# On unit weighted zero-sum constants of $\mathbb{Z}_{n}$

Santanu Mondal  Ramakrishna Mission Vivekananda Educational and Research
Institute, Belur, Dist. Howrah, 711202, India Krishnendu Paul  Ramakrishna
Mission Vivekananda Educational and Research Institute, Belur, Dist. Howrah,
711202, India Shameek Paul E-mail addresses:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur, Dist. Howrah, 711202, India

###### Abstract

Given $A\subseteq\mathbb{Z}_{n}$, the constant $C_{A}(n)$ is defined to be the
smallest natural number $k$ such that any sequence of $k$ elements in
$\mathbb{Z}_{n}$ has an $A$-weighted zero-sum subsequence having consecutive
terms. The value of $C_{U(n)}(n)$ is known when $n$ is odd. We give a
different argument to determine the value of $C_{U(n)}(n)$ for any $n$. A
$C$-extremal sequence for $U(n)$ is a sequence in $\mathbb{Z}_{n}$ whose
length is $C_{U(n)}(n)-1$ and which does not have any $U(n)$-weighted zero-sum
subsequence having consecutive terms. We characterize the $C$-extremal
sequences for $U(n)$ when $n$ is a power of 2. For any $n$, we determine the
value of $C_{A}(n)$ where $A$ is the set of all odd (or all even) elements of
$\mathbb{Z}_{n}$ and also when $A=\\{1,2,\ldots,r\\}$ where $r<n$.

##  1 Introduction

We denote the number of elements in a finite set $S$ by $|S|$. In this paper,
$R$ will denote a ring with unity. The next three definitions are given in
[8].

######  Definition 1.1.

Let $M$ be an $R$-module and $A\subseteq R$. A subsequence $T$ of a sequence
$S=(x_{1},x_{2},\ldots,x_{k})$ in $M$ is called an $A$-weighted zero-sum
subsequence if the set $I=\\{i:x_{i}\in T\\}$ is non-empty and for each $i\in
I$, there exist $a_{i}\in A$ such that $\sum_{i\in I}a_{i}x_{i}=0$.

######  Definition 1.2.

Given a finite $R$-module $M$ and $A\subseteq R$, the $A$-weighted Davenport
constant $D_{A}(M)$ is the least positive integer $k$ such that any sequence
in $M$ of length $k$ has an $A$-weighted zero-sum subsequence.

######  Definition 1.3.

Given a finite $R$-module $M$ and $A\subseteq R$, the constant $C_{A}(M)$ is
the least positive integer $k$ such that any sequence in $M$ of length $k$ has
an $A$-weighted zero-sum subsequence of consecutive terms.

The next constant has been studied when $M=R=\mathbb{Z}_{n}$. Some results are
given in [1], [3], [6], [7] and [13].

######  Definition 1.4.

Given a finite $R$-module $M$ and $A\subseteq R$, the $A$-weighted Gao
constant $E_{A}(M)$ is the least positive integer $k$ such that any sequence
in $M$ of length $k$ has an $A$-weighted zero-sum subsequence of length $|M|$.

######  Remark 1.5.

We denote the above three constants by $D(M),C(M)$ and $E(M)$ when
$A=\\{1\\}$. For a finite $R$-module $M$, we have $D_{A}(M)\leq
C_{A}(M)\leq|M|$ (see Section 1 of [8]). Also, it follows easily that
$E_{A}(M)\geq D_{A}(M)+|M|-1$.

When $A\subseteq\mathbb{Z}_{n}$, we denote the constants
$D_{A}(\mathbb{Z}_{n})$, $E_{A}(\mathbb{Z}_{n})$ and $C_{A}(\mathbb{Z}_{n})$
by $D_{A}(n)$, $E_{A}(n)$ and $C_{A}(n)$ respectively. Yuan and Zeng have
shown in [14] that $E_{A}(n)=D_{A}(n)+n-1$.

######  Definition 1.6.

Let $A\subseteq\mathbb{Z}_{n}$. A sequence in $\mathbb{Z}_{n}$ of length
$E_{A}(n)-1$ which does not have any $A$-weighted zero-sum subsequence of
length $n$, is called an $E$-extremal sequence for $A$. A sequence in
$\mathbb{Z}_{n}$ of length $C_{A}(n)-1$ which does not have any $A$-weighted
zero-sum subsequence of consecutive terms, is called a $C$-extremal sequence
for $A$.

We denote the ring $\mathbb{Z}/n\mathbb{Z}$ by $\mathbb{Z}_{n}$. Let $U(n)$
denote the group of units in $\mathbb{Z}_{n}$ and $U(n)^{k}=\\{\,x^{k}:x\in
U(n)\,\\}$. When $n=p_{1}^{r_{1}}\ldots p_{s}^{r_{s}}$ where the $p_{i}$’s are
distinct primes, we let $\Omega(n)=r_{1}+\cdots+r_{s}$. We use the notation
$v_{p}(n)=r$ to mean $p^{r}\mid n$ and $p^{r+1}\nmid n$. For
$a,b\in\mathbb{Z}$, we denote the set $\\{k\in\mathbb{Z}:a\leq k\leq b\\}$ by
$[a,b]$.

In Corollary 4 of [8] it has been shown that $C_{U(n)}(n)=2^{\Omega(n)}$ when
$n$ is odd. In Theorem 3.1 we show that $C(\mathbb{Z}_{2}^{{}^{a}})=2^{a}$ and
as a consequence of this in Theorem 3.8 we show that for any $n$ we have
$C_{U(n)}(n)=2^{\Omega(n)}$. Thus, we obtain a new proof of Corollary 4 of [8]
while also generalizing it.

In Theorem 4.1 we rederive the result that $D(\mathbb{Z}_{2}^{{}^{a}})=a+1$.
This result is a special case of the main result of Olson in [11]. We have
included it here as our proof is very elementary. As a consequence of this we
rederive the result that $D_{U(n)}=\Omega(n)+1$. This can also be arrived at
by using the result of Yuan and Zeng in [14], as the value of $E_{U(n)}(n)$
has been obtained in [6] and [7].

We feel that it is of interest to derive the value of $D_{U(n)}(n)$ without
using the value of $E_{U(n)}(n)$ since for other weight-sets
$A\subseteq\mathbb{Z}_{n}$, the values of $D_{A}(n)$ have been determined
independently of the corresponding values of $E_{A}(n)$. Some of the other
results in this article are the following.

  * •

For a finite abelian group $G$ we have that $C(G)=|G|$.

  * •

We characterize the $C$-extremal and $E$-extremal sequences for $U(n)$ when
$n$ is a power of 2.

  * •

Let $n$ be even, $v_{2}(n)=r$ and $m=n/2^{r}$. Let $B$ denote the set of all
odd elements of $\mathbb{Z}_{n}$ and $\\{m\\}\subseteq A\subseteq B$. Then we
have that $C_{A}(n)=2^{r}$.

  * •

Let $r\in[1,n-1]$ and $A=\\{1,2,\ldots,r\\}$. We show that $C_{A}(n)=\lceil
n/r\rceil$.

  * •

If $p$ and $k$ are odd primes such that $p\equiv 1~{}(mod~{}k)$ and
$p\not\equiv 1~{}(mod~{}k^{2})$, then $D_{U(p)^{k}}(p)\leq k$.

##  2 Some general results

######  Theorem 2.1.

Let $M$ be a finite $R$-module and $N$ be a finite $R^{\prime}$-module.
Suppose $A\subseteq R$ and $B\subseteq R^{\prime}$. Then $M\times N$ is a
module over $R\times R^{\prime}$ and we have $C_{A\times B}(M\times N)\geq
C_{A}(M)\,C_{B}(N)$.

###### Proof.

Let $C_{A}(M)=k+1$ and $C_{B}(N)=l+1$. Suppose $S_{1}=(x_{1},\ldots,x_{k})$ is
a sequence in $M$ which does not have any $A$-weighted zero-sum subsequence of
consecutive terms and $S_{2}=(y_{1},\ldots,y_{l})$ is a sequence in $N$ which
does not have any $B$-weighted zero-sum subsequence of consecutive terms.
Consider the sequence $S$ in $M\times N$ defined as

$\big{(}(x_{1},0),\ldots,(x_{k},0),(0,y_{1}),(x_{1},0),\ldots,(x_{k},0),(0,y_{2}),(x_{1},0),\ldots,(x_{k},0),(0,y_{3}),$
$.\,.\,.\,.\,,(x_{1},0),\ldots,(x_{k},0),(0,y_{l-1}),(x_{1},0),\ldots,(x_{k},0),(0,y_{l}),(x_{1},0),\ldots,(x_{k},0)\big{)}.$

Suppose $S$ has an $A\times B$-weighted zero-sum subsequence $T$ having
consecutive terms. If no term of $T$ is of the form $(0,y_{j})$ for any
$j\in[1,l]$, then we get the contradiction that $S_{1}$ has an $A$-weighted
zero-sum subsequence of consecutive terms. If $T$ has a term of the form
$(0,y_{j})$ for some $j\in[1,l]$, then by taking the projection to the second
coordinate, we get the contradiction that $S_{2}$ has a $B$-weighted zero-sum
subsequence of consecutive terms. Thus, we see that $S$ does not have any
$A\times B$-weighted zero-sum subsequence of consecutive terms. Hence, we get
that $C_{A\times B}(M\times N)\geq C_{A}(M)\,C_{B}(N)$. ∎

By a similar argument we get the next result.

######  Theorem 2.2.

Let $M$ and $N$ be $R$-modules and $A\subseteq R$. Then $M\times N$ is an
$R$-module and we have $C_{A}(M\times N)\geq C_{A}(M)\,C_{A}(N)$.

######  Theorem 2.3.

Let $M$ be a finite $R$-module and $N$ be a finite $R^{\prime}$-module.
Suppose $A\subseteq R$ and $B\subseteq R^{\prime}$. Then $M\times N$ is a
module over $R\times R^{\prime}$ and we have $D_{A\times B}(M\times N)\geq
D_{A}(M)+D_{B}(N)-1$.

###### Proof.

Let $D_{A}(M)=k+1$ and $D_{B}(N)=l+1$. Suppose $S_{1}=(x_{1},\ldots,x_{k})$ is
a sequence in $M$ which does not have any $A$-weighted zero-sum subsequence
and $S_{2}=(y_{1},\ldots,y_{l})$ is a sequence in $N$ which does not have any
$B$-weighted zero-sum subsequence. Let $S$ be the sequence in $M\times N$
defined as

$\big{(}(x_{1},0),(x_{2},0),\ldots,(x_{k-1},0),(x_{k},0),(0,y_{1}),(0,y_{2}),\ldots,(0,y_{l-1}),(0,y_{l})\big{)}.$

Suppose $S$ has an $A\times B$-weighted zero-sum subsequence $T$. If no term
of $T$ is of the form $(0,y_{j})$ for any $j\in[1,l]$, then we get the
contradiction that $S_{1}$ has an $A$-weighted zero-sum subsequence. If $T$
has a term of the form $(0,y_{j})$ for some $j\in[1,l]$, then by taking the
projection to the second coordinate, we get the contradiction that $S_{2}$ has
a $B$-weighted zero-sum subsequence. Thus, we see that $S$ does not have any
$A\times B$-weighted zero-sum subsequence. Hence, we get that $D_{A\times
B}(M\times N)\geq D_{A}(M)+D_{B}(N)-1$. ∎

By a similar argument we get the next result.

######  Theorem 2.4.

Let $M$ and $N$ be $R$-modules and $A\subseteq R$. Then $M\times N$ is an
$R$-module and we have $D_{A}(M\times N)\geq D_{A}(M)+D_{A}(N)-1$.

##  3 $C_{U(n)}(n)$

######  Theorem 3.1.

$C(\mathbb{Z}_{2}^{{}^{a}})=2^{a}$.

###### Proof.

We will prove that $C(\mathbb{Z}_{2}^{{}^{a}})\geq 2^{a}$ by using induction
on $a$. This is clear when $a=1$. For $a>1$, by the induction hypothesis we
have $C(\mathbb{Z}_{2}^{{}^{a-1}})\geq 2^{a-1}$. Taking $A=\\{1\\}$,
$M=\mathbb{Z}_{2}^{{}^{a-1}}$ and $N=\mathbb{Z}_{2}$ in Theorem 2.2 gives us
$C(\mathbb{Z}_{2}^{{}^{a}})\geq 2^{a}$. Also, from Theorem 1 of [8] we get
$C(\mathbb{Z}_{2}^{{}^{a}})\leq 2^{a}$. ∎

######  Theorem 3.2.

For any finite abelian group $G$, we have that $C(G)=|G|$.

###### Proof.

By induction on the rank of $G$, Corollary 1 of [8] and Theorem 2.2 we can
show that $C(G)\geq|G|$. Also, from Theorem 1 of [8] we get that
$C(G)\leq|G|$. ∎

######  Remark 3.3.

Theorem 3.2 is in contrast to Theorem 1 of [12] which says that for a finite
abelian group $G$ we have that $D(G)=|G|$ if and only if $G$ is a finite
cyclic group.

Let $x=(x_{1},\ldots,x_{m}),y=(y_{1},\ldots,y_{m})\in\mathbb{Z}_{2}^{{}^{m}}$.
We define the dot product of $x$ and $y$ to be $x\cdot
y=x_{1}y_{1}+\ldots+x_{m}y_{m}$.

######  Lemma 3.4.

Let $v_{1},\ldots,v_{a}\in\mathbb{Z}_{2}^{{}^{m}}$ where $m\geq 2^{a}$. Then
there exists a non-zero vector $w\in\mathbb{Z}_{2}^{{}^{m}}$ such that for
each $i\in[1,a]$ we have $w\cdot v_{i}=0$ and the coordinates of $w$ which are
equal to one occur in consecutive positions.

