lambda_{1}}{\lambda_{2}}-a_{1}q_{2}\right|$
$\displaystyle=\left|\frac{a_{2}}{\lambda_{2}\alpha}(q_{1}\lambda_{1}\alpha-
a_{1})-\frac{a_{1}}{\lambda_{2}\alpha}(q_{2}\lambda_{2}\alpha-a_{2})\right|$
$\displaystyle\ll q_{2}|q_{1}\lambda_{1}\alpha-
a_{1}|+q_{1}|q_{2}\lambda_{2}\alpha-a_{2}|$ $\displaystyle\ll
Q_{2}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{1}}\right)^{4}+Q_{1}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4}.$

Remembering that
$Q_{i}\ll\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{i}}\right)^{4}$,
$Z_{i}\gg X^{\frac{1}{2}-u+\varepsilon}$,

$\displaystyle\left|a_{2}q_{1}\frac{\lambda_{1}}{\lambda_{2}}-a_{1}q_{2}\right|\ll\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{X^{\frac{1}{2}-u+\varepsilon}}\right)^{4}X^{-1}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{X^{\frac{1}{2}-u+\varepsilon}}\right)^{4}\ll\frac{X^{3+2\varepsilon}}{X^{4-8u+8\varepsilon}}\ll
X^{-1+8u-6\varepsilon}.$ (18)

We recall that $q=X^{1-8u}$ is a denominator of a convergent of
$\lambda_{1}/\lambda_{2}$. Hence by (18) Legendre’s law of best approximation
implies that $|a_{2}q_{1}|\geq q$ and by the same token, for any pair
$\alpha$, $\alpha^{\prime}$ having distinct associated products $a_{2}q_{1}$
(see [26], Lemma 2),

$|a_{2}(\alpha)q_{1}(\alpha)-a_{2}(\alpha^{\prime})q_{1}(\alpha^{\prime})|\geq
q;$

thus, by the pigeon-hole principle, there is at most one value of $a_{2}q_{1}$
in the interval $[rq,(r+1)q)$ for any positive integer $r$. Hence $a_{2}q_{1}$
determines $a_{2}$ and $q_{1}$ to within $X^{\varepsilon}$ possibilities (the
upper bound for the divisor function) and consequently also $a_{2}q_{1}$
determines $a_{1}$ and $q_{2}$ to within $X^{\varepsilon}$ possibilities from
(18).

Hence we got a lower bound for $q_{1}q_{2}$, remembering that in our shorthand
$Q_{j}\leq q_{j}\leq 2Q_{j}$:

$\displaystyle
q_{1}q_{2}=a_{2}q_{1}\frac{q_{2}}{a_{2}}\gg\frac{rq}{|\alpha|}\gg rqy^{-1}$

for the quadruple under consideration. As a consequence we obtain from (17),
that the total length of the interval $I(Z_{1},Z_{2},y,Q_{1},Q_{2})$ with
$a_{2}q_{1}\in[rq,(r+1)q)$ does not exceed

$\displaystyle\mu(I)\ll
X^{1+2\varepsilon}{Z_{1}^{-2}Z_{2}^{-2}}r^{-\frac{1}{2}}q^{-\frac{1}{2}}y^{\frac{1}{2}}.$

Now we need a bound for $r$: inside the interval $[rq,(r+1)q)$,
$rq\leq|a_{2}q_{1}|$ and, in turn from (16), $a_{2}\ll q_{2}|\alpha|$, then

$\displaystyle rq\ll
q_{1}q_{2}|\alpha|\ll\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{1}}\right)^{4}\left(\frac{X^{\frac{1}{2}+\varepsilon/4}}{Z_{2}}\right)^{4}y\ll
yX^{4+2\varepsilon}Z_{1}^{-4}Z_{2}^{-4}$ $\displaystyle\Rightarrow r\ll
q^{-1}yX^{4+2\varepsilon}Z_{1}^{-4}Z_{2}^{-4}.$

Now, we sum on every interval to get an upper bound for the measure of
$\mathcal{A}$:

$\displaystyle\mu(\mathcal{A})\ll
X^{1+2\varepsilon}{Z_{1}^{-2}Z_{2}^{-2}}q^{-\frac{1}{2}}y^{\frac{1}{2}}\sum_{1\leq
r\ll q^{-1}yX^{4+2\varepsilon}Z_{1}^{-4}Z_{2}^{-4}}r^{-\frac{1}{2}}.$

By standard estimation we obtain

$\displaystyle\sum_{1\leq r\ll
q^{-1}yX^{4+2\varepsilon}Z_{1}^{-4}Z_{2}^{-4}}r^{-\frac{1}{2}}\ll(q^{-1}yX^{4+2\varepsilon}Z_{1}^{-4}Z_{2}^{-4})^{\frac{1}{2}}$

then

$\displaystyle\mu(\mathcal{A})\ll
yX^{3+3\varepsilon}Z_{1}^{-4}Z_{2}^{-4}q^{-1}\ll
yX^{3+3\varepsilon}Z_{1}^{-4}Z_{2}^{-4}X^{-1+8u}\ll
yX^{2+8u+3\varepsilon}Z_{1}^{-4}Z_{2}^{-4}.$

This concludes the proof of Lemma 6. ∎

Using Lemma 6 we finally are able to get a bound for
$\mathcal{I}(\eta,\omega,\mathcal{A})$:

