Twisted recurrence for dynamical systems with exponential decay of
correlations
Jiajie Zheng

###### Abstract

We study the set of points returning infinitely often to a sequence of targets
dependent on the starting points. With an assumption of decay of correlations
for $L^{1}$ against bounded variations, we prove a generalized quantitative
recurrence result under Lipschitz twists.

##  1 Introduction

Let $(X,d)$ be a separable and compact metric space, and let $(X,\mu,T)$ be a
Borel probability measure-preserving system. The Poincaré Recurrence Theorem,
see e.g. [EW] states that almost all points in measurable dynamical systems
return close to themselves under a measure-preserving map; i.e.,

$\liminf_{n\to\infty}d(T^{n}x,x)=0\quad\text{for almost every }x\in X.$

Boshernitzan quantified the speed of recurrence in [Bo]. Namely, if the
$\alpha$-dimensional Hausdorff measure of $X$ is $\sigma$-finite for some
$\alpha>0$, then

$\liminf_{n\to\infty}n^{1/\alpha}d(T^{n}x,x)<\infty\quad\text{for almost every
}x\in X.$

A natural generalization of the recurrence speed is to consider the following
set

$R(\psi):=\\{x\in X:d(T^{n}x,x)<\psi(n)\text{ for infinitely many }n\\}$

given a function $\psi:\mathbb{N}\to(0,\infty)$. Much has been done on the
quantitative recurrence theory since then; for example, see [BF, CWW, DFL,
HLSW, KZ, KKP, Pe].

A topic closely related to recurrence theory is the so-called shrinking target
problem, which is concerned with determining the speed at which the orbit of a
$\mu$-generic point accumulates near a fixed point $y\in X$. More precisely,
for $\psi:\mathbb{N}\to(0,\infty)$ and a given point $y\in X$, one can define
the set

$A(\psi,y):=\\{x\in X:d(T^{n}x,y)<\psi(n)\text{ for infinitely many }n\\}.$

There have been plenty of results concerning the zero-one laws for
$\mu(A(\varphi,y))$ in specific systems; for example, see [CK, FMP, HNPV, KM,
LLVZ, Ph].

A more general setting called twisted recurrence, which can specialize into
shrinking target problem and quantitative recurrence, was introduced in [KZ,
LWW]. For $\psi:\mathbb{N}\to(0,\infty)$ and a Borel measurable function
$f:X\to X$, one can consider the set

$R(\psi,f):=\\{x\in X:d(T^{n}x,f(x))<\psi(n)\text{ for infinitely many }n\\}.$

Clearly $R(\psi,f)=R(\psi)$ if $f$ is the identity function and
$R(\psi,f)=A(\psi,y)$ if $f$ is the constant function with value $y$. When $f$
is Lipschitz, the zero-one laws for $\mu(R(\psi,f))$ were proved for some
special systems in [KZ, LWW], including classical dynamical systems like
$\beta$-transformations, Gauss transformations and left shift on Cantor sets.
Specifically, in these cases

$\mu(R(\psi,f))=\begin{cases}0&\text{if
}\sum_{n=1}^{\infty}\psi(n,x)^{\delta}<\infty\\\ 1&\text{if
}\sum_{n=1}^{\infty}\psi(n,x)^{\delta}=\infty,\end{cases}$

where $\delta$ is the Hausdorff dimension of the support of $\mu$.

In this paper, we prove quantitative Lipschitz recurrent properties of
dynamical systems with exponential decay of correlations. To state the main
result of the paper, we need to adapt and modify the settings and assumptions
from [KKP]. For the rest of the paper, let $X=[0,1]$ and $d$ be the standard
metric, and assume for any sequence of positive real number $\\{M_{n}\\}_{n}$
contained in $(0,1)$, there exists a sequence of functions
$\\{r_{n}:X\to(0,1)\\}_{n}$ such that $r_{n}(x)=\inf\\{r:\mu(B(x,r))=M_{n}\\}$
for all $x\in X$. For a function $f:X\to X$, the $f$-twisted recurrence set we
are interested is defined by

$R(\\{M_{n}\\}_{n},f):=\\{x\in X:T^{n}x\in B(f(x),r_{n}(x))\text{ for
infinitely many }n\\}.$

We say that $\mu$ is Ahlfors regular if there exist constants $c,\delta>0$
such that