###### Proof.

Let $P$ be the matrix of size $m\times a$ whose columns are the vectors
$v_{i}$. If $w\in\mathbb{Z}_{2}^{{}^{m}}$, then $wP$ is the vector in
$\mathbb{Z}_{2}^{{}^{a}}$ which is the sum of those rows of the matrix $P$
which correspond to the coordinates of $w$ which are 1. From Theorem 3.1 we
have $C(\mathbb{Z}_{2}^{{}^{a}})=2^{a}$. As $m\geq 2^{a}$, the sum of some
consecutive rows of $P$ is zero. Thus, we can find a vector
$w\in\mathbb{Z}_{2}^{{}^{m}}$ as in the statement of the lemma. ∎

######  Lemma 3.5.

Let $X_{1},\ldots,X_{a}$ be subsets of $[1,m]$ where $m\geq 2^{a}$. Then there
exists $Y\subseteq\\{1,\ldots,m\\}$ such that the elements of $Y$ are
consecutive numbers and for each $i\in[1,a]$ we have $|Y\cap X_{i}|$ is even.

###### Proof.

We identify a subset $A\subseteq[1,m]$ with the vector
$x_{A}\in\mathbb{Z}_{2}^{{}^{m}}$ whose $j^{th}$ coordinate is one if and only
if $j\in A$. For any two subsets $A$ and $B$ of $[1,m]$ we observe that
$x_{A}\cdot x_{B}=0$ if and only if $|A\cap B|$ is even. So we see that Lemma
3.5 follows from Lemma 3.4. ∎

For a divisor $m$ of $n$, we define the natural map
$f_{n,m}:\mathbb{Z}_{n}\to\mathbb{Z}_{m}$ to be the map given by
$f_{n,m}\big{(}x+n\mathbb{Z}\big{)}=x+m\mathbb{Z}$. Let $p$ be a prime divisor
of $n$ with $v_{p}(n)=r$. For a sequence $S$ in $\mathbb{Z}_{n}$, we denote
its image under $f_{n,p^{r}}$ by $S^{(p)}$.

######  Observation 3.6.

A sequence $S$ is a $U(n)$-weighted zero-sum sequence in $\mathbb{Z}_{n}$ if
and only if for every prime divisor $p$ of $n$ the sequence $S^{(p)}$ is a
$U(p^{r})$-weighted zero-sum sequence in $\mathbb{Z}_{p^{r}}$ where
$v_{p}(n)=r$.

This is Observation 2.2 in [6]. The next result follows from Lemma 2.4 in [6]
along with the remark which follows it.

######  Lemma 3.7.

Let $p$ be a prime, $n=p^{r}$ for some $r$ and $S=(x_{1},\ldots,x_{m})$ a
sequence in $\mathbb{Z}_{n}$. Suppose for each $i\in[1,r]$, the size of the
set $X_{i}$ is even where $X_{i}=\\{j:x_{j}\not\equiv 0~{}(mod~{}p^{i})\\}$.
Then $S$ is a $U(n)$-weighted zero-sum sequence.

######  Theorem 3.8.

$C_{U(n)}(n)=2^{\Omega(n)}$.

###### Proof.

By Corollary 2 of [8] we have $C_{U(n)}(n)\geq 2^{\Omega(n)}$. Let
$a=\Omega(n)$ and $S=(x_{1},\ldots,x_{m})$ be a sequence in $\mathbb{Z}_{n}$
of length $m=2^{a}$. If we show that $S$ has a $U(n)$-weighted zero-sum
subsequence of consecutive terms, it will follow that $C_{U(n)}(n)\leq
2^{\Omega(n)}$. For any prime divisor $p$ of $n$ and for each
$i\in[1,v_{p}(n)]$ we let

$X_{i}^{(p)}=\\{\,j:x_{j}\not\equiv 0~{}(mod~{}p^{i})\,\\}.$

We observe that we have $a$ sets in the collection

$\big{\\{}\,X_{i}^{(p)}:p~{}\textrm{is a prime divisor
of}~{}n~{}\textrm{and}~{}i\in[1,v_{p}(n)]\,\big{\\}}.$

By Lemma 3.5 we have a subset $Y\subseteq[1,m]$ such that all the elements of
$Y$ are consecutive numbers and for each prime divisor $p$ of $n$ and for each
$i\in[1,v_{p}(n)]$ we have $|\,Y\cap X_{i}^{(p)}\,|$ is even. Let $T$ be the
subsequence of $S$ such that $x_{j}$ is a term of $T$ if and only if $j\in Y$.

Let $q$ be a prime divisor of $n$ and $v_{q}(n)=r$. By Lemma 3.7 the sequence
$T^{(q)}$ is a $U(q^{r})$-weighted zero-sum sequence in $\mathbb{Z}_{q^{r}}$
as for each $i\in[1,r]$ the set

$Y\cap X_{i}^{(q)}=\\{\,j\in Y:x_{j}\not\equiv 0~{}(mod~{}q^{i})\,\\}$

has even size. Thus, by Observation 3.6 it follows that the sequence $T$ is a
$U(n)$-weighted zero-sum sequence. Hence, we see that $S$ has a
$U(n)$-weighted zero-sum subsequence of consecutive terms. ∎

##  4 $D_{U(n)}(n)$

The next result is a special case of the main result in [11]. We are able to
give a very simple argument and have hence included the proof. We can also
prove Theorem 4.1 by using Theorem 2.4 and the easy observation that
$D(\mathbb{Z}_{2})=2$.

######  Theorem 4.1.

$D(\mathbb{Z}_{2}^{{}^{a}})=a+1$.

###### Proof.

Let $B=\\{x_{1},\ldots,x_{a}\\}$ be a basis of the $\mathbb{Z}_{2}$-vector
space $\mathbb{Z}_{2}^{{}^{a}}$. As $B$ is linearly independent, the sequence
$S=(x_{1},\ldots,x_{a})$ does not have any zero-sum subsequence and so we see
that $D(\mathbb{Z}_{2}^{{}^{a}})\geq a+1$. Any sequence of length $a+1$ in
$\mathbb{Z}_{2}^{{}^{a}}$ has a zero-sum subsequence as any set of $a+1$
vectors is linearly dependent. Hence, we see that
$D(\mathbb{Z}_{2}^{{}^{a}})\leq a+1$. ∎

######  Lemma 4.2.

Let $v_{1},\ldots,v_{a}\in\mathbb{Z}_{2}^{{}^{m}}$ where $m\geq a+1$. Then
there exists a non-zero vector $w\in\mathbb{Z}_{2}^{{}^{m}}$ such that for
each $i\in[1,a]$ we have $w\cdot v_{i}=0$.

###### Proof.

Let $P$ be the matrix of size $m\times a$ whose columns are the vectors
$v_{i}$. If $w\in\mathbb{Z}_{2}^{{}^{m}}$, then $wP$ is the vector in
$\mathbb{Z}_{2}^{{}^{a}}$ which is the sum of those rows of the matrix $P$
which correspond to the coordinates of $w$ which are 1. From Theorem 4.1 we
have $D(\mathbb{Z}_{2}^{{}^{a}})=a+1$. As $m\geq a+1$, the sum of some rows of
$P$ is zero. Thus, we can find a vector $w\in\mathbb{Z}_{2}^{{}^{m}}$ as in
the statement of the lemma. ∎

The proof of the next result is similar to the proof of Lemma 3.5.

######  Lemma 4.3.

Let $X_{1},\ldots,X_{a}$ be subsets of $\\{1,\ldots,m\\}$, where $m\geq a+1$.
Then there exists a non-empty subset $Y\subseteq\\{1,\ldots,m\\}$ such that
for each $i\in[1,a]$, we have $|\,Y\cap X_{i}\,|$ is even.

######  Theorem 4.4.

$D_{U(n)}(n)=\Omega(n)+1$.

###### Proof.

For a prime $p$, the argument in the proof of Theorem 2 of [8] shows that
$D_{U(p)}(p)=2$. So from Lemma 1.8 of [10] we see that
$D_{U(n)}(n)\geq\Omega(n)+1$.

Let $S=(x_{1},\ldots,x_{m})$ be a sequence in $\mathbb{Z}_{n}$ of length
$m=\Omega(n)+1$. If we show that $S$ has a $U(n)$-weighted zero-sum
subsequence, then it will follow that $D_{U(n)}(n)\leq\Omega(n)+1$. For any
prime divisor $p$ of $n$ and for each $i\in[1,v_{p}(n)]$ let
$X_{i}^{(p)}=\\{\,j:x_{j}\not\equiv 0~{}(mod~{}p^{i})\,\\}$. The collection

$\\{\,X_{i}^{(p)}:p~{}\textrm{is a prime divisor
of}~{}n~{}\textrm{and}~{}i\in[1,v_{p}(n)]\,\\}$

has $\Omega(n)$ sets. As $m=\Omega(n)+1$, by Lemma 4.3 we have a non-empty
subset $Y\subseteq\\{1,2,\ldots,m\\}$ such that for each prime divisor $p$ of
$n$ and for each $i\in[1,v_{p}(n)]$ we have $|\,Y\cap X_{i}^{(p)}\,|$ is even.
Let $T$ be the subsequence of $S$ such that $x_{j}$ is a term of $T$ if and
only if $j\in Y$.

Let $p$ be a prime divisor of $n$ and let $v_{p}(n)=r$. By Lemma 3.7 the
sequence $T^{(p)}$ is a $U(p^{r})$-weighted zero-sum sequence in
$\mathbb{Z}_{p^{r}}$, as for each $i\in[1,r]$ the set

$Y\cap X_{i}^{(p)}=\\{\,j\in Y:x_{j}\not\equiv 0~{}(mod~{}p^{i})\,\\}$

has even size. Thus, by Observation 3.6 the sequence $T$ is a $U(n)$-weighted
zero-sum sequence. Hence, it follows that $S$ has a $U(n)$-weighted zero-sum
subsequence. ∎

##  5 Zero-sum subsequences of length $m$ in $\mathbb{Z}_{2}^{{}^{a}}$

If $T$ is a subsequence of a sequence $S$, then $S-T$ denotes the sequence
which is obtained by removing the terms of $T$ from $S$.

######  Observation 5.1.

Let $(G,+)$ be a finite abelian group of order $n$ and $a\in G$. Let $S$ be a
sequence of length $n$ in $G$ and $S-a$ denote the sequence which is obtained
by subtracting $a$ from each term of $S$. If $S-a$ is a zero-sum sequence then
$S$ is a zero-sum sequence.

We now rederive a weaker version of Corollary 3.2 of [6]. The proof given in
[6] uses Theorem 1.1 of [4]. We have given a different proof which uses the
value of $D(\mathbb{Z}_{2}^{{}^{a}})$.

######  Theorem 5.2.

Let $S$ be a sequence in $\mathbb{Z}_{2}^{{}^{a}}$ of length $m+a$ where $m$
is even and $m\geq 2^{a}$. Then $S$ has a zero-sum subsequence of length $m$.

###### Proof.

Let $m\geq 2^{a}$ and $S=(x_{1},\ldots,x_{k})$ be a sequence in
$\mathbb{Z}_{2}^{{}^{a}}$ of length $k=m+a$. Suppose each term of $S$ occurs
an even number of times in $S$. As $x+x=0$ for any
$x\in\mathbb{Z}_{2}^{{}^{a}}$, it follows that $S$ has a zero-sum subsequence
of any even length and hence also of length $m$. So we may assume that some
term of $S$ occurs an odd number of times. By Observation 5.1 we can assume
that that term is $0$.

Let $S^{\prime}$ be the unique subsequence of $S$ whose terms are all distinct
and such that each term of $S-S^{\prime}$ occurs an even number of times in
$S-S^{\prime}$. Let the length of $S^{\prime}$ be $k^{\prime}$. Suppose
$k^{\prime}\leq a$. It follows that $k-k^{\prime}=(m+a)-k^{\prime}$ and so
$m\leq k-k^{\prime}$. As $m$ is even, we see that $S-S^{\prime}$ (and hence
$S$) has a zero-sum subsequence of length $m$.

So we may assume that $k^{\prime}\geq a+1$. By Theorem 4.1 we get
$D(\mathbb{Z}_{2}^{{}^{a}})=a+1$ and so we see that $S^{\prime}$ has a zero-
sum subsequence. Let $T$ be a zero-sum subsequence of $S^{\prime}$ having
largest length. As $0$ occurs an odd number of times in $S$, it is a term of
$S^{\prime}$ and so $0$ is a term of $T$. Let the length of $T$ be $l$. As
$D(\mathbb{Z}_{2}^{{}^{a}})=a+1$ and $S^{\prime}-T$ does not have any zero-sum
subsequence, it follows that $k^{\prime}-l\leq a$. We now claim that

$l\leq m\leq(k-k^{\prime})+l.$ (1)

As all terms of $S^{\prime}$ are distinct, it follows that $k^{\prime}\leq
2^{a}$. So as $l\leq k^{\prime}$ and $2^{a}\leq m$, we see that $l\leq m$. As
$k^{\prime}-l\leq a$ and $m=k-a$, we have that $m\leq(k-k^{\prime})+l$. This
proves our claim (1).

Suppose $l$ is even. From (1) we see that $m-l\leq k-k^{\prime}$. As $m-l$ is
even and each term of $S-S^{\prime}$ occurs an even number of times, it
follows that $S-S^{\prime}$ has a zero-sum subsequence having length $m-l$. As
$T$ is a zero-sum subsequence of $S^{\prime}$ having length $l$, we get a
zero-sum subsequence of $S$ having length $m$.