$\displaystyle\mathcal{I}(\eta,\omega,\mathcal{A})=$
$\displaystyle\int_{\mathfrak{m}^{*}}|\widetilde{S}_{2}(\lambda_{1}\alpha)||\widetilde{S}_{2}(\lambda_{2}\alpha)||S(\lambda_{3}\alpha)||S_{k}(\lambda_{4}\alpha)|K_{\eta}(\alpha)\mathrm{d}\alpha$
$\displaystyle\ll$
$\displaystyle\left(\int_{\mathcal{A}}|\widetilde{S}_{2}(\lambda_{1}\alpha)\widetilde{S}_{2}(\lambda_{2}\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha\right)^{\frac{1}{4}}\left(\int_{\mathcal{A}}|S_{2}(\lambda_{3}\alpha)|^{4}K_{\eta}(\alpha)\mathrm{d}\alpha\right)^{\frac{1}{4}}$
$\displaystyle\left(\int_{\mathcal{A}}|S_{k}(\lambda_{4}\alpha)|^{2}K_{\eta}(\alpha)\mathrm{d}\alpha\right)^{\frac{1}{2}}$
$\displaystyle\ll$
$\displaystyle\left(\min\left(\eta^{2},\frac{1}{y^{2}}\right)\right)^{\frac{1}{4}}\left((Z_{1}Z_{2})^{4}\mu(\mathcal{A})\right)^{\frac{1}{4}}\left(\eta
X\log^{2}X\right)^{\frac{1}{4}}\left(\eta
X^{\frac{1}{k}}\log^{3}X\right)^{\frac{1}{2}}$ $\displaystyle\ll$
$\displaystyle\left(\min\left(\eta^{2},\frac{1}{y^{2}}\right)\right)^{\frac{1}{4}}Z_{1}Z_{2}(yX^{2+8u+4\varepsilon}Z_{1}^{-4}Z_{2}^{-4})^{\frac{1}{4}}\eta^{\frac{3}{4}}X^{\frac{1}{4}+\frac{1}{2k}+\varepsilon}$
$\displaystyle\ll$
$\displaystyle\left(\min\left(\eta^{2},\frac{1}{y^{2}}\right)\right)^{\frac{1}{4}}y^{\frac{1}{4}}\eta^{\frac{3}{4}}X^{\frac{3}{4}+2u+\frac{1}{2k}+\varepsilon}$
$\displaystyle\ll$ $\displaystyle\eta
X^{\frac{3}{4}+u+\frac{1}{2k}+\varepsilon}$

and this must be $o\left(X^{{1+\frac{1}{k}}-\varepsilon}\right)$.

The condition on $\eta$ is

$\displaystyle\eta=\infty\left(X^{{\frac{1}{4}}-\frac{1}{2k}+2u+\varepsilon}\right).$
(19)

Collecting all the conditions (5), (19) and the condition given by Lemma 5, we
get the following linear optimization system: setting $x=\frac{1}{k}$ and let
$w$ be the exponent of $\eta$ we would like to optimize,

$\displaystyle\begin{cases}&x\leq 1;\ w\geq 0;\ u\leq\frac{1}{14}\\\
&-w\geq\frac{1}{2}-\frac{x}{2}-u\\\ &-w\geq\frac{1}{4}-\frac{x}{2}+2u.\\\
\end{cases}$

Solving the system, it turns out that $u=\frac{1}{14}$ (and consequently
$X=q^{7/3}$) are the optimal values; unfortunately, condition (19) does not
affect linear optimization for values of $u\leq 1/14$. Then, the maximum
$k$-range is $\left(1,\frac{7}{6}\right)$ and

$\displaystyle\eta=\left(\max_{j}p_{j}\right)^{-\frac{7-6k}{14k}+\varepsilon}.$

We thank the anonymous referees for an extremely careful reading of a previous
version of this paper and the fruitful suggestions.

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# Off-centre supermassive black holes in bright central galaxies

Aline Chu 1, Pierre Boldrini 1,2 and Joe Silk 1,3,4
1Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98
bis bd Arago, 75014 Paris, France
2Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
3Department of Physics and Astronomy, The Johns Hopkins University, Baltimore
MD 21218, USA
4Beecroft Institute for Particle Astrophysics and Cosmology, Department of
Physics, University of Oxford, Oxford OX1 3RH, UK
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Supermassive black holes (SMBHs) are believed to reside at the centre of
massive galaxies such as brightest cluster galaxies (BCGs). However, as BCGs
experienced numerous galaxy mergers throughout their history, the central BH
can be significantly kicked from the central region by these dynamical
encounters. By combining the TNG300 simulations and orbital integrations, we
demonstrate that mergers with satellite galaxies on radial orbits are a main
driver for such BH displacements in BCGs. BHs can get ejected to distances
varying between a few parsecs to hundreds of kiloparsecs. Our results clearly
establish that SMBH offsets are common in BCGs and more precisely a third of
our BHs are off-centred at $z=0$. This orbital offset can be sustained for up
to at least 6 Gyr between $z=2$ and $z=0$ in half of our BCGs. Since the dense
gas reservoirs are located in the central region of galaxies, we argue that
the consequences of off-center SMBHs in BCGs are to quench any BH growth and
BH feedback.

###### keywords:

galaxy clusters – supermassive black hole – brightest cluster galaxy – off-
centre black holes – orbital integrations

††pubyear: 2015††pagerange: Off-centre supermassive black holes in bright
central galaxies–B

##  1 Introduction

Galaxy clusters are the largest virialized structures known today. These rich
systems of hundreds to thousands of galaxies are the perfect targets to study
how environment can impact the formation of galaxies and their evolution.
Located at the intersection of the filaments which compose the cosmic web,
clusters are believed to form mainly by accreting small galaxies and groups
which fall along the filaments, or by mergers with other systems (Kravtsov &
Borgani, 2012). Galaxies, gas clumps and globular clusters, which are captured
by the cluster potential, would sink to the central region of the cluster.
Indeed, these satellites are bound to merge with the central galaxy, which is
often referred to as the Brightest Cluster Galaxy (BCG) (Crawford et al.,
1999; Bernardi et al., 2007).