Suppose $l$ is odd. As both $m$ and $k-k^{\prime}$ are even, from (1) we see
that $m\leq(k-k^{\prime})+l-1$. As $m-(l-1)$ is even and each term of
$S-S^{\prime}$ occurs an even number of times, it follows that $S-S^{\prime}$
has a zero-sum subsequence having length $m-(l-1)$. As $0$ is a term of $T$
which has length $l$, we get a zero-sum subsequence of $S^{\prime}$ having
length $l-1$. So we get a zero-sum subsequence of $S$ having length $m$. ∎

######  Lemma 5.3.

Let $v_{1},\ldots,v_{a}\in\mathbb{Z}_{2}^{{}^{m+a}}$ where $m$ is even and
$m\geq 2^{a}$. Then there exists a non-zero vector
$w\in\mathbb{Z}_{2}^{{}^{m+a}}$ such that $w$ has exactly $m$ terms which are
one and for $i\in[1,a]$ we have $w\cdot v_{i}=0$.

###### Proof.

The proof of this lemma is similar to the proof of Lemma 4.2, where we use
Theorem 5.2 in place of Theorem 4.1. ∎

The next result follows from this lemma.

######  Lemma 5.4.

Let $X_{1},\ldots,X_{a}$ be subsets of $\\{1,\ldots,m+a\\}$ where $m$ is even
and $m\geq 2^{a}$. Then there exists a non-empty subset
$Y\subseteq\\{1,\ldots,m+a\\}$ such that $|Y|=m$ and for each $i\in[1,a]$ we
have $|\,Y\cap X_{i}\,|$ is even.

######  Theorem 5.5.

Let $a=\Omega(n)$ and $m$ be an even number such that $m\geq 2^{a}$. Suppose
$S$ is a sequence in $\mathbb{Z}_{n}$ having length $m+a$. Then $S$ has a
$U(n)$-weighted zero-sum subsequence of length $m$.

###### Proof.

The proof of this theorem is similar to the proof of Theorem 4.4, where we use
Lemma 5.4 in place of Lemma 4.3. ∎

######  Remark 5.6.

Theorem 1.3 of [6] is a generalization of Theorem 5.5. The argument which is
given in [6] for even $n$, uses a weaker version of Theorem 1.1 of [4]. We
give a different argument in the proof of Theorem 5.5 which is more direct.

######  Corollary 5.7.

Let $m$ be even and $m\geq 2^{a}$. Then every $m$-dimensional subspace of
$\mathbb{Z}_{2}^{{}^{m+a}}$ has a vector $v$ such that exactly $m$ coordinates
of $v$ are equal to one.

###### Proof.

Let $W$ be an $m$-dimensional subspace of $\mathbb{Z}_{2}^{{}^{m+a}}$. Then
the dimension of $W^{\perp}$ is $a$. Let $\\{v_{1},\ldots,v_{a}\\}$ be a basis
of $W^{\perp}$ and $A$ be the $a\times(m+a)$ matrix whose rows are the basis
vectors of $W^{\perp}$. Then $A$ gives a map
$\mathbb{Z}_{2}^{{}^{m+a}}\to\mathbb{Z}_{2}^{{}^{a}}$ whose kernel is
$(W^{\perp})^{\perp}$ which can be shown to be $W$. For each $j\in[1,m+a]$,
let $C_{j}$ denote the $j^{th}$ column of $A$. By Theorem 5.2 we can find a
subset $I\subseteq[1,m+a]$ such that $|I|=m$ and $\sum_{i\in I}C_{i}=0$. Let
$v=(x_{1},\ldots,x_{m+a})\in\mathbb{Z}_{2}^{{}^{m+a}}$ be such that $x_{i}=1$
if and only if $i\in I$. As we have that $Av=\sum_{i\in I}C_{i}$, it follows
that $v\in ker~{}A=W$. Hence, we see that $W$ has a vector $v$ such that
exactly $m$ coordinates of $v$ are equal to one. ∎

######  Definition 5.8.

For a vector $v\in\mathbb{Z}_{2}^{{}^{n}}$ we define the weight of $v$ to be
the number of coordinates of $v$ which are equal to one. Let $l(n,\bar{m})$
denote the largest integer $k$ such that there is a subspace of
$\mathbb{Z}_{2}^{{}^{n}}$ of dimension $k$ which does not have any vector of
weight $m$. The notation $l(n,\bar{m})$ was introduced in [4].

######  Corollary 5.9.

Let $m$ and $n$ be such that $m$ be even, $m>n$ and $m\geq 2^{n-m}$. Then we
have that $l(n,\bar{m})=m-1$.

###### Proof.

For each $i\in[1,n]$ let $e_{i}$ denote the vector in
$\mathbb{Z}_{2}^{{}^{n}}$ whose only non-zero coordinate is one which occurs
in the $i^{th}$ position. The subspace of $\mathbb{Z}_{2}^{{}^{n}}$ which is
spanned by $\\{e_{1},\ldots,e_{m-1}\\}$ does not have any vector of weight $m$
and hence it follows that $l(n,\bar{m})\geq m-1$. So by using Corollary 5.7 we
see that if $m$ is even, $m>n$ and $m\geq 2^{n-m}$, we have that
$l(n,\bar{m})=m-1$. ∎

Corollary 5.9 is a weaker version of Theorem 1.1 of [4]. We have rederived the
result in Corollary 5.9 here as it is an easy consequence of Theorem 5.2.

##  6 Extremal sequences for $U(n)$ where $n=2^{k}$

######  Definition 6.1.

Let $A$ be a subgroup of $U(n)$. Suppose $S=(x_{1},\ldots,x_{k})$ and
$T=(y_{1},\ldots,y_{k})$ are sequences in $\mathbb{Z}_{n}$. We say that $S$
and $T$ are $A$-equivalent if there is a unit $c\in U(n)$, a permutation
$\sigma\in S_{k}$ and $a_{1},\ldots,a_{k}\in A$ such that for each $i\in[1,k]$
we have $c\,y_{\sigma(i)}=a_{i}x_{i}$.

######  Remark 6.2.

Let $A$ be a subgroup of $U(n)$. If $S$ is an $E$-extremal sequence (a
$D$-extremal sequence) for $A$ and if $S$ and $T$ are $A$-equivalent, then $T$
is also an $E$-extremal sequence (a $D$-extremal sequence) for $A$.

When $n=p^{r}$ where $p$ is an odd prime, in Theorem 3 of [2] it was shown
that a sequence in $\mathbb{Z}_{n}$ is an $E$-extremal sequence for $U(n)$ if
and only if it is $U(n)$-equivalent to the sequence

$(\underbrace{0,0,\ldots,0}_{\text{$(n-1)$ times}},1,p,p^{2},\ldots,p^{r-1}).$

In Theorem 4 of [2] it was shown that when $n=2^{r}$, a sequence in
$\mathbb{Z}_{n}$ is a $D$-extremal sequence for $U(n)$ if and only if it is
$U(n)$-equivalent to the sequence $(1,2,2^{2},\ldots,2^{r-1})$.

The next result has been shown in [6] and [7]. We can also derive this as a
consequence of Theorem 5.5.

######  Theorem 6.3.

For any $n$ we have $E_{U(n)}(n)=n+\Omega(n)$.

######  Theorem 6.4.

A sequence in $\mathbb{Z}_{2^{r}}$ is an $E$-extremal sequence for $U(2^{r})$
if it is $U(2^{r})$-equivalent to a sequence of length $n+r-1$ in which
$2^{i}$ occurs exactly once for each $i\in[0,r-2]$ and there exists an odd
number $m\in[1,2^{r}-1]$ such that $2^{r-1}$ occurs exactly $m$ times and the
remaining terms are zero.

###### Proof.

Let $S$ be a sequence as in the statement of the theorem. Suppose $T$ is a
$U(2^{r})$-weighted zero-sum subsequence of $S$ of length $2^{r}$. Consider
the set $J=\\{\,i\in[0,r-2]:2^{i}\in T\,\\}$. Suppose $J\neq\emptyset$. Then
$J$ has a least element $i_{0}$. As $T$ cannot have only one non-zero term, we
see that $2^{i}$ is a term of $T$ for some $i\in[i_{0}+1,r-1]$. As
$2^{i_{0}+1}$ divides all the terms of $T$ except $2^{i_{0}}$ and as $T$ is a
$U(2^{r})$-weighted zero-sum sequence, we get the contradiction that a unit is
divisible by 2.

Thus we see that $J=\emptyset$. It follows that $T$ is a sequence of length
$2^{r}$ in which there are an odd number of non-zero terms all of which are
equal to $2^{r-1}$. As $T$ is a $U(2^{r})$-weighted zero-sum sequence, we see
that an odd multiple of $2^{r-1}$ is zero. This gives the contradiction that
$2^{r-1}=0$. Hence, we see that $S$ is a sequence of length $2^{r}+r-1$ which
does not have any $U(2^{r})$-weighted zero-sum subsequence of length $2^{r}$.
From Theorem 6.3 we get that $E_{U(2^{r})}(2^{r})=2^{r}+r$. So we see that $S$
is an $E$-extremal sequence for $U(2^{r})$. It follows that a sequence which
is $U(2^{r})$-equivalent to $S$ is also an $E$-extremal sequence for
$U(2^{r})$. ∎

The next result is Lemma 1 (ii) of [7].

######  Lemma 6.5.

If a sequence in $\mathbb{Z}_{2^{r}}$ has a non-zero even number of units,
then it is a $U(2^{r})$-weighted zero-sum sequence.

######  Theorem 6.6.

If a sequence in $\mathbb{Z}_{2^{r}}$ is an $E$-extremal sequence for
$U(2^{r})$, then it is $U(2^{r})$-equivalent to a sequence in which $2^{i}$
occurs exactly once for each $i\in[0,r-2]$ and there is an odd number
$m\in[1,2^{r}-1]$ such that $2^{r-1}$ occurs exactly $m$ times and the
remaining terms are zero.

###### Proof.

Let $n=2^{r}$ and $S$ be a sequence in $\mathbb{Z}_{n}$ which is an
$E$-extremal sequence for $U(n)$. By Theorem 6.3 we have that
$E_{U(n)}(n)=n+r$. So any $E$-extremal sequence for $U(n)$ has length $n+r-1$.
Suppose $S$ has at least two units. By Lemma 6.5 we see that $S$ has a zero-
sum subsequence of length $t$ for any even $t$ which is at most $k-1$. We may
assume that $r\geq 2$ and so we get that $k-1=n+r-2\geq n$. Thus, we get the
contradiction that $S$ has a $U(n)$-weighted zero-sum subsequence of length
$n$. Hence, we see that $S$ has at most one unit.

Let $s\leq r-2$. Suppose for each $i\in[0,s-1]$, the sequence $S$ has at most
one term which is a unit multiple of $2^{i}$. We claim that $S$ has at most
one term which is a unit multiple of $2^{s}$. By our assumption, we see that
$S$ has at least $k-s$ terms which are divisible by $2^{s}$. If our claim is
not true, we can find a subsequence of $S$ having length $k-s-1$ which has an
even number of terms which are a unit multiple of $2^{s}$.

So given any even number $t$ which is at most $k-s-1$, by Lemma 6.5 we see
that $S$ has a $U(n)$-weighted zero-sum subsequence having length $t$. As
$n=k-(r-1)$ and $s\leq r-2$, it follows that $n\leq k-1-s$. Thus, we get the
contradiction that $S$ has a $U(n)$-weighted zero-sum subsequence of length
$n$. Hence, our claim must be true. Thus, we see by induction that for each
$i\in[0,r-2]$ the sequence $S$ can have at most one term which is a unit
multiple of $2^{i}$.

We now claim that for each $i\in[0,r-2]$ the sequence $S$ has exactly one term
which is a unit multiple of $2^{i}$. If not, there are at most $r-2$ such
terms and so $S$ will have at least $k-(r-2)=n+1$ terms which are either zero
or a unit multiple of $2^{r-1}$. We can find a subsequence of $S$ having
length $n$ which has an even number of terms which are a unit multiple of
$2^{r-1}$. So by Lemma 6.5 we get the contradiction that $S$ has a
$U(n)$-weighted zero-sum subsequence of length $n$. Hence, we see that our
claim is true.

Thus, we see that $S$ has $k-(r-1)=n$ terms which are either zero or a unit
multiple of $2^{r-1}$. By Lemma 6.5 we see that the number of terms of $S$
which are a unit multiple of $2^{r-1}$ must be odd. Thus, $S$ is
$U(n)$-equivalent to a sequence in which $2^{i}$ occurs exactly once for each
$i\in[0,r-2]$ and there is an odd number $m\in[1,n-1]$ such that $2^{r-1}$
occurs exactly $m$ times and the remaining terms are zero. ∎

For example, a sequence in $\mathbb{Z}_{8}$ is an $E$-extremal sequence for
$U(8)$ if and only if it is $U(8)$-equivalent to one of the following
sequences:

$(\,1,\,2,\,4,\,0,\,0,\,0,\,0,\,0,\,0,\,0\,),~{}(\,1,\,2,\,4,\,4,\,4,\,0,\,0,\,0,\,0,\,0\,),$
$(\,1,\,2,\,4,\,4,\,4,\,4,\,4,\,0,\,0,\,0\,),~{}(\,1,\,2,\,4,\,4,\,4,\,4,\,4,\,4,\,4,\,0\,).$

We will now characterize the $C$-extremal sequences for $U(2^{r})$. The
following result is Lemma 5 in [8].