BCGs are among the most massive galaxies observed in the Universe (Dubinski,
1998). These peculiar elliptical galaxies, which generally lie at the bottom
of the cluster potential well, are more likely to encounter other objects due
to their special location, and thus, to undergo many mergers during their
lifetimes which promotes their growth. BCGs form via environmental processes
(Castignani et al., 2020, and references therein), and suffer and reflect all
the processes which formed and shaped their host clusters. Hence, BCGs have
properties which are closely linked to those of their host clusters (see Lauer
et al., 2014; Sohn et al., 2022).

In the local Universe, black holes (BHs) are nearly ubiquitous in galaxies
(Kormendy & Ho, 2013; Saglia et al., 2016; van den Bosch, 2016). In
particular, BCGs host among the most massive BHs ever detected, the
supermassive black holes (SMBH). Recently, measurements were reported of SMBH
masses in excess of $10^{10}$ M☉ (McConnell et al., 2011; Hlavacek-Larrondo et
al., 2012, 2015; McConnell & Ma, 2013a; Thomas et al., 2016). We expect the
central BHs to grow at the same time as their host galaxies by accreting
matter brought in by infalling galaxies. That is the reason why central BH
masses seem to be correlated with the stellar and bulge masses of their host
galaxies (Magorrian et al., 1998; Gültekin et al., 2009; Marleau et al., 2013;
McConnell & Ma, 2013b; Sahu et al., 2019). A prominent example is the SMBH of
Holm 15A, the BCG of the galaxy cluster Abell 85, which is the most massive BH
directly detected via stellar dynamics with a mass of about $4\times 10^{10}$
M☉, i.e. 2$\%$ of the total stellar mass of the central galaxy (Mehrgan et
al., 2019).

However, it has been demonstrated that BHs are not necessarily located exactly
at the bottom of the cluster potential, inside and at the center of the
central galaxy. In particular, tens of thousands of active galactic nuclei
(AGN) are not located at the centers of their host galaxies. These
observations exhibit BH offsets between tens of parsecs to a few kiloparsecs
(Menezes et al., 2014, 2016; Reines et al., 2020; Shen et al., 2019). Many
scenarios have been proposed to explain these off-centre BHs (Sundararajan et
al., 2010; Tremmel et al., 2018; Barth et al., 2009; Comerford & Greene, 2014;
Merritt & Milosavljević, 2005; Volonteri & Perna, 2005; Loeb, 2007; Komossa,
2012; Boldrini et al., 2020). It is important to notice that most of these
scenarios need to invoke interactions and mergers with other galaxies or dark
matter (DM) substructures (Bellovary et al., 2019; Pfister et al., 2019;
Bellovary et al., 2021, 2018; Boldrini et al., 2020).

This BH feature was also found in more massive galaxies such as Andromeda
(M31). It has been shown that its SMBH is offset by 0.26 pc from the centre of
the galaxy (Kormendy & Bender, 1999). As there are several indications of a
recent merger activity in M31, it was demonstrated that the accretion of a
satellite on a highly eccentric orbit naturally explains this off-centre BH in
our neighbor galaxy (Boldrini, 2020). As BCGs are the results of numerous
galaxy mergers throughout their history, we suspect that some mergers could
have significantly offset the SMBH located at the center of BCGs to distances
even larger than of the order of a kiloparsec.

In this paper, we aim to examine the impact of mergers between BCGs and
satellite galaxies, within galaxy clusters, on the SMBH of BCGs. In fact, we
track the orbit of the central SMBH in 370 BCGs by using orbital integration
methods after analysing the merger history of the central galaxies provided by
the TNG300 simulation. The paper is organized as follows. Our method is based
on the coupling between TNG300 data and orbital integration methods and is
described in details in Section 2. We present our results in Section 3, and
discuss the implications of this BH behaviour in Section 4. Section 5 presents
our conclusions and perspectives.

##  2 METHOD

Using a large sample of 370 galaxy clusters detected by Barnes et al. (2018)
in Illustris-TNG300 (Nelson et al., 2018; Pillepich et al., 2017; Springel et
al., 2017; Nelson et al., 2017; Naiman et al., 2018; Marinacci et al., 2018),
we track the orbit of a test SMBH throughout the merger history of the BCG, by
integrating its orbit in the fixed potential of the BCG using the publicly
available code galpy111Available at https://github.com/jobovy/galpy (Bovy,
2015). The sample is limited to masses $M_{500}$ $>$ 1013.75 M⊙ and has a
median mass $M_{500}$ = 8.8 $\times$ 1013 M⊙ at $z$ = 0. The following
describes our approach.

###  2.1 Merger history of BCGs with TNG300

For each cluster in TNG300, we identify the BCG at $z=0$, which is assumed to
be the most massive galaxy in the cluster at $z=0$. Going back through its
merger history tree until $z=2$, we search for all the main mergers that the
BCG has undergone. Indeed, BCGs were shown to have photometric properties and
shapes that do not change much in that redshift range, hinting at an earlier
formation epoch (Chu et al., 2021; Chu et al., 2022). Moreover, other studies
such as De Lucia & Blaizot (2007) and Thomas et al. (2010) have shown evidence
that the stellar population in such galaxies has most likely settled since
$z=2$ as most of the stars in BCGs were already formed in-situ a long time
ago, and the stellar masses measured today are found not to be significantly
different from their masses 9 billion years ago (Collins et al., 2009). More
precisely, BCGs only grow a factor of 1.8 in mass between $z=1$ and $z=0$
(Burke & Collins, 2013).

We identify the BCG at snapshot $n$ and go back one snapshot to $n-1$ to look
for its first progenitor (FP) (the most massive galaxy in the cluster at
snapshot $n-1$). If a merger happened, we also identify the most massive
satellite galaxy with which the BCG has merged, which is referred to as the
next progenitor (NP) in the TNG300 simulation. At snapshot $n-1$, the NP will
also have the BCG as a descendant. We thus retrieve all FPs and NPs at each
snapshot present in the parent halo (or cluster). If two galaxies are merging,
we retrieve their positions, velocities, half-mass radii, stellar and DM
masses, which will be used to integrate the orbit of the satellite in the
potential of the BCG. We find that BCGs in this sample have undergone between
12 and 130 mergers since $z=2$.

###  2.2 Orbital integrations with galpy

For each satellite galaxy (or NP), we integrate its orbit forward in time in
the potential of the BCG (or FP) from $z=2$ until $z=0$. The potential of the
BCG consists in a Hernquist and Plummer profile for the DM and stellar
components respectively, which is computed considering the properties found in
TNG300. All masses and scale radii are taken from the cosmological simulation.
The DM and stellar masses of the BCG remain fixed along all the orbital
integrations. In fact, we neglect the mass contribution of satellites from
galaxy mergers. The satellite is assumed to be a point mass, which suffered
the dynamical friction from the BCG. Its total half-mass radius $r^{\rm
NP}_{\rm hm}$ is incorporated in the dynamical friction force. We used the
Chandrasekhar dynamical friction force, which is already implemented in galpy
and roughly follows Petts et al. (2016). The impact of the dynamical friction
modeling on our findings is discussed in Appendix A. Besides, we also neglect
the mass loss of the satellite, which could reduce the impact of the merger
with the BCG. The assumption that the satellite mass remains constant during
integrations is addressed in Appendix B.

Figure 1: Dynamical heating from satellites of BCG: Scheme illustrating the
radial orbit of a satellite (green points), falling into the potential of the
BCG, which kicks away the central BH (black point) after its passage. It
results in an off-centered SMBH. The maroon circle represents the region of
the BCG where Equations (1) and (2) are satisfied. We note $T_{d}$ the time at
which both of these conditions are met. The dotted orange circle represents
the half-mass radius of the BCG stellar component. Figure 2: Black hole kicks
in BCGs: Orbital radius of a M${}_{\rm SMBH}=2.6\times 10^{10}$ M☉ SMBH in a
BCG potential composed of a DM halo of M${}_{\rm DM}=$8.1$\times 10^{12}$ M☉
and a stellar component of M${}_{*}=$4.4$\times 10^{11}$ M☉ between $z=2$ and
$z=0$. The SMBH has experienced 17 mergers since $z=2$ but only 4 mergers have
satisfied our kicking criteria described by Equations (1), (2) and (6), showed
by green arrows. The BH can be ejected to hundreds of kiloparsecs. Between
mergers, we observe the orbital decay of the BH due to dynamical friction. At
$z=0$, the BH is still significantly off-centered by about 10 kpc.

We aim to determine if the satellite galaxy, via the merger, dynamically heats
the central region of the BCG and more particularly its SMBH. The satellite
needs to be massive enough and pass close enough to the center of the BCG on
its first orbit to be able to transfer energy to the central region of the
galaxy during the merger. Two conditions need to be met:

  * •

The distance $d$ between the centers of the satellite and the BCG must be
smaller than the half-mass radius of the NP $r^{\rm sat}_{\rm hm}$:

$d\leq r^{\rm sat}_{\rm hm}.$ (1)

This condition needs to be true on the satellite’s first passage in the
potential of the BCG (radial merger). Indeed, through each orbit, the
satellite will transfer its mass to the total mass of the BCG with which it is
merging until all its mass has been consumed by the BCG. In order to simplify
the problem, we neglect the mass loss of the satellite, which is valid only on
its first passage.

  * •

The total mass of the satellite galaxy must be bigger than the total mass of
the FP contained in a radius $d$:

$M^{\rm sat}_{\rm tot}\geq M^{\rm BCG}_{\rm int}(d).$ (2)

Only under this condition, the BCG potential can be displaced due to the mass
deposit from the satellite (Goerdt et al., 2010; Read et al., 2006). Then the
BH can vacate the central region of the galaxy. In our mass calculations, we
take into account the DM and stellar components.

The two previous conditions are equivalent to checking if the satellite has a
radial orbit. Indeed, the satellite must pass by the center or very close to
the center of the BCG in order to affect the central SMBH. We note $T_{d}$ the
time at which both of these conditions are met. We update at each $T_{d}$ the
stellar and DM masses of the BCG given by the TNG300 simulation. Figure 1
illustrates a satellite which falls in the potential of the BCG passing by its
center, in a radius $r_{\rm hm}^{\rm sat}$ as described in Equation 1. The
SMBH is significantly kicked by the satellite galaxy (see Figure 1).