######  Lemma 6.7.

Let $S$ be a sequence in $\mathbb{Z}_{n}$ and $p$ be a prime divisor of $n$
such that every term of $S$ is divisible by $p$. Suppose $n^{\prime}=n/p$ and
$S^{\prime}$ is the sequence in $\mathbb{Z}_{n^{\prime}}$ whose terms are
obtained by dividing the terms of $S$ by $p$ and taking their images under
$f_{n,n^{\prime}}$. If $S^{\prime}$ is a $U(n^{\prime})$-weighted zero-sum
sequence, then $S$ is a $U(n)$-weighted zero-sum sequence.

######  Theorem 6.8.

Let $n=2^{r}$, $n^{\prime}=n/2$ and $S=(x_{1},\ldots,x_{n-1})$ be a sequence
in $\mathbb{Z}_{n}$. Suppose $S_{1}=(x_{1},\ldots,x_{n^{\prime}-1})$ and
$S_{2}=(x_{n^{\prime}+1},\ldots,x_{n-1})$. Then $S$ is a $C$-extremal sequence
for $U(n)$ if and only if all the terms of $S$ are even except the ‘middle’
term $x_{n^{\prime}}$ and $S_{1}^{\prime},\,S_{2}^{\prime}$ are $C$-extremal
sequences for $U(n^{\prime})$ in $\mathbb{Z}_{n^{\prime}}$ where
$S_{1}^{\prime}$ and $S_{2}^{\prime}$ denote the sequences in
$\mathbb{Z}_{n^{\prime}}$ which are obtained by dividing the terms of $S_{1}$
and $S_{2}$ by two and then taking their images under $f_{n,n^{\prime}}$.

###### Proof.

Let $n=2^{r}$, $n^{\prime}=n/2$ and $S=(x_{1},\ldots,x_{n-1})$ be a
$C$-extremal sequence for $U(n)$. Suppose $S$ has at least two odd terms. We
can find a subsequence $T$ of consecutive terms of $S$ which has exactly two
odd terms. By Lemma 6.5 we get the contradiction that $T$ is a $U(n)$-weighted
zero-sum sequence. Thus, we see that at most one term of $S$ can be odd.

Suppose the ‘middle’ term $x_{n^{\prime}}$ is even. As at most one term of $S$
is odd, we can find a subsequence $T$ of consecutive terms of $S$ of length
$n^{\prime}$ all of whose terms are even. Let $T^{\prime}$ be the sequence in
$\mathbb{Z}_{n^{\prime}}$ whose terms are obtained by dividing the terms of
$T$ by two and then taking their images under $f_{n,n^{\prime}}$. As
$n^{\prime}=2^{r-1}$ by Theorem 3.8 we have that
$C_{U(n^{\prime})}(n^{\prime})=n^{\prime}$. Hence, as $T^{\prime}$ has length
$n^{\prime}$ we see that $T^{\prime}$ has an $U(n^{\prime})$-weighted zero-sum
subsequence of consecutive terms. So by Lemma 6.7 we get the contradiction
that $T$ (and hence $S$) has a $U(n)$-weighted zero-sum subsequence of
consecutive terms.

Thus, we see that $S$ has a unique odd term which is $x_{n^{\prime}}$. For
$i=1,2$ consider the sequences $S_{i}$ and $S_{i}^{\prime}$ which are as
defined in the statement of the theorem. Suppose $S_{1}^{\prime}$ has a
$U(n^{\prime})$-weighted zero-sum subsequence of consecutive terms. By Lemma
6.7 we get the contradiction that $S_{1}$ (and hence $S$) has a
$U(n)$-weighted zero-sum subsequence of consecutive terms. As $S_{1}^{\prime}$
has length $n^{\prime}-1$, it follows that $S_{1}^{\prime}$ is a $C$-extremal
sequence for $U(n^{\prime})$ in $\mathbb{Z}_{n^{\prime}}$. A similar argument
shows that $S_{2}^{\prime}$ is also a $C$-extremal sequence for
$U(n^{\prime})$ in $\mathbb{Z}_{n^{\prime}}$.

The proof of the converse part is similar to the proof of Theorem 5 of [9] and
so we will omit it. ∎

For example, a sequence $S$ in $\mathbb{Z}_{8}$ is a $C$-extremal sequence for
$U(8)$ if and only if there exist units $a_{1},\ldots,a_{7}\in\mathbb{Z}_{8}$
such that

$S=(4a_{1},2a_{2},4a_{3},a_{4},4a_{5},2a_{6},4a_{7}).$

##  7 Some other weight-sets

For a real number $x$, we denote the smallest integer which is greater than or
equal to $x$ by $\lceil x\rceil$. When $A=\\{1,2,\ldots,r\\}$ where $r<n$, it
is shown in Theorem 3 (i) of [13] that $D_{A}(n)=\lceil n/r\rceil$. As we have
that $C_{A}(n)\geq D_{A}(n)$, it follows that $C_{A}(n)\geq\lceil n/r\rceil$.

######  Theorem 7.1.

Let $A=\\{1,2,\ldots,r\\}$ where $r<n$. Then we get $C_{A}(n)=\lceil
n/r\rceil$.

###### Proof.

It suffices to show that $C_{A}(n)\leq\lceil n/r\rceil$. Let
$S=(x_{1},\ldots,x_{m})$ be a sequence in $\mathbb{Z}_{n}$ of length $m=\lceil
n/r\rceil$. Consider the sequence

$S^{\prime}=(\,\overbrace{x_{1},\,\ldots,\,x_{1}}^{\text{$r$
times}},\,\overbrace{x_{2},\,\ldots,\,x_{2}}^{\text{$r$
times}},\,\ldots,\,\overbrace{x_{m},\,\ldots,\,x_{m}}^{\text{$r$ times}}\,).$

We observe that the length of $S^{\prime}$ is $mr$ which is at least $n$. By
Theorem 1 of [8] we have $C_{\\{1\\}}(n)\leq n$ and so we see that
$S^{\prime}$ has a zero-sum subsequence of consecutive terms. Thus, we obtain
an $A$-weighted zero-sum subsequence of $S$ having consecutive terms. Hence,
it follows that $C_{A}(n)\leq\lceil n/r\rceil$. ∎

######  Theorem 7.2.

Let $n$ be even, $v_{2}(n)=r$ and $m=n/2^{r}$. Let $B$ denote the set of all
odd elements of $\mathbb{Z}_{n}$ and $\\{m\\}\subseteq A\subseteq B$. Then we
have that $C_{A}(n)=2^{r}$.

###### Proof.

Let $S=(x_{1},\ldots,x_{k})$ be a sequence in $\mathbb{Z}_{n}$ having length
$k=2^{r}$. Let $r=v_{2}(n)$, $m=n/2^{r}$ and $S^{\prime}$ denote the image of
the sequence $S$ under the map $f_{n,2^{r}}$. By Theorem 1 of [8] we have that
$C_{\\{1\\}}(2^{r})\leq 2^{r}$. So there is a subsequence $T$ of $S$ having
consecutive terms such that the image $T^{\prime}$ of $T$ under $f_{n,2^{r}}$
is a zero-sum subsequence. It follows that the sum of the terms of $T$ is
divisible by $2^{r}$. So we see that $T$ is an $\\{m\\}$-weighted zero-sum
sequence. Thus, we see that $C_{\\{m\\}}(n)\leq 2^{r}$.

By Corollary 2 of [8] we have that $C_{U(2^{r})}(2^{r})\geq 2^{r}$. So there
is a sequence $S^{\prime}$ of length $2^{r}-1$ in $\mathbb{Z}_{2^{r}}$ which
has no $U(2^{r})$-weighted zero-sum subsequence of consecutive terms. As the
map $f_{n,2^{r}}$ is onto, we can find a sequence $S$ in $\mathbb{Z}_{n}$
whose image under $f_{n,2^{r}}$ is $S^{\prime}$. Suppose $S$ has a
$B$-weighted zero-sum subsequence having consecutive terms. As $n$ is even, we
see that the image of $B$ under $f_{n,2^{r}}$ is contained in $U(2^{r})$. So
we get the contradiction that $S^{\prime}$ has a $U(2^{r})$-weighted zero-sum
subsequence having consecutive terms. Thus, we see that $C_{B}(n)\geq 2^{r}$.

As we have that $\\{m\\}\subseteq A\subseteq B$, it follows that $C_{B}(n)\leq
C_{A}(n)\leq C_{\\{m\\}}(n)$ and so we get that $C_{A}(n)=2^{r}$. ∎

Our next result generalizes a result of [13]. We follow a similar argument as
in Theorem 7.2.

######  Theorem 7.3.

Let $n$ be even, $v_{2}(n)=r$ and $m=n/2^{r}$. Let $B$ denote the set of all
odd elements of $\mathbb{Z}_{n}$ and $\\{m\\}\subseteq A\subseteq B$. Then we
have that $D_{A}(n)=r+1$.

###### Proof.

Let $S=(x_{1},\ldots,x_{k})$ be a sequence in $\mathbb{Z}_{n}$ having length
$k=r+1$. Let $r=v_{2}(n)$, $m=n/2^{r}$ and $S^{\prime}$ denote the image of
the sequence $S$ under the map $f_{n,2^{r}}$. By Theorem 4.4 we have that
$D_{U(2^{r})}(2^{r})=r+1$. So there is a subsequence $T$ of $S$ such that the
image $T^{\prime}$ of $T$ under $f_{n,2^{r}}$ is a $U(2^{r})$-weighted zero-
sum subsequence. By Lemma 7 of [9] we see that the map $f_{n,2^{r}}$ maps
$U(n)$ onto $U(2^{r})$. So if $I$ denotes the set $\\{i:x_{i}$ is a term of
$T\\}$, we see that for each $i\in I$ there exists $a_{i}\in U(n)$ such that
$f_{n,2^{r}}\big{(}\sum_{i\in I}a_{i}x_{i}\big{)}=0$. Hence, it follows that
$\sum_{i\in I}a_{i}x_{i}$ is divisible by $2^{r}$ and so we see that $T$ is an
$\\{m\\}$-weighted zero-sum sequence. Thus, we see that $D_{\\{m\\}}(n)\leq
r+1$.

By Theorem 4.4 we have that $D_{U(2^{r})}(2^{r})=r+1$. So there is a sequence
$S^{\prime}$ of length $r$ in $\mathbb{Z}_{2^{r}}$ which has no
$U(2^{r})$-weighted zero-sum subsequence. The rest of the proof is very
similar to the argument given in the second and third paragraphs of the proof
of Theorem 7.2. Hence, we get that $D_{A}(n)=r+1$. ∎

######  Remark 7.4.

When $n$ is even and $A$ is the set of all even elements of $\mathbb{Z}_{n}$,
in [13] it is shown that $D_{A}(n)=2$ and so it follows that $C_{A}(n)=2$.

When $n$ is odd and $A$ is the set of all odd (or all even) elements of
$\mathbb{Z}_{n}$, in [13] it is shown that $D_{A}(n)=3$. From the proof of
this result we see that $C_{A}(n)=3$.

######  Theorem 7.5.

Let $p$ and $k$ be odd primes such that $p\equiv 1~{}(mod~{}k)$ and
$p\not\equiv 1~{}(mod~{}k^{2})$. Then we have that $D_{U(p)^{k}}(p)\leq k$.

###### Proof.

As $p\equiv 1~{}(mod~{}k)$ there exists $c\in U(p)$ such that $c$ has order
$k$. The subgroup $U(p)^{k}$ is the image of the map $U(p)\to U(p)$ given by
$x\mapsto x^{k}$. The kernel of this map has at most $k$ elements and contains
$c$. As $c$ has order $k$, it follows that the kernel is
$\langle\,c\,\rangle$. Thus, we have that $|\,U(p)\,|=k\,|U(p)^{k}|$ and so we
see that $U(p)/U(p)^{k}$ has order $k$.

Suppose $c\in U(p)^{k}$. Then there exists $a\in U(p)$ such that $c=a^{k}$ and
so $a^{k^{2}}=c^{k}=1$. As $k$ is a prime and $a^{k}=c\neq 1$, we see that the
order of $a$ is $k^{2}$. However, as $p\not\equiv 1~{}(mod~{}k^{2})$ there is
no element of order $k^{2}$ in $U(p)$. Thus, we see that $c\notin U(p)^{k}$.
For $x\in U(p)$ if we denote the coset $xU(p)^{k}$ by $[x]$ we see that
$[c]\neq[1]$. As $k$ is prime we see that $[c]$ has order $k$. Hence, we get a
partition of $U(p)$ by the cosets $[1],\,[c],\,\ldots,\,[c^{k-1}]$.

Let $S=(x_{1},\ldots,x_{k})$ be a sequence in $U(p)$. Suppose we show that $S$
has a $U(p)^{k}$-weighted zero-sum subsequence. It will follow that
$D_{U(p)^{k}}(p)\leq k$.

######  Case 7.5.1.

There exist two elements of $S$ which are in the same coset.

As there exists $l\in[0,k-1]$ such that $x_{i},x_{j}\in[c^{\,l}]$, it follows
that there exist $a,b\in U(p)^{k}$ such that $x_{i}=a\,c^{\,l}$ and
$x_{j}=b\,c^{\,l}$. Then we have that $(-b)\,x_{i}+a\,x_{j}=0$. As $k$ is odd,
we see that $-1\in U(p)^{k}$. Hence, it follows that $(x_{i},x_{j})$ is a
$U(p)^{k}$-weighted zero-sum subsequence of $S$.

######  Case 7.5.2.

No two elements of $S$ are in the same coset.