Using the simulation information, we are able to follow the orbit of
satellites throughout the merger history of the BCG with galpy, by considering
their initial positions, velocities, and masses. Then, a SMBH with a constant
mass is placed at the centre of each BCG potential. As we are not using the BH
as a PartType5 particle from the simulation, the mass of the SMBH was
calculated based on the Magorrian et al. (1998) relation: MSMBH = 0.06
M${}^{\rm BCG}_{*}$, with M${}^{\rm BCG}_{*}$ the stellar mass of the BCG at
the first Td. We suppose that the SMBH is initially on a circular orbit at a
arbitrary radius, which is 100 times its Schwarzschild radius, as the SMBH
should lie in the central region of its host galaxy. After a careful check,
choosing a smaller radius does not significantly affect our results. If both
Equations (1) and (2) are satisfied, the SMBH is kicked at Td with a velocity
based on the analytic expansion derived in Naab et al. (2009):

$v^{\rm first}_{\rm kick}=\sqrt{\frac{(1+\eta\epsilon^{2})}{(1+\eta)}}v^{\rm
SMBH}_{\rm c}(100\times R_{\rm s}),$ (3)

with $v^{\rm SMBH}_{\rm c}$, which is the circular velocity of the SMBH at 100
times its Schwarzschild radius $R_{\rm s}$,

$\eta=M^{\rm sat}_{\rm tot}/M^{\rm BCG}_{\rm int}(d),$ (4)

and

$\epsilon=\frac{v^{\rm sat}}{v^{\rm SMBH}_{\rm c}},$ (5)

where $v^{\rm sat}$ the velocity norm of the satellite galaxy at Td (see
Figure 2). Our criterion (2) is established to maximise the energy transferred
to the SMBH via the velocity kick, described by Equations (3) and (4). All the
velocities are calculated relative to the BCG. We then integrate the BH orbit
with its new velocity by applying dynamical friction to it with a constant
mass.

At each subsequent merger, we check if the SMBH, which might have been kicked
from the center of the galaxy, passes close to the inner region of the BCG
heated by the satellite. We, therefore, use the orbital radii of the SMBH and
distance $d$ between the centers of the satellite and the BCG to define a
simple criterion in order to establish if the BH can be affected after this
new merger:

$r^{\rm SMBH}_{\rm orb}\leq d.$ (6)

In order to satisfy this condition, the BH must have had sufficient time to
come back to the BCG centre before the next merger happens. Then, the SMBH is
kicked once again with a new velocity:

$v^{\rm sub}_{\rm kick}=\sqrt{\frac{(1+\eta\epsilon)}{(1+\eta)}}v^{\rm
SMBH}_{\rm N}.$ (7)

The only difference with Equation (3) is that the SMBH is no longer in a
circular orbit and now has a velocity norm $v^{\rm SMBH}_{\rm N}$.

First, we plot the orbit of the SMBH within a BCG which went through multiple
mergers throughout its history. As an example, Figure 2 depicts the four kicks
that a M${}_{\rm SMBH}=2.6\times 10^{10}$ M☉ BH has experienced in a BCG
potential composed of a DM halo of M${}_{\rm DM}=$8.1$\times 10^{12}$ M☉ and a
stellar component of M${}_{*}=$4.4$\times 10^{11}$ M☉ between $z=2$ and $z=0$.
The maximum distance reached by the SMBH is about 150 kpc. The SMBH has been
heated by a first merger between $z=2$ and $z=1$ and by three more mergers
between $z=1$ and $z=0$. We emphasize that the number of mergers shown in
Figure 2 may not be equal to the total number of mergers experienced by the
BCG. Indeed, the BCG has experienced 17 mergers since $z=2$ but only 4 mergers
have satisfied our kicking criteria described by Equations (1), (2) and (6).
In this example, the BCG undergoes its last merger almost 2 Gyr ago, the SMBH
was kicked up to 100 kpc away from the center, which is to say beyond $r^{\rm
BCG}_{\rm hm}=32$ kpc at $z=0$. After being kicked, we observe the orbital
decay of the SMBH in the BCG potential due to dynamical friction. We argue
that the SMBH needs to return to the central region to be kicked again (see
Figure 2). Chances of encounter between an offcenter BH and a falling
satellite are very small. In the case of dynamical interactions out of the
central regions of BCGs, the collision would have little impact. Indeed, it is
really hard to satisfy the criterion (2) in these regions of BGCs as the major
impact of satellites takes place especially in the central region where the
enclosed mass of BCG is smaller (at large distances d, M${}^{\rm sat}_{\rm
tot}$ would need to be comparable to the total mass of the BCG M${}^{\rm
BCG}_{\rm int}(d)$). We stress that at $z=0$, the BH is still significantly
off-centered by about 10 kpc.

We apply this procedure to our full sample of 370 galaxy clusters from TNG300.