Without loss of generality, we can assume that for each $i\in[1,k]$ there
exist $a_{i}\in U(p)^{k}$ such that $x_{i}=a_{i}\,c^{\,i-1}$. Then
$a_{1}^{-1}\,x_{1}+a_{2}^{-1}\,x_{2}+\ldots+a_{k}^{-1}\,x_{k}=1+c+\ldots+c^{k-1}=0$
as $c$ satisfies $X^{k}-1=(X-1)\,(X^{k-1}+\ldots+X+1)$ and $c\neq 1$. Thus, it
follows that $S$ is a $U(p)^{k}$-weighted zero-sum sequence. ∎

From Theorem 3 of [3] we see that $D_{U(p)^{2}}(p)=3$ where $p$ is an odd
prime. This shows that Theorem 7.5 is not true when $k=2$. The next result is
Corollary 1 of [9].

######  Lemma 7.6.

Let $F$ be a field and $A$ be a subgroup of $F^{*}$. A sequence $S=(x,y)$ in
$F$ does not have an $A$-weighted zero-sum subsequence if and only if $x$ and
$-y$ are in different cosets of $A$ in $F^{*}$.

######  Corollary 7.7.

Let $F$ be a field and $A$ be a proper subgroup of $F^{*}$. Then we have that
$D_{A}(F)\geq 3$.

######  Corollary 7.8.

Let $k\geq 2$ and $p$ be a prime such that $p\equiv 1~{}(mod~{}k)$. Then we
have that $D_{U(p)^{k}}(p)\geq 3$.

###### Proof.

As $p\equiv 1~{}(mod~{}k)$, there is an element in $U(p)$ of order $k$. So as
in the first paragraph of the proof of Theorem 7.5 we see that the index of
the subgroup $U(p)^{k}$ of $U(p)$ is $k$. As $k\geq 2$ we see that $U(p)^{k}$
is a proper subgroup of $U(p)$ and so the result follows from Corollary 7.7. ∎

##  8 Concluding remarks

It will be interesting to characterize the extremal sequences for $U(n)$ for
the $U(n)$-weighted zero-sum constants $D_{U(n)}(n)$, $C_{U(n)}(n)$ and
$E_{U(n)}(n)$, when $n$ is an even number which is not a power of 2. In [2]
the $D$-extremal sequences for $U(n)$ have been characterized when $n$ is odd
and the $E$-extremal sequences for $U(n)$ have been characterized when $n$ is
a power of an odd prime. In [9] the $C$-extremal sequences for $U(n)$ have
been characterized when $n$ is odd.

In Theorem 3 of [13] and Remark 7.4 it is shown that $C_{A}(n)=D_{A}(n)=3$
when $n$ is odd and $A$ is the set of all odd elements of $\mathbb{Z}_{n}$. In
Theorem 3 of [3] and Theorem 4 of [8] it is shown that
$C_{Q_{p}}(p)=D_{Q_{p}}(p)=3$ where $p$ is an odd prime. This makes us curious
to see if we can find a weight-set $A\subseteq\mathbb{Z}_{n}$ where $n$ is odd
such that $A$ has size $(n-1)/2$ and we have that $D_{A}(n)>3$ or
$C_{A}(n)>3$.

Acknowledgement. Santanu Mondal would like to acknowledge CSIR, Govt. of
India, for a research fellowship.

## References

  * [1] S. D. Adhikari, The relation between two combinatorial group invariants: History and ramifications Math. Student 77 nos. 1-4 (2008), 131-143.
  * [2] S. D. Adhikari, I. Molla and S. Paul, Extremal sequences for some weighted zero-sum constants for cyclic groups, CANT IV, Springer Proc. Math. Stat. 347 (2021), 1-10.
  * [3] S. D. Adhikari and P. Rath, Davenport constant with weights and some related questions, Integers 6 (2006), #A30.
  * [4] H. Enomoto, P. Frankl, N. Ito and K. Nomura, Codes with given distances, Graphs Combin. 3 (1987), 25-38.
  * [5] W. D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996), 100-103.
  * [6] S. Griffiths, The Erdős-Ginzberg-Ziv theorem with units, Discrete Math. 308 no. 23 (2008), 5473-5484.
  * [7] F. Luca, A generalization of a classical zero-sum problem, Discrete Math. 307 no. 13 (2007), 1672-1678.
  * [8] S. Mondal, K. Paul and S. Paul, On a different weighted zero-sum constant, Discrete Math. 346 no. 6 (2023), 113350.
  * [9] S. Mondal, K. Paul and S. Paul, Extremal sequences for a weighted zero-sum constant, Integers 22 (2022), #A93.
  * [10] S. Mondal, K. Paul and S. Paul, Zero-sum constants related to the Jacobi symbol. arxiv:2111.14477v3.
  * [11] J.E. Olson, A combinatorial problem on finite abelian groups, I, J. Number Theory 1 (1969), 8-10.
  * [12] J.E. Olson, A combinatorial problem on finite abelian groups, II, J. Number Theory 1 (1969), 195-199.
  * [13] X. Xia and Z. Li, Some Davenport constants with weights and Adhikari and Rath’s conjecture, Ars Combin. 88 (2008), 83-95.
  * [14] P. Yuan and X. Zeng, Davenport constant with weights, European J. Combin. 31 (2010), 677-680.


# Review of the online analyses of multi-messenger alerts and electromagnetic
transient events with the ANTARES neutrino telescope

A. Albert  S. Alves  M. André  M. Ardid  S. Ardid  J.-J. Aubert  J. Aublin  B.
Baret  S. Basa  B. Belhorma  M. Bendahman  F. Benfenati  V. Bertin  S. Biagi
M. Bissinger  J. Boumaaza  M. Bouta  M.C. Bouwhuis  H. Brânzaş  R. Bruijn  J.
Brunner  J. Busto  B. Caiffi  D. Calvo  S. Campion  A. Capone  L. Caramete  J.
Carr  V. Carretero  S. Celli  M. Chabab  T. N. Chau  R. Cherkaoui El Moursli
T. Chiarusi  M. Circella  J.A.B. Coelho  A. Coleiro  R. Coniglione  P. Coyle
A. Creusot  A. F. Díaz  B. De Martino  C. Distefano  I. Di Palma  A. Domi  C.
Donzaud  D. Dornic  D. Drouhin  T. Eberl  T. van Eeden  D. van Eijk  N. El
Khayati  A. Enzenhöfer  M. Fasano  P. Fermani  G. Ferrara  F. Filippini  L.
Fusco  S. Gagliardini  J. García  P. Gay  N. Geißelbrecht  H. Glotin  R.
Gozzini  R. Gracia Ruiz  K. Graf  C. Guidi  L. Haegel  S. Hallmann  H. van
Haren  A.J. Heijboer  Y. Hello  J.J. Hernández-Rey  J. Hößl  J. Hofestädt  F.
Huang  G. Illuminati  C. W. James  B. Jisse-Jung  M. de Jong  P. de Jong  M.
Kadler  O. Kalekin  U. Katz  A. Kouchner  I. Kreykenbohm  V. Kulikovskiy  R.
Lahmann  M. Lamoureux  D. Lefèvre  E. Leonora  G. Levi  S. Le Stum  D. Lopez-
Coto  S. Loucatos  L. Maderer  J. Manczak  M. Marcelin  A. Margiotta  A.
Marinelli  J.A. Martínez-Mora  K. Melis  P. Migliozzi  A. Moussa  R. Muller
L. Nauta  S. Navas  E. Nezri  B. Ó Fearraigh  A. Păun  G.E. Păvălaş  C.
Pellegrino  M. Perrin-Terrin  V. Pestel  P. Piattelli  C. Pieterse  C. Poirè
V. Popa  T. Pradier  N. Randazzo  D. Real  S. Reck  G. Riccobene  A. Romanov
A. Saina  A. Sánchez-Losa  F. Salesa Greus  D. F. E. Samtleben  M. Sanguineti
P. Sapienza  J. Schnabel  J. Schumann  F. Schüssler  J. Seneca  M. Spurio  Th.
Stolarczyk  M. Taiuti  Y. Tayalati  S.J. Tingay  B. Vallage  V. Van Elewyck
F. Versari  S. Viola  D. Vivolo  J. Wilms  S. Zavatarelli  A. Zegarelli  J.D.
Zornoza  J. Zúñiga

###### Abstract

By constantly monitoring at least one complete hemisphere of the sky, neutrino
telescopes are well designed to detect neutrinos emitted by transient
astrophysical events. Real-time searches with the ANTARES telescope have been
performed to look for neutrino candidates coincident with gamma-ray bursts
detected by the Swift and Fermi satellites, high-energy neutrino events
registered by IceCube, transient events from blazars monitored by HAWC,
photon-neutrino coincidences by AMON notices and gravitational wave candidates
observed by LIGO/Virgo. By requiring temporal coincidence, this approach
increases the sensitivity and the significance of a potential discovery.
Thanks to the good angular accuracy of neutrino candidates reconstructed with
the ANTARES telescope, a coincident detection can also improve the positioning
area of non-well localised triggers such as those detected by gravitational
wave interferometers. This paper summarises the results of the follow-up
performed by the ANTARES telescope between 01/2014 and 02/2022, which
corresponds to the end of the data taking period.

##  1 Introduction

Multi-messenger approaches consisting in concomitant searches for the same
sources with neutrino telescopes, gravitational wave interferometers and/or
multi-wavelength facilities constitute a privileged way of identifying
astrophysical cosmic-ray accelerators. Neutrino astronomy allows the study of
the most energetic non-thermal processes in the Universe and provides insight
into source characteristics not accessible through other messengers. By
constantly monitoring at least one complete hemisphere of the sky, neutrino
telescopes are well designed to detect neutrinos emitted by transient
phenomena. Real-time searches with the ANTARES telescope have been performed
to look for neutrino candidates coincident with gamma-ray bursts detected by
the Swift and Fermi satellites, high-energy neutrino events registered by
IceCube, transient events from blazars monitored by HAWC, photon-neutrino
coincidences by AMON notices and gravitational wave candidates observed by
LIGO/Virgo. Requiring a spatial and temporal coincidence with other messengers
increases the sensitivity and the significance of a potential discovery with
respect to a solely neutrino based search.

The ANTARES telescope, completed in 2008, was the first operating neutrino
telescope in the Mediterranean Sea [1]. It was composed of 12 detection lines
of about 500 m height anchored at 2475 m depth offshore Toulon (42∘48’N,
6∘10’E). The mean distance between lines was about 65 m. Each line was made by
25 storeys with an inter-storey distance of 14.5 m. Every storey holded three
optical modules housing a single 10-inch diameter photomultiplier tube (PMT)
looking downward at an angle of 45∘. In total, a $\sim$ 10 Mt mass of water
was instrumented with 885 optical modules. ANTARES had an average data taking
efficiency larger than 94 % with an effective area decreasing in time due to a
loss of optical modules in operation. The data losses can be due to the
shutdown of data taking, calibration periods or too high bioluminescence
activities. The ANTARES data acquisition was switched off mid February 2022,
when KM3NeT [2] reached a comparable instantaneous sensitivity to cosmic
neutrinos.

In the online analysis framework, a dedicated real-time pipeline was developed
to look for neutrino events in both time and space coincidences with transient
events announced by public alerts distributed through the Gamma-ray
Coordinates Network (GCN https://gcn.gsfc.nasa.gov) or by private alerts
transmitted via special channels. This analysis framework also hosted a
neutrino alert sending program [3]. The online selection was optimised to
yield a neutrino sample (atmospheric and cosmic) with a minimal contamination
from atmospheric muons. Similar cuts as the ones designed for the ANTARES
standard offline point-like source search [4] were applied here: only upgoing
track events with a good reconstruction quality are used in the analysis to
ensure a median angular resolution of about 0.5∘. This leads to an atmospheric
muon contamination lower than 10 %. The typical rate of neutrino candidates
after selection in ANTARES is shown in Figure 1, revealing a slow continuous
efficiency loss of the detector [5]. The online analysis used an ideal static
detector both for the trigger and the reconstruction. It did not include the
knowledge of the dynamical positioning and the precise charge and time
calibration sets, which were made available a few months later for the offline
analysis.

Figure 1: Evolution of the average number of neutrinos per day (averaged
during a month period) between 2014 and 2022 selected for the ANTARES online
analysis.

For interesting cases, more optimised offline analyses, using the most precise
knowledge of the detector, are then performed to improve the online search
[6]. Offline searches are also performed for neutrino counterparts to
catalogued transients [7, 8].

This paper focuses on the outcomes of the real-time follow-up program of
ANTARES, in operation since 2014. Sections 2 and 3 summarise the results for
the triggers provided by IceCube and LIGO/Virgo. Sections 4 and 5 present the
follow-up of electromagnetic (EM) transients for gamma-ray bursts (GRB) and
transient alerts reported by HAWC, respectively. Conclusions and outlooks are
drawn in Section 6.

##  2 Follow-up of IceCube neutrino alerts

A detection of neutrinos by ANTARES and IceCube telescopes within a close
temporal window and with compatible directions (coincidence) would be a
significant proof of their astrophysical origin and would point directly to
the position of the source in the sky. An alert for a neutrino coincidence
would be so rare to induce the astronomy community to perform a prompt and
multi-frequency EM follow-up.

Since 2016, IceCube has been sending public triggers [9] for high-energy
starting events (HESE) and extremely high-energy track candidates (EHE). The
events are received by the Astrophysical Multi-messenger Observatory Network
(AMON [10]) and distributed to the community via an alert of the GCN. In June
2019, IceCube substituted these alerts with two new very-high energy track
event samples: gold (with a probability to be astrophysical $>$ 50 %) and
bronze ($>$ 30 %) samples [9]. In July 2020, IceCube provided another alert
stream based on very high-energy cascades with a typical resolution
uncertainty of 15-20∘ (50 % radius) and a typical rate of 8 events per year.
The list of triggers is available at this address:
https://gcn.gsfc.nasa.gov/amon.html.