##  3 Results

Figure 3: Mergers in our full sample: Satellite stellar mass as a function of
the BCG stellar mass. We distinguish two different categories of mergers in
our full sample of 370 BCGs: mergers which have kicked the central BH (green
points, 229 mergers), and mergers which have not affected the BH (orange
points, 6628 mergers). The dotted line represents a constant ratio of 100 in
mass. The majority of mergers leave the BH unaffected. BHs are mainly kicked
by satellites which have stellar masses $M^{\rm sat}_{*}>M^{\rm BCG}_{*}/100$.
Figure 4: Off-centered black holes in BCGs: Cumulative histograms showing the
BH offset at $z=0$ in yellow, and the maximum offset measured since $z=2$ in
maroon. In almost half of our sample (170 BCGs), the SMBH is still off-
centered ($r>0.01$ kpc) at $z=0$. About 60$\%$ of these SMBHs are located at
$r>0.1$ kpc at present time, and more than 80$\%$ of these SMBHs were off-
centered by at least 1 kpc during their dynamical history. Figure 5: Black
hole offset time: Cumulative histogram of the total duration of our 170 BHs
with an offset $>$ 10, 100 and 1000 pc since $z=2$. A huge percentage 85$\%$
(60$\%$) of the BCGs exhibit a SMBH off-centered with $r>$ 10 pc ($r>$ 100 pc)
for at least 6 Gyr. Besides, these BHs spend less than 3 Gyr above 1 kpc.
Figure 6: Offset epochs: Cumulative histograms showing the times when the
first (yellow), the maximum (maroon) and the last (green) kick happened. The
vertical black dashed lines show $z=2$, 1 and 0. 70$\%$ of our 170 BHs
experienced their first kick before $z=1$. The merger which kicked the central
SMBH to the maximum distance does not take place at a particular time since
$z=2$, but still happens much earlier than the last merger. 65$\%$ of the
clusters have undergone their last mergers after $z=1$. That is the reason why
it is very likely that SMBHs in BCGs are still off-centered at $z=0$. Figure
7: Dynamical friction on black holes: BH offset as a function of its mass at
$z=0$ (yellow dots) and at the time of the maximum offset (maroon dots). More
massive BH exhibits larger offsets than less massive ones because dynamical
friction acting on BHs becomes significantly weaker as the total galaxy mass
grows.

Here we present our results by considering our full sample of 370 BCGs and
thus 370 orbiting BHs, tracked with orbital integrations.

As depicted by Figure 3, not all mergers undergone by the BCGs perturb the
central BH. We derive that BHs are mainly kicked by satellites which have
stellar masses $M^{\rm sat}_{*}>M^{\rm BCG}_{*}/100$. We confirm that massive
satellites on a radial orbit are rare (only 229 mergers out of all 6628
mergers accounted for all 370 clusters between $z=2$ and $z=0$ meet Equations
(1) and (2)), but their rareness does not invalidate their efficiency in
displacing the BH of BCGs. Almost half of BHs ($\sim$170) of our sample were
off-centered during their orbital history. Some of the more massive satellites
also appear to not have decentered the BH, as indicated by the orange island
of points above the line. Out of all mergers with a stellar mass ratio $M^{\rm
sat}_{*}>M^{\rm BCG}_{*}/100$, 220 out of 3500 satellites kicked away the BHs
from their central positions. Out of the 3280 satellites that did not manage
to move the BH, 95% are massive satellites which did not satisfy Equation (1),
i.e. their orbits are not radial. The remaining 5% are massive satellites that
do have a radial orbit, but the BH was still off-centered at the moment of the
merger and was thus not affected. Thus, we predict two different populations
of BHs: those that have always resided at the centre of their BCGs and those
that have been off-centered. Hence, we expect that these BHs exhibit
differences in accretion, growth and feedback.

For all the 170 BHs heated by satellite galaxies, we analyse their orbit
between $z=2$ and $z=0$. Figure 4 shows the cumulative distribution of the
SMBH offset at $z=0$ and the maximum offset measured since $z=2$. In almost
all these BCGs, the SMBH is still off-centered at $z=0$. About $60\%$ of these
SMBHs are located at $r>0.1$ kpc at present time, and almost all SMBHs were
off-centered by at least 1 kpc during their dynamical history. The maximum
offset reached by these BHs is about 200 kpc (see Figure 4). We also
demonstrate that even if BHs have been significantly displaced at a specific
redshift, BHs have sufficient time to decrease their offset. We stress that
below 10 pc, the determination of SMBH orbits is not accurate as the dynamical
friction becomes inefficient and three-body interactions with surrounding
stars or other orbiting BHs can cause the BH orbit to further decay (Sesana &
Khan, 2015; Dosopoulou & Antonini, 2017; Ryu et al., 2018).

We now evaluate the time spent by the SMBH at distances longer than 10, 100
and 1000 pc from the BCG center. Figure 5 illustrates the cumulative histogram
of the duration of the SMBH offset. We state that 85$\%$ (60$\%$) of the BCGs
exhibit a SMBH with an offset of 10 pc (100 pc) for more than 6 Gyr since
$z=2$. This means that SMBHs in BCGs spent more than half of their lifetime
off-centered. We also notice that these BHs spend less than 3 Gyr above 1 kpc.

Besides, we show in Figure 6 the cumulative distribution of the specific times
at which the first merger, the merger which has led to the maximum offset, and
the last merger happened. We establish that 70$\%$ of our 170 BHs experienced
their first kick before $z=1$. Hence, SMBHs do not inhabit the center of the
BCG potential already at early times of the Universe. Most BHs have undergone
their most important merger in terms of dynamical heating, i.e. the one that
displaces the most our BHs, after $z=1$, but much earlier than their last
mergers. We then establish that the energy given by subsequent mergers is one
of the key ingredient that permits BHs to reach these far distances of the
order of a tens of kpc. More precisely, as a result of the first merger, BHs
gain energy from the satellite by changing its velocity according to Equation
(3). Before the second kick, BHs have eccentric orbits and thus are more
energetic. This dynamical behaviour allows them to reach velocities which are
higher than the escape velocity computed at their position, until large radius
from the galaxy centre. Finally, we demonstrate that 65$\%$ of the clusters
have undergone their last mergers after $z=1$. This phenomenon naturally
explains why it is very likely that SMBHs in BCGs are still off-centered at
$z=0$. Indeed, these BHs do not have a sufficient time to loose their angular
momentum via dynamical friction and then reach the central region of the
galaxy, where they are expected to reside.

Finally, Figure 7 displays the BH offset as a function of the BH mass, both at
$z$ = 0 and at the time the SMBH suffered its most significant kick. We recall
that MBH was fixed at the mass derived from the BCG mass at the first $T_{d}$.
We report a correlation between BH mass and its offset. Indeed, more massive
BH exhibits larger offsets than less massive ones at $z$ = 0. As a result, it
is more likely to find off-centered SMBHs in the most massive BCGs. In the
hierarchical structure formation model, the density at the centre decreases as
the total galaxy mass grows. In fact, dynamical friction acting on BHs becomes
significantly weaker and then BHs take more time to sink towards the centre of
their BCGs, as demonstrated in Figure 7.

##  4 Discussion

BH offsets of a few dozen parsecs were shown to have huge consequences on the
physics that reign at the center of galaxies (Smith et al., 2018; Bahé et al.,
2022). Even at such small distances, Boldrini et al. (2020) have pointed out
that accretion by the SMBH in dwarf galaxies would become inefficient, due to
the lower density of gas at higher radii from the center. Smith et al. (2018)
indicated that this might also be the case for more massive galaxies, such as
BCGs.

Our results clearly establish that SMBH offsets are common in BCGs as they
have undergone frequent dynamical perturbations. Indeed, about a third of our
SMBH sample are off-centred at the present time, and half of SMBHs were off-
centered by at least 1 kpc during their dynamical history. Similarly to what
was shown for dwarf galaxies (Bellovary et al., 2021), we demonstrate that the
reason for off-center locations is also mainly due to galaxy-galaxy mergers
and more precisely mergers with satellites on radial orbits. The orbital
offset in BCGs can sustain up to at least 6 Gyr between $z=2$ and $z=0$ in
half of our BCGs. Our result reinforces the prediction of a population of off-
centered BHs, not only in dwarf galaxies (Governato et al., 1994; Rashkov &
Madau, 2014; Micic et al., 2011; Islam et al., 2004; Volonteri et al., 2003;
Schneider et al., 2002; Bellovary et al., 2018; Boldrini et al., 2020), but
also in very massive galaxies such as BCGs.

The fact that half of our SMBHs spend almost all their dynamical history
offset from the BCG center should have major consequences on the BH and galaxy
formation and evolution as there is an interplay between the feeding/feedback
mechanisms of the SMBH and the host galaxy (Silk & Rees, 1998). In particular,
it was pointed out that BHs must intersect with a highly accretable clump at
some time in order to accrete gas efficiently (Smith et al., 2018). Hence, as
off-centered BHs seems to be very common in BCGs, we expect that they accrete
gas very inefficiently as gas clumps are centrally located. As a result, off-
centered BHs should experience essentially no growth at all. It was already
demonstrated that in cosmological simulations, SMBH offset has a dramatic
effect on BH growth (Bahé et al., 2022). It is established that SMBHs gain
mass via gas accretion (Soltan, 1982) and via BH mergers. Even if BH mergers
are strongly subdominant to gas accretion for SMBHs (Ni et al., 2022; Zhang et
al., 2021), we pointed out that both mechanisms are effectively halted by
mergers of satellites on radial orbits in BCGs.

According to our results, we expect that growth and feedback for a non-
negligible SMBH population in BCGs was quenched between $z=2$ and $z=1$ until
$z=0$. For this reason, we predict the presence of a population of SMBHs with
very different masses. It will result in a larger scatter in mass compared to
what cosmological simulations predict. However, observations of BHs show the
presence of a scatter around the MBH \- MBCG relation (Gaspari et al., 2019;
Bogdán et al., 2018; Savorgnan et al., 2016; Main et al., 2017; McConnell &
Ma, 2013a). Such scatter is evidence of the complex co-evolution of SMBHs with
their host galaxies in clusters (Volonteri et al., 2021). In particular, we
establish that the growth of these SMBHs seems to be governed by one property
of their host clusters, i.e. the number of satellites on radial orbits.

The absence of gas accretion by the SMBHs will also obviously influence their
feedback (Heckman & Best, 2014). If the SMBHs are not accreting, BH feedback
also becomes inefficient. Nevertheless, it is well established that the BH
feedback can alter the DM distribution of galaxies. In fact, the repeating
episodes of gas ejection and recycling can flatten the cuspy DM profile until
the formation of a constant density in the central region of galaxies, called
the DM core. This effect on the DM profile has been studied in massive
galaxies in the Horizon-AGN simulation (Peirani et al., 2017, 2019) and in the
NIHAO-AGN simulations (Macciò et al., 2020).They showed that BH feedback can
very slightly flatten the DM profile of BCGs. Our result, i.e the quenching of
BH feedback due to the BH offset, reinforces the prediction concerning the
inefficiency of this mechanism to create DM cores in massive galaxies such as
BCGs.

This work highlights a major problem encountered in simulations which pin the
SMBH to the center of the potential of the cosmological box (Bahé et al.,
2021). Doing so has many advantages: keeping the BH at the center renders sub-
grid AGN models effective, and this ensures that the initial merger-driven
growth of BHs can proceed rapidly as this early phase is highly uncertain.
This also avoids the question of dynamical friction, which is still treated
poorly within simulations (Morton et al., 2021). However this results in
higher accretion rates, overestimated BH growth and higher counts of BH-BH
mergers (see Bahé et al., 2021; Barausse et al., 2020, and references
therein). We confirm that BH repositioning in simulations (see Section 2.2 of
Weinberger et al., 2017, which details the repositioning of the BH in the TNG
simulation,) is nonphysical and show the need of new repositioning and
dynamical friction models in order to correctly model the dynamics in galaxy
clusters. There have now been many efforts on testing and incorporating
dynamical friction from gas and stars in cosmological simulations (e.g Tremmel
et al., 2015; Pfister et al., 2019; Chen et al., 2022), which do produce off-
center SMBHs.