Figure 2: Sky map in equatorial coordinates showing in blue the visible
ANTARES field of view at the time of IC170922. The direction of the IC event
is drawn as a red cross.

In this context, follow-up analyses have been performed for each IceCube event
whose position on the sky was below the horizon of ANTARES (which could
consequently yield an upgoing event at the time of the alert). ANTARES has
received 115 neutrino triggers from the IceCube alert system and has followed
37 alerts (7 HESE, 3 EHE, 10 gold and 17 bronze). The rest of the triggers was
either retracted by the IceCube Collaboration or located in the opposite
hemisphere. As an illustration, Figure 2 shows the direction of the IceCube
event IC170922 and the ANTARES visibility (i.e., the visible solid angle
yielding upgoing events) at the time of the event. No neutrino candidates were
found within a cone of 3∘ centred on the IceCube event coordinates and a time
window of $\pm$1 hour, further extended to $\pm$1 day. These non-detections
have been used to derive preliminary 90 % confidence level (C.L.) upper limits
on the radiant neutrino fluence111The radiant fluence, F, is computed with the
formula:
$F=\int_{E{min}}^{E_{max}}E\phi_{0}\left(\frac{E}{E_{0}}\right)^{-\gamma}dE$,
where $\phi_{0}$ refers to the normalisation of the neutrino spectrum,
$\gamma$ is the spectral index and $E_{0}=1$ GeV. $[E_{min};E_{max}]$
corresponds to the 5–95% energy range of the detectable neutrino flux. of the
possible sources originating these events of the order of $\sim$ 15 GeV cm-2
and $\sim$ 30 GeV cm-2 for the assumed $E^{-2}$ and the $E^{-2.5}$
differential neutrino fluxes, respectively (see columns 3 and 4 in Table 1).
These results have been published as GCN circulars and Astronomer’s Telegrams
typically one day after the alerts (columns 5 and 6 in Table 1).

IceCube event | Elevation | Fluence U.L. (GeV cm-2) at 90 % C.L. | GCN | ATels
---|---|---|---|---
|  | $dN/dE\propto E^{-2}$ | $dN/dE\propto E^{-2.5}$ | Id | Id
IC160731A (EHE/HESE) | -28∘ | 14 (2.8 TeV - 3.1 PeV) | 27 (0.4 - 280 TeV) | / | 9324
IC160814A (HESE) |  $-$26∘ | 16 (2.9 TeV - 3.3 PeV) | 43 (0.5 - 250 TeV) | 19885 | 9440
IC161103A (HESE) |  $-$26∘ | 13 (3.8 TeV - 3.8 PeV) | 22 (0.7 - 370 TeV) | 20134 | 9715
IC170321A (EHE) |  $-$57∘ | 16 (2.5 TeV - 2.5 PeV) | 26 (0.5 - 220 TeV) | 20926 | 10189
IC170922A (EHE) |  $-$14∘ | 15 (3.3 TeV - 3.4 PeV) | 34 (0.5 - 280 TeV) | 21923 | 10773
IC171015A (HESE) |  $-$45∘ | 14 (2.7 TeV - 2.9 PeV) | 27 (0.4 - 240 TeV) | 22019 | 10854
IC180908A (EHE) |  $-$41∘ | 18 (2.4 TeV - 2.6 PeV) | 36 (0.4 - 250 TeV) | 23218 | 12024
IC190104A (HESE) |  $-$39∘ | 16 (3.2 TeV - 3.5 PeV) | 30 (0.6 - 320 TeV) | 23611 | 12359
IC190124A (HESE) |  $-$44∘ | 15 (3.1 TeV - 3.6 PeV) | 25 (0.6 - 320 TeV) | 23793 | 12423
IC190504A (HESE) |  $-$18∘ | 16 (3.1 TeV - 3.5 PeV) | 32 (0.6 - 320 TeV) | 24400 | 12731
IC190619A (gold) |  $-$19∘ | 13 (3.9 TeV - 3.9 PeV) | 33 (0.7 - 320 TeV) | 24866 | 12878
IC190712A (bronze) |  $-$13∘ | 16 (4.6 TeV - 4.3 PeV) | 40 (0.8 - 420 TeV) | 25064 | 12937
IC191119A (gold) |  $-$37∘ | 16 (3.4 TeV - 3.6 PeV) | 28 (0.7 - 340 TeV) | 26266 | 13295
IC191231A (bronze) |  $-$17∘ | 15 (5.3 TeV - 5.0 PeV) | 32 (1.0 - 470 TeV) | 26623 | 13380
IC200127A (bronze) |  $-$18∘ | 15 (6.8 TeV - 6.3 PeV) | 29 (1.0 - 610 TeV) | 26811 | 13409
IC200421A (bronze) |  $-$25∘ | 15 (3.9 TeV - 4.9 PeV) | 27 (0.7 - 380 TeV) | 27619 | 13654
IC200530A (gold) |  $-$0.04∘ | 80 (6.0 TeV - 6.0 PeV) | 110 (1 - 560 TeV) | 27871 | 13770
IC200620A (bronze) |  $-$32∘ | 15 (5.0 TeV - 4.0 PeV) | 30 (0.8 - 400 TeV) | 28002 | 13820
IC200911A (bronze) |  $-$7∘ | 14 (10.0 TeV - 8.0 PeV) | 34 (1.5 - 740 TeV) | 28415 | 14008
IC200916A (bronze) |  $-$29∘ | 18 (4.0 TeV - 4.5 PeV) | 33 (1 - 430 TeV) | 28446 | 14025
IC200926B (bronze) |  $-$13∘ | 15 (8.0 TeV - 7.0 PeV) | 35 (1 - 690 TeV) | 28515 | 14045
IC200929A (gold) |  $-$10∘ | 13 (3.0 TeV - 4.0 PeV) | 45 (0.7 - 340 TeV) | 28535 | 14054
IC201014A (bronze) |  $-$30∘ | 18 (5.0 TeV - 4.5 PeV) | 30 (0.8 - 430 TeV) | 28624 | 14095
IC201021A (bronze) |  $-$14∘ | 19 (5.0 TeV - 5.0 PeV) | 48 (0.8 - 430 TeV) | 28738 | 14110
IC201114A (bronze) |  $-$41∘ | 19 (4.0 TeV - 4.0 PeV) | 31 (0.7 - 370 TeV) | 28890 | 14176
IC201115A (gold) |  $-$7∘ | 17 (3.2 TeV - 3.2 PeV) | 68 (0.6 - 330 TeV) | 28901 | 14181
IC201209A (gold) |  $-$34∘ | 16 (3.0 TeV - 3.0 PeV) | 30 (0.5 - 280 TeV) | 29023 | 14259
IC210210A (gold) |  $-$18∘ | 16 (3.5 TeV - 3.7 PeV) | 40 (0.7 - 360 TeV) | 29475 | /
IC210922A (gold) |  $-$37∘ | 16 (3.0 TeV - 3.3 PeV) | 30 (0.6 - 300 TeV) | 30875 | 14935
IC210926A (cascade) |  $-$71∘ | 21 (2.3 TeV - 3.2 PeV) | 30 (0.4 - 240 TeV) | 30887 | 14938
IC211023A (bronze) |  $+$0.8∘ | 12 (3.0 TeV - 3.0 PeV) | 48 (0.6 - 300 TeV) | 30971 | 14995
IC211116A (bronze) |  $-$47∘ | 16 (3.0 TeV - 3.5 PeV) | 26 (0.7 - 320 TeV) | 31090 | 15042
IC211117A (gold) |  $-$12∘ | 15 (3.0 TeV - 3.5 PeV) | 43 (0.6 - 320 TeV) | 31094 | 15044
IC211125A (bronze) |  $-$6∘ | 12 (5.0 TeV - 5.0 PeV) | 35 (1 - 500 TeV) | 31128 | 15065
IC211208A (bronze) |  $-$10∘ | 17 (5.0 TeV - 5.0 PeV) | 43 (1 - 500 TeV) | 31225 | 15106
IC211216A (bronze) |  $-$8∘ | 16 (5.0 TeV - 5/0 PeV) | 49 (1 - 450 TeV) | 31252 | 15121
IC211216B (bronze) |  $-$4∘ | 17 (5.0 TeV - 5.0 PeV) | 40 (1 - 450 TeV) | 31262 | 15127
IC220205B (gold) |  $-$51∘ | 16 (3.0 TeV - 3.3 PeV) | 30 (0.6 - 300 TeV) | 31556 | 15207
Table 1: Upper limits (at 90 % C.L.) on neutrino fluence as derived from the
non-observation of ANTARES coincidences for each IceCube neutrino candidate.
For each upper limit, the energy range in which 90 % of events are observed
(excluding the 5 % of events with the lower/higher energies) is given. The
publication reference number (Id) in GCN and in Astronomer’s Telegram of the
ANTARES follow-up of IceCube public triggers are given in the last two
columns.

Given the importance of some of the IceCube alerts, dedicated offline analyses
have been performed for the following events: IC170922A and the blazar
TXS0506+056 [11], IC191001A / IC200530A and the tidal disruption events
AT2019dsg / AT2019fdr [12]. An offline time and space correlation analysis for
54 IceCube high-energy track-like neutrino events was performed with the
ANTARES neutrino offline data set, resulting in no significant coincidence
[13].

##  3 Follow-up of LIGO/Virgo gravitational wave alerts

Current modeling of the binary black-hole merger evolution does not imply EM
or neutrino counterparts. However, in a sufficiently dense circumbinary
region, an accretion disk might form and/or a relativistic jet connected to
the accretion could be released. Accreting black holes can drive relativistic
outflows [14]. In this case, the process might lead to gamma-ray emission with
a potential high-energy neutrino counterpart if a hadronic component is
present [15, 16, 17, 18, 19, 20]. More GW detections will allow to probe
poorly known systems, with asymmetric masses, or very large masses, thus
leaving room for possible discoveries. An EM counterpart, presumably
associated with a hadronic emission is more likely from neutron star/black
hole (NSBH) or neutron star/neutron star (BNS) mergers. Most of the models are
based either on the formation of a gamma-ray burst [21, 22, 23] or a magnetar
[24]. The other advantage provided by the neutrino follow-up is that the
angular resolution of ANTARES [3] ($\sim$ 0.5∘ at $\sim$ 10 TeV) compared to
the size of the gravitational wave error box (a few hundreds of square degrees
on the sky) offers the possibility to drastically reduce the size of the
region of interest in case of a coincident neutrino detection.

During the first observing run O1 in 2015, three GW events coming from binary
black hole (BBH) mergers were detected by the LIGO interferometers [25]. As
the GW online analysis was not ready at that time, only offline analyses have
been performed by the ANTARES Collaboration [26, 27]. About one year later,
during the second observing run O2 (November 30, 2016 to August 25, 2017), the
upgraded LIGO and Virgo detectors observed GWs from seven binary black hole
mergers (plus 3 additional sources found in the offline analyses) and the BNS
merger GW170817. Only for this last event, EM counterparts have been
identified as a short gamma-ray burst followed by a kilonova [28, 6]. During
the run O2, the LIGO/Virgo Collaboration triggered 15 alerts identified by the
online analysis using a loose false-alarm-rate threshold of one per month.
These triggers were shared with partner collaborations having signed a
Memorandum of Understanding with LIGO/Virgo. Each of these alerts were
followed by the ANTARES neutrino telescope by searching for a potential
neutrino counterpart. The online analysis consists in looking for (i) temporal
coincidences within $\pm$500 seconds and $\pm$1 hour time windows around the
GW alert [29] and (ii) spatial overlap between the 90 % probability contour
from GW interferometers and the ANTARES visibility region at the time of the
GW event. Figure 3 illustrates the principle of the real-time GW analysis.
LIGO and Virgo are sending a few notices for each GW candidate with updated
information (Preliminary, Initial, Update, Retraction). Each new type is
processed as a new GW trigger. At the end, the results of the last stable
revision are provided. This analysis scheme has been applied to all the GW
candidate triggers: no upgoing neutrino candidates temporally coincident with
any of the GW candidates were found. The results of the nearly real-time
analyses have been transmitted to LIGO/Virgo follow-up community via the GCN.
Table 2 lists the different GCN circulars sent on behalf the ANTARES
Collaboration. In general, the online analyses performed for each GW candidate
have been followed by a more optimised all-sky analysis [6, 7].

GW alert | Confirmed GW name | Type | GCN Id
---|---|---|---
G268556 | GW170104 | BBH | 20517
G270580 |  | BBH | 20621
G274296 |  | BBH | 20704
G275404 |  | BBH | 20751
G275697 |  | BBH | 20765
G277583 |  | BBH | 20866
G284239 |  | BBH | 21066
G288732 | GW170608 | BBH | 21223
G296853 | GW170809 | BBH | 21433
G297595 | GW170814 | BBH | 21479
G298048 | GW170817 | BNS | 21522 / 21601 / 21631
G298936 | GW170823 | BBH | 21659
G299232 |  | BBH | 21696 / 21769
Table 2: Characteristics of the GW candidates distributed by LIGO/Virgo during
run O2. The last column gives the references of the GCN circular published by
the ANTARES collaboration for each GW candidates. Figure 3: Principle of the
online GW analysis. T0, Tn and T${}_{n^{2}}$ correspond to the time of the
detection, the time of the reception of the first notice and the time of the
successive notices for one GW trigger. At the time Tn and when the results of
the two searches are available, one email is sent to the ANTARES GW subgroup.
An SMS is also sent to speed up the communication within the dedicated
analysis group.

The third observing run O3 started on April 1st, 2019, with even more upgraded
interferometers. Until the end of March 2020, 78 alerts were distributed
publicly, 22 retracted by LIGO/Virgo after further investigations. Among the
56 events, 37 are classified as BBH, 5 Mass Gap, 4 NSBH, 6 BNS, 1 unmodeled
and 3 probably coming from terrestrial noise. The real-time analysis has been
performed for 51 GW triggers (Figure 4). Two triggers have not been analysed
since the GW were quickly classified as terrestrial noise, 2 had their allowed
provenience regions totally outside the ANTARES field of view while for the
other events, ANTARES was in maintenance. As an example, Figure 5 illustrates
the probability contours of the GW event S190602aq together with the ANTARES
visibility at the time of the event. The main characteristics of the 51 GW
candidates and the results of the neutrino search are summarised in Table 3.

Figure 4: Cumulative number of the GW candidates detected during run O3 as a
function of the date: all GW triggers (black); analysable triggers, i.e., not
terrestrial, nor retracted at the time of the analysis (blue); analysed GW
candidates (red) by ANTARES. Figure 5: Sky map in equatorial coordinates
showing the 99 % (gray), 90 % (blue) and 50 % (red) probability contours for
the allowed provenience region of S190602aq together with the ANTARES field of
view at the event time (blue part of the map).

Figure 6 shows the distribution of the fraction of the 90 % C.L. allowed
provenience region visible by the ANTARES detector at the GW detection time,
T0, as upgoing directions for all the GW triggers. Unfortunately, during this
run O3, most of the GW candidates were reconstructed with a large allowed
provenience region, typically above 1000 deg2. Only a few events were
reconstructed with a provenience region of less than 100 deg2. This makes the
EM follow-up even more difficult and the detection of a neutrino more
relevant.

For those events detected during the O3 period of GW interferometers, similar
analyses to the ones for the run O2 events have been performed, but in a
completely automated way. No tracks induced by a muon neutrino have been found
in time and space coincidence with any alert of run O3 (Table 3). All the
results have been reported via the publication of a circular to the GCN. The
provided information contain a sky map of the visible region of ANTARES (as
upgoing) at the time of the GW candidate together with the GW allowed
provenience regions (see Figure 5 as an example), the fraction of the GW 90 %
allowed provenience region covered by the ANTARES field of view, the number of
detected events in time/space coincidence and the expected number of
atmospheric background events in the region visible by ANTARES. This false
alarm rate is computed directly from the data using an off-region area before
the GW trigger. The results are reported for two search time windows: $\pm$500
s and $\pm$1 hour centred on the time of the GW alert. The typical delay
between the detection time of each GW candidate and the time of the ANTARES
circular is of about 4.5 hours (Figure 7). Note that one necessary condition
to submit our results was to receive a confirmation circular by LIGO/Virgo. If
this time is used as a reference, the results have been published on average
in less than 2 h (Figure 7). The offline analysis has already been done for a
few selected events published by LIGO/Virgo [30, 31, 32].

Table 3: Characteristics of the GW candidates distributed by LIGO/Virgo during run O3. The coverage indicates the fraction of the 90 % GW allowed provenience region falling in the visibility of ANTARES at the time of the event. The allowed area is the size of the region provided by GW interferometers at the time of the analysis. Some more refined GW parameters may have arrived later. The GW are ordered by DATE (YYMMDD), contained in the name of the GW (as in GraceDB https://gracedb.ligo.org/superevents/public/O3/). The GCN references are the ones published by the ANTARES Collaboration. GW name | Type | Allowed area | Distance | Coverage | GCN
---|---|---|---|---|---
|  | (deg2) | (Mpc) | (%) | Id
S190408an | BBH | 387 | 1473 |  |
S190412m | BBH | 156 | 812 | 9.3 | 24105
S190421ar | BBH | 1917 | 1628 | 52 | 24156
S190425z | BNS | 10183 | 156 |  |
S190426c | BNS | 1932 | 377 | 45 | 24271
S190503bf | BBH | 448 | 421 | 98 | 24387
S190510g | Terrestrial | 3462 | 237 |  |
S190512at | BBH | 399 | 1388 | 83 | 24516
S190513bm | BBH | 691 | 1987 | 55 | 24539
S190517h | BBH | 939 | 2950 | 83.3 | 24581
S190519bj | BBH | 967 | 3154 | 34 | 24602
S190521g | BBH | 1163 | 3931 | 56 | 24628
S190521r | BBH | 388 | 1136 | 30 | 24634
S190602aq | BBH | 1172 | 797 | 84 | 24719
S190630ag | BBH | 8493 | 926 | 68.6 | 24924
S190701ah | BBH | 67 | 1849 | 99.9 | 24952
S190706ai | BBH | 1100 | 5263 | 48.7 | 25009
S190707q | BBH | 1375 | 874 | 58.3 | 25013
S190718y | Terrestrial | 7246 | 227 | 77.5 | 25091
S190720a | BBH | 1599 | 869 | 41.6 | 25120
S190727h | BBH | 1357 | 2839 | 55.1 | 25168
S190728q | BBH | 977 | 874 | 38.1 | 25194
S190814bv | NSBH | 772 | 267 | 99.9 | 25330
S190828j | BBH | 603 | 1946 | 53.1 | 25508
S190828l | BBH | 948 | 1528 | 56.8 | 25507
S190901ap | BNS | 13613 | 241 | 54.1 | 25611
S190910d | NSBH | 3829 | 632 | 50.9 | 25700
S190910h | BNS | 24226 | 230 | 50 | 25711
S190915ak | BBH | 528 | 1584 | 45.6 | 25758
S190923y | MassGap | 2107 | 438 | 41.7 | 25816
S190930s | MassGap | 1998 | 709 | 25.9 | 25881
S190930t | NSBH | 24220 | 108 | 50 | 25882
S191105e | BBH | 1253 | 1183 | 66.4 | 26189
S191109d | BBH | 1487 | 1810 | 79.1 | 26210
S191110af | unmodeled | 1261 |  | 56 | 26230
S191129u | BBH | 1011 | 742 | 51.6 | 26307
S191204r | BBH | 433 | 678 | 92.9 | 26336
S191205ah | NSBH | 6378 | 385 | 28 | 26352
S191213g | BNS | 1393 | 201 | 75.1 | 26404
S191215w | BBH | 923 | 1770 | 42.3 | 26443
S191216ap | BBH | 300 | 376 | 15.9 | 26458
S191222n | BBH | 2324 | 2518 | 53.1 | 26550
S200105ae | Terrestrial | 7719 | 283 | 35.6 | 26643
S200112r | BBH | 6199 | 1125 | 38.4 | 26718
S200114f | Unmodeled | 403 |  | 6 | 26742
S200115j | MassGap | 920 | 340 | 76 | 26762
S200128d | BBH | 2521 | 3702 | 50.1 | 26912
S200129m | BBH | 53 | 755 |  |
S200208q | BBH | 1120 | 2142 | 68.5 | 27016
S200213t | BNS | 2587 | 201 | 31.9 | 27049
S200219ac | BBH | 1251 | 3533 | 55.5 | 27135
S200224ca | BBH | 69 | 1575 |  |
S200225q | BBH | 403 | 995 | 42.4 | 27201
S200302c | BBH | 6704 | 1820 | 49.5 | 27284
S200311bg | BBH | 52 | 1115 |  |
S200316bj | MassGap | 1117 | 1178 | 26.6 | 27390
Figure 6: Distribution of the fraction of the 90 % C.L. allowed provenience
region of the GW candidates visible by ANTARES at T0 as upgoing directions for
neutrino candidates..  Figure 7: Cumulative distribution of time delays
between the time of the 51 GW candidates and the reception time of the first
notice (black) and the submission times of the circular with the ANTARES
results (red). The blue curve corresponds to the time difference between the
reception of the GW confirmation circular and the ANTARES circulars.

##  4 Follow-up of gamma-ray bursts

Gamma-ray bursts are mainly detected by X-ray and gamma-ray satellites such as
Swift and Fermi. Once a GRB is detected, an alert message is sent publicly via
the GCN within a few tens of seconds. Figure 8 left shows the delay of the
alert sending for all GRBs detected by Swift and Fermi selected in the ANTARES
analysis from 01/2014 to 02/2022 (see below for the details). ANTARES is able
to react in real time to this type of alerts. Only the bursts with directions
below the ANTARES horizon are analysed online. A dedicated search for
neutrino-induced muons in the online data set is performed in real time within
a time window [–250 s; +750 s] around the detection time and in a cone centred
on the GRB position. The radius of the cone is determined by taking the
maximum between 2∘ (containing about 90 % of the point spread function, see
Figure 3 in Ref. [33]) and the size of the error box provided by Fermi (Figure
8 right). In the case of Swift triggers, a 2∘ cone is always used. For a cone
radius of 2∘, the detection of one event yields a p-value (i.e., a probability
that the coincidence is due to background) in the range of 2–5$\times$10-5.
The analysis is performed automatically. To ensure the quality of the data at
the alert time, the detector stability is monitored over several hours before
the alert, i.e., the reconstructed event rates should follow a Gaussian
distribution. This analysis has been operational since the beginning of 2014
and $\sim$ 98 % of the alerts have been processed. Over more than 8 years of
operation (01/2014–02/2022), there were 317 Swift and 770 Fermi-GBM bursts.
The bursts detected at the same time by both satellites are tagged with the
information provided by Swift. Figure 9 shows the directions of the GRBs of
both samples. No online neutrino signals have been detected in this search.

Figure 8: GRBs detected by Swift (black histogram) and Fermi (red histogram)
selected in the ANTARES analysis from 01/2014 to 02/2022. (Left) Time delays
between the burst detection by Swift and Fermi and the received notice.
(Right) Error in the position of the GRBs detected by Swift and Fermi. The
information are extracted directly from the GCN notices. Figure 9: Sky map in
Galactic coordinates with the positions of the Fermi (red triangles) and Swift
(blue triangles) GRBs followed by ANTARES in the full analysed period (01/2014
to 02/2022). The shade of grey indicates the ANTARES visibility. The darkest
region indicates the maximum visibility.

In case of a coincident neutrino detection (never found in our data), a
dedicated offline analysis would have been foreseen to confirm the result and
to compute its significance (expected to be higher than 3$\sigma$ in most of
the cases). Using the most precise knowledge of the detector, offline
individual and stacked analyses on GRB catalogues have been performed with
improved event selections [8, 34, 35].

##  5 Follow-up of HAWC alerts for transient phenomena

Since mid 2019, the HAWC Collaboration has been issuing alerts of short TeV
transients lasting from 0.2 s to 100 s, targeting in particular GRBs. HAWC
shares the same advantage as of ANTARES to monitor half of the sky in
permanence with a high-duty cycle. The quest of TeV gamma rays originated by
transient astrophysical sources is particularly interesting for high energy
neutrino telescopes. First, the gamma ray detection proves that the sources
generating the events are powerful cosmic accelerators. Second, in hadronic
production scenarios, these gamma rays have almost the same flux and energy
spectrum as the accompanying neutrinos, within the energy range in which the
telescope is most sensitive. The alerts are channeled via the AMON framework
and then distributed by the GCN. Up to Feb. 2022, the HAWC Collaboration sent
22 triggers, 7 of them with a direction within the ANTARES field of view at
the time of the alert. The alert parameters are available at this address:
https://gcn.gsfc.nasa.gov/amon_hawc_events.html. Figure 10 shows the direction
of the HAWC alerts together with the integrated ANTARES visibility. The same
analysis strategy as for the IceCube neutrino alerts is applied and the
results are then published as a circular to the GCN and/or Astronomer’s
Telegram. No online neutrinos have been identified in coincidence with the
HAWC transients. Table 4 summarises the main alert parameters and lists the
GCN published for this analysis.

Figure 10: Sky map in Galactic coordinates with the positions of the HAWC alerts (triangles) up to Feb 2022. The blue triangles indicate the analysed alerts. The shade of grey indicates the ANTARES visibility. The darkest region indicates the maximum visibility. Event Id | Direction | Error |  $\delta$T | Visibility | GCN
---|---|---|---|---|---
| RA, Dec (deg) | (arcmin) | (s) | (%) | Id
HAWC-211123A (110400_42) | 34.12, $-$8.04 | 36 | 1 | 53 | 31111
HAWC-210507A (1010067_1345) | 257.24, $+$8.09 | 24 | 10 | 45 | 29967
HAWC-201019A (1009678_72) | 203.15, $+$29.70 | 36 | 100 | 32 | 28729
HAWC-200709A (1009500_793) | 252.38, $+$15.20 | 24 | 100 | 42 | 28074
HAWC-200226A (19170_50) | 182.80, $-$0.61 | 48 | 1 | 50 | 27242
HAWC-191210A (9066_1171) | 210.80, $-$1.52 | 48 | 0.2 | 35 | 26391
HAWC-191019A (8991_1097) | 217.50, $+$25.81 | 48 | 0.2 | 50 | 26049
Table 4: Summary of HAWC transient triggers and ANTARES publication in GCN.
"Error" refers to the location uncertainty (radius, 50 % containment).
$\delta$T is the trigger duration interval. "Visibility" refers to the ANTARES
visibility fraction of the alert direction during one day. The last column
gives the GCN circular references of the ANTARES follow-up.