##  5 Conclusion

Making use of the work of Barnes et al. (2018), we study how satellites affect
the dynamics in the central region of 370 different BCGs from the TNG300
simulation, between redshift $z$ = 2 and $z$ = 0. By using orbital integration
methods, we show how galaxies which fall in the potential of the BCG can heat
dynamically the center of BCGs, and kick away the central SMBH to distances
from a few parsecs up to a hundred kiloparsecs. Our study demonstrates that
half of SMBHs in BCGs, more particularly those in the most massive galaxies,
were kicked away since $z$ = 2, but only a third are still off-centered by
more than 200 pc today. These BHs spent most of their lives outside the
central region of their host, which has consequences on the physics that reign
at the center of clusters.

As gas is mostly condensed in the center of galaxies, BHs, if offcentered, do
not encounter as many gas clumps. As a result, BHs can not accrete gas
efficiently. Consequently, the growth of SMBH may have been overestimated.
Indeed, the lack of efficient accretion means that they do not gain much mass
in their lifetime. As the central SMBH can not accrete matter efficiently, AGN
feeback also becomes inefficient.

Following this study, analyzing how mergers can affect the distribution of DM
in the central regions of BCGs might also be an interesting subject of
research. Indeed, the multiple passage of satellites in the central region of
BCGs may be responsible for ejecting DM out of the center of the galaxy,
resulting in the flattening of the central DM density.

The presence of such a population of offcentered BHs in BCGs brings up the
question of their detections and the identification of BCGs which may host a
SMBH which has been kicked away in the recent past. Traces in the morphology
of the BCG such as stellar tails or streams may hint at the passage of a
massive satellite and thus at a recent merger. Observations of the central
regions of these galaxies with integral field units such as the Multi Unit
Spectroscopic Explorer (MUSE) may enable us to pinpoint the location of a
potential offcentered BH by looking for the presence of a broadline region,
characteristic of Seyfert galaxies or associated with the presence of an AGN
(Gaskell, 2009). This is still a challenging project as Elitzur et al. (2014)
show that the broadline emission decreases as the accretion rate of the BH
decreases as well. If offcentered, we would thus not expect an intense
emission.

In this paper, we only considered the central BH of the BCG. However,
satellites may harbor a SMBH as well in their centers. We could thus expect a
population of wandering BHs of different masses in the BCG. The
characterization of this population of BHs and the knowledge of their orbits
in the potential of the BCGs may give us a better idea of the rate of BH-BH
mergers observed in the future.

## Acknowledgements

We thank Josh Borrow and Mark Vogelsberger for providing the Illustris data,
and for useful comments. We also thank Florence Durret, Warren Massonneau and
Yohan Dubois for fruitful discussions concerning this project. Finally, we
thank the referee for his constructive comments and suggestions. This work was
supported by the EXPLORAGRAM Inria AeX grant.

## Data Availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

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##  Appendix A Dynamical friction model

In this appendix, we discuss the sensitivity of the dynamical friction model
chosen in our orbital integrations to track the orbit of central SMBHs in BCG
potentials. The dynamical friction model is described in section 2.2. In
addition, we evaluate here the impact of the $\gamma$ parameter, defined as
the absolute value of the logarithmic slope of the density profile of BCGs.
Indeed, this parameter controls the amplitude of the dynamical friction force
during orbital integrations. In our study, $\gamma$ was set to 1. According to
Figure 8, we observe a very slight deviation in both cumulative histograms by
decreasing the $\gamma$ parameter. This figure demonstrates that modifying
this dynamical friction parameter does not significantly affect our results.

Figure 8: Impact of the $\gamma$ parameter: Cumulative histograms showing the
BH offset at $z=0$ in yellow, and the maximum offset measured since $z=2$ in
maroon. About 55$\%$ (60$\%$) of these SMBHs are located at $r>0.1$ kpc at
present time, and more than 80$\%$ (80$\%$) of these SMBHs were off-centered
by at least 1 kpc during their dynamical history for $\gamma=0.5$
($\gamma=1$).

##  Appendix B Assumption of fixed satellite mass

In order to test our assumption of fixed satellite mass during the orbital
integrations, we measure the mass loss of satellite galaxies between $z=2$ and
the time of the first BH kick ($T_{d}$) in Illustris-TNG simulation. Figure 9
depicts that below $10^{10}$ M☉ (above $10^{11}$ M☉) for the satellite mass at
$z=2$, we over(under)estimated the satellite mass during our integrations by a
factor of 10. Considering higher(lower) masses for satellites will affect
their orbit by accelerating(delaying) their infall towards the BCG center.
Taking into account the mass loss in orbital integration remains a challenging
problem as it is computationally costly and also difficult to estimate due to
the lack of resolution of the Illustris-TNG300 snapshot.

Figure 9: Absence of mass loss during integrations: Mean satellite mass at
$T_{d}$ as function the satellite mass at $z=2$ for all the first mergers in
BCGs. We display the mean over the satellite mass range at $z=2$, for better
visualisation. Error bars represent 1-$\sigma$ Poisson uncertainties.