AMON is also issuing some alerts for the significant coincidence between
ANTARES neutrinos and Fermi/LAT photons and between IceCube neutrinos and HAWC
transients. The ANTARES Collaboration has performed the follow-up of the
IceCube + HAWC coincidences (NuEM) with a similar analysis method as for the
IceCube alerts. The NuEM alert parameters are available at this address:
https://gcn.gsfc.nasa.gov/amon_nu_em_coinc_events.html. The event
characteristics are presented in Table 5, together with the reference of the
GCN circulars where the results have been published.

Alert | Trigger | RA,Dec | Rerr | Duration | Visibility | GCN
---|---|---|---|---|---|---
| Id | (deg) | (deg) | (sec) | (%) | Id
NuEM-220116A | 0_105322 | 322.13, $+$27.26 | 0.17 | 23130.4 | 34 | 31476
NuEM-211209A | 0_101674 | 12.03, $-$5.75 | 0.18 | 18273.3 | 53 | 31198
NuEM-211020A | 0_96720 | 99.76, $+$9.07 | 0.17 | 21670.1 | 45 | 30954
NuEM-210515B | 0_85791 | 93.93, $+$12.51 | 0.2 | 22165.2 | 42 | 30024
NuEM-210515A | 0_85790 | 93.64, $+$14.66 | 0.15 | 22443 | 42 | 30024
NuEM-210111A | 0_73310 | 162.34, $-$19.46 | 0.37 | 22742.5 | 39 | 29294
NuEM-201124A | 0_68186 | 135.0, $+$7.74 | 0.23 | 21531.2 | 46 | 28953
NuEM-201107A | 0_66291 | 140.2, $+$29.76 | 0.15 | 23105.4 | 30 | /
NuEM-200717A | 0_54519 | 118.5, $-$1.62 | 0.38 | 19395.5 | 50 | 28144
Table 5: Summary of the ANTARES follow-up of the NuEM triggers (IceCube / HAWC
coincidence). Rerr is the 50 % allowed region. "Duration" corresponds to the
duration of the coincidence. The last column gives the GCN circular references
of the ANTARES follow-up.

##  6 Conclusions

As a coincident observation by two experiments significantly decreases the
probability of false alerts, fast confirmation is essential to allow
observatories with limited follow-up capabilities, e.g., due to limited sky
coverage or observation time, to efficiently prioritise and schedule their
resources. A further advantage is that a subsequent offline analysis of data
collected by different instruments upon an alert may yield a statistically
relevant result from a combination of signals that by themselves would not be
considered significant enough for reporting.

Public alerts are common for EM transients, especially gamma-ray bursts, soft-
gamma-ray repeaters, supernovae, etc. To study the parameters of physical
processes inherent to these astrophysical sources, it is necessary to collect
as much as possible wide multi-wavelength and multi-probe information. This
can only happen with a synergy between different instruments based on
efficient, fast and reliable communication between them. The GCN is in the
centre of this strategy as a fast dispatcher of triggers and results.
Recently, multi-messenger actors have also adopted a strategy similar to the
public alert distribution: IceCube in 2016, LIGO/Virgo in 2019, HAWC in 2020.
With more than one hundred triggers in one year from the run O3 of LIGO/Virgo
and the new alert selection of IceCube, some maturity has been reached. An
online analysis framework has been implemented in the ANTARES Collaboration,
fully automatised, that looks for time/space coincidences with the
time/direction of the EM, neutrino and GW transient external triggers. All the
received public alerts have been followed provided that at the time of the
trigger, their position in the sky was below the horizon for the ANTARES
detector. Despite the fact that no coincidences have been found in the online
analysis, this effort has highlighted the multi-messenger program of the
ANTARES neutrino telescope to a broad community. Most of the remaining work
consists in the writing and validation of each GCN circular. This step can
also be automatised in the future.

KM3NeT [2] is starting to take data with a sensitivity larger than ANTARES,
and this new detector will allow multi-flavor neutrino detection in real time
with an unprecedented angular resolution [36]. For the muon-neutrino golden
channel, the angular precision can be as low as 0.1∘ at very high energies. In
KM3NeT, all-flavor neutrino events will be used for online follow-up studies
in a large energy range from a few GeV to a few PeV.

## Acknowledgments

The authors acknowledge the financial support of the funding agencies: Centre
National de la Recherche Scientifique (CNRS), Commissariat à l’énergie
atomique et aux énergies alternatives (CEA), Commission Européenne (FEDER fund
and Marie Curie Program), LabEx UnivEarthS (ANR-10-LABX-0023 and
ANR-18-IDEX-0001), Région Alsace (contrat CPER), Région Provence-Alpes-Côte
d’Azur, Département du Var and Ville de La Seyne-sur-Mer, France;
Bundesministerium für Bildung und Forschung (BMBF), Germany; Istituto
Nazionale di Fisica Nucleare (INFN), Italy; Nederlandse organisatie voor
Wetenschappelijk Onderzoek (NWO), the Netherlands; Executive Unit for
Financing Higher Education, Research, Development and Innovation (UEFISCDI),
Romania; Ministerio de Ciencia e Innovación: Programa Estatal para Impulsar la
Investigación Científico-Técnica y su Transferencia (refs.
PID2021-124591NB-C41, -C42, -C43) (MCIU/FEDER), Programa de Planes
Complementarios I+D+I (refs. ASFAE/2022/023, ASFAE/2022/014) and Programa
María Zambrano (Spanish Ministry of Universities, funded by the European
Union, NextGenerationEU), Generalitat Valenciana: Prometeo
(PROMETEO/2020/019), and GenT (refs. CIDEGENT/2018/034, /2019/043, /2020/049.
/2021/23) programs, Junta de Andalucía (ref. SOMM17/6104/UGR, P18-FR-5057),
EU: MSC program (ref. 101025085), Spain; Ministry of Higher Education,
Scientific Research and Innovation, Morocco, and the Arab Fund for Economic
and Social Development, Kuwait. We also acknowledge the technical support of
Ifremer, AIM and Foselev Marine for the sea operation and the CC-IN2P3 for the
computing facilities.

## References

  * [1] M. Ageron et al., “ANTARES: the first undersea neutrino telescope,” Nucl. Instrum. Meth. A, vol. 656, pp. 11–38, 2011.
  * [2] S. Adrian-Martinez et al., “Letter of intent for KM3NeT 2.0,” J. Phys. G, vol. 43, no. 8, p. 084001, 2016.
  * [3] S. Adrián-Martínez et al., “Optical and X-ray early follow-up of ANTARES neutrino alerts,” JCAP, vol. 02, p. 062, 2016.
  * [4] A. Albert et al., “ANTARES Search for Point Sources of Neutrinos Using Astrophysical Catalogs: A Likelihood Analysis,” Astrophys. J., vol. 911, no. 1, p. 48, 2021.
  * [5] A. Albert et al., “Long-term monitoring of the ANTARES optical module efficiencies using${}^{40}\mathrm{{K}}$ decays in sea water,” Eur. Phys. J. C, vol. 78, no. 8, p. 669, 2018.
  * [6] A. Albert et al., “Search for High-energy Neutrinos from Binary Neutron Star Merger GW170817 with ANTARES, IceCube, and the Pierre Auger Observatory,” Astrophys. J. Lett., vol. 850, no. 2, p. L35, 2017.
  * [7] A. Albert et al., “Search for neutrino counterparts of gravitational-wave events detected by LIGO and Virgo during run O2 with the ANTARES telescope,” Eur. Phys. J. C, vol. 80, no. 5, p. 487, 2020.
  * [8] S. Adrián-Martínez et al., “Stacked search for time shifted high energy neutrinos from gamma ray bursts with the ANTARES neutrino telescope,” Eur. Phys. J. C, vol. 77, no. 1, p. 20, 2017.
  * [9] R. Abbasi et al., “Follow-up of Astrophysical Transients in Real Time with the IceCube Neutrino Observatory,” Astrophys. J., vol. 910, no. 1, p. 4, 2021.
  * [10] M. Smith et al., “The Astrophysical Multimessenger Observatory Network (AMON),” Astropart. Phys., vol. 45, pp. 56–70, 2013.
  * [11] A. Albert et al., “The Search for Neutrinos from TXS 0506+056 with the ANTARES Telescope,” Astrophys. J. Lett., vol. 863, no. 2, p. L30, 2018\.
  * [12] A. Albert et al., “Search for Neutrinos from the Tidal Disruption Events AT2019dsg and AT2019fdr with the ANTARES Telescope,” Astrophys. J., vol. 920, no. 1, p. 50, 2021.
  * [13] A. Albert et al., “ANTARES neutrino search for time and space correlations with IceCube high-energy neutrino events,” Astrophys. J., vol. 879, no. 2, p. 108, 2019.
  * [14] P. Meszaros, “Gamma-Ray Bursts,” Rept. Prog. Phys., vol. 69, pp. 2259–2322, 2006.
  * [15] R. Moharana, S. Razzaque, N. Gupta, and P. Meszaros, “High Energy Neutrinos from the Gravitational Wave event GW150914 possibly associated with a short Gamma-Ray Burst,” Phys. Rev. D, vol. 93, no. 12, p. 123011, 2016.
  * [16] K. Kotera and J. Silk, “Ultrahigh Energy Cosmic Rays and Black Hole Mergers,” Astrophys. J. Lett., vol. 823, no. 2, p. L29, 2016.
  * [17] R. Perna, D. Lazzati, and B. Giacomazzo, “Short Gamma-Ray Bursts from the Merger of Two Black Holes,” Astrophys. J. Lett., vol. 821, no. 1, p. L18, 2016.
  * [18] I. Bartos, B. Kocsis, Z. Haiman, and S. Márka, “Rapid and Bright Stellar-mass Binary Black Hole Mergers in Active Galactic Nuclei,” Astrophys. J., vol. 835, no. 2, p. 165, 2017.
  * [19] N. C. Stone, B. D. Metzger, and Z. Haiman, “Assisted inspirals of stellar mass black holes embedded in AGN discs: solving the ‘final au problem’,” Mon. Not. Roy. Astron. Soc., vol. 464, no. 1, pp. 946–954, 2017.
  * [20] K. Murase, K. Kashiyama, P. Mészáros, I. Shoemaker, and N. Senno, “Ultrafast Outflows from Black Hole Mergers with a Minidisk,” Astrophys. J. Lett., vol. 822, no. 1, p. L9, 2016.
  * [21] S. S. Kimura, K. Murase, I. Bartos, K. Ioka, I. S. Heng, and P. Mészáros, “Transejecta high-energy neutrino emission from binary neutron star mergers,” Phys. Rev. D, vol. 98, no. 4, p. 043020, 2018.
  * [22] S. S. Kimura, K. Murase, P. Mészáros, and K. Kiuchi, “High-Energy Neutrino Emission from Short Gamma-Ray Bursts: Prospects for Coincident Detection with Gravitational Waves,” Astrophys. J. Lett., vol. 848, no. 1, p. L4, 2017.
  * [23] S. S. Kimura, “High-energy emissions from neutron star mergers,” EPJ Web Conf., vol. 210, p. 03001, 2019.
  * [24] K. Fang and B. D. Metzger, “High-Energy Neutrinos from Millisecond Magnetars formed from the Merger of Binary Neutron Stars,” Astrophys. J., vol. 849, no. 2, p. 153, 2017.
  * [25] B. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett., vol. 116, no. 6, p. 061102, 2016\.
  * [26] S. Adrian-Martinez et al., “High-energy Neutrino follow-up search of Gravitational Wave Event GW150914 with ANTARES and IceCube,” Phys. Rev. D, vol. 93, no. 12, p. 122010, 2016.
  * [27] A. Albert et al., “Search for High-energy Neutrinos from Gravitational Wave Event GW151226 and Candidate LVT151012 with ANTARES and IceCube,” Phys. Rev. D, vol. 96, no. 2, p. 022005, 2017.
  * [28] B. Abbott et al., “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J. Lett., vol. 848, no. 2, p. L12, 2017.
  * [29] B. Baret et al., “Bounding the Time Delay between High-energy Neutrinos and Gravitational-wave Transients from Gamma-ray Bursts,” Astropart. Phys., vol. 35, pp. 1–7, 2011.
  * [30] B. Abbott et al., “GW190425: Observation of a Compact Binary Coalescence with Total Mass $\sim$3.4 M⊙.,” Astrophys. J. Lett., vol. 892, p. L3, 2020.
  * [31] R. Abbott et al., “GW190412: Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses,” Phys. Rev. D, vol. 102, no. 4, p. 043015, 2020.
  * [32] M. Colomer Molla, B. Baret, A. Coleiro, D. Dornic, and T. Pradier, “Search for neutrino counterparts of cataloged gravitational-wave events detected by Advanced-LIGO and Virgo during run O2 with ANTARES,” PoS, vol. ICRC2019, p. 856, 2020.
  * [33] A. Albert et al., “First all-flavor neutrino pointlike source search with the ANTARES neutrino telescope,” Phys. Rev. D, vol. 96, no. 8, p. 082001, 2017.
  * [34] A. Albert et al., “The search for high-energy neutrinos coincident with fast radio bursts with the ANTARES neutrino telescope,” Mon. Not. Roy. Astron. Soc., vol. 482, no. 1, pp. 184–193, 2019.
  * [35] A. Albert et al., “Constraining the contribution of Gamma-Ray Bursts to the high-energy diffuse neutrino flux with 10 yr of ANTARES data,” Mon. Not. Roy. Astron. Soc., vol. 500, no. 4, pp. 5614–5628, 2020.
  * [36] W. Assal, D. Dornic, F. Huang, E. Le Guirriec, M. Lincetto, and G. Vannoye, “Real-time multi-messenger analysis framework for KM3NeT,” JINST, vol. 16, no. 09, p. C09034, 2021.