$\frac{1}{c}r^{s}\leq\mu(B(x,r))\leq cr^{s}\quad\forall x\in\text{Supp}X\text{
and balls }B(x,r)\subset X$

and that $\mu$ is upper Ahlfors regular if there exist constants $c,\delta>0$
such that

$\mu(B(x,r))\leq cr^{s}\quad\forall x\in\text{Supp}X\text{ and balls
}B(x,r)\subset X.$ (1)

If $\mu$ is Ahlfors regular, then $R(\\{M_{n}\\}_{n},f)=R(\psi,f)$ where
$\psi(n)=r_{n}(x)$. In general, these definitions are different. Before
stating our main theorems, we now specify the class of functions $f$ which we
deal with by our technique. $f:X\to X$ is said to be Lipschitz if

$\sup_{x,y\in X,x\neq y}\frac{d(f(x),f(y))}{d(x,y)}<\infty.$ (2)

######  Definition 1.

Let $(X,\mu,T)$ be a measure-preserving system and
$p:\mathbb{N}\to\mathbb{R}^{+}$ be a sequence. We say that the correlations
for the system decay as $p$ for $L^{1}$ against bounded variation (BV), if

$\left|\int(f\circ T^{n})\cdot g\,d\mu-\int f\,d\mu\int
g\,d\mu\right|\leq||f||_{L^{1}}\cdot||g||_{\text{BV}}\cdot p(n)$

for all $n\in\mathbb{N}$ and for all functions $f$ with
$||f||_{1}:=\int|f|\,d\mu<\infty$ and $g$ with
$||g||_{\text{BV}}:=\sup_{x_{i}}\sum|g(x_{i+1})-g(x_{i})|+\sup|g|<\infty$.

######  Remark 1.

Definition 1 is weaker than the uniform mixing condition found in [KZ], as for
any balls $E,F\subset X$, we can take $f=\chi_{E}$ and $g=\chi_{F}$ and we
will get

$\left|\mu(T^{-n}E\cap F)-\mu(E)\mu(F)\right|\leq 3\mu(E)p(n).$

######  Remark 2.

For a non-increasing function $\psi:\mathbb{N}\to\mathbb{R}_{>0}$, we know
that there exists some

$\alpha\in\mathcal{W}(\psi):=\\{\alpha\in[0,1]:|q\alpha-p|<\psi(q)\text{ for
infinitely many natural numbers }p,q\\}.$

Then consider the system

$T:[0,1]\to[0,1],\quad x\mapsto x+\alpha\,(\bmod{1})$

together with the Lebesgue measure. Then $R(\psi,\text{Id})=[0,1]$, where
$\text{Id}:[0,1]\to[0,1]$ is the identity function. The convergence case of
Theorem 1 would fail; hence some mixing condition is need for a zero-one law
for $R(\psi,f)$ to hold. For more details, see [KZ, §2].

We first state the sufficient condition for $R(\\{M_{n}\\}_{n},f)=0$, which
depends on the convergence of $\sum_{n=1}^{\infty}M_{n}$. If there exists a
countable partition of subinterval $\\{X_{i}\\}_{i\in\mathcal{I}}$ so that
$f|_{X_{i}}$ is Lipschitz for all $i\in\mathcal{I}$, then we say $f$ is
piecewise Lipschitz. If there exists a countable partition of subinterval
$\\{X_{i}\\}_{i\in\mathcal{I}}$ so that $f|_{X_{i}}$ is monotone for all
$i\in\mathcal{I}$, then we say $f$ is piecewise monotone.

######  Theorem 1.

Let $p:\mathbb{N}\to\mathbb{R}^{+}$ be a function and assume the correlations
for $(X,\mu,T)$ decay as $p$ for $L^{1}$ against BV with
$\sum_{n=1}^{\infty}p(n)<\infty$. Let $\\{M_{n}\\}_{n}$ be a sequence
contained in $(0,1)$. Suppose $f:X\to X$ is piecewise Lipschitz and piecewise
monotone. If $\sum_{n=1}^{\infty}M_{n}<\infty$, then
$\mu(R(\\{M_{n}\\}_{n},f))=0$.

For the full measure part,

######  Theorem 2.

Let $p:\mathbb{N}\to\mathbb{R}^{+}$, $(X,\mu,T)$, $\\{M_{n}\\}_{n}$ and
$f:X\to X$ be as in Theorem 1. Additionally, suppose that

  * •

There exist $C>0$ and $0<\gamma<1$ such that $p(n)=C\gamma^{n}$.

  * •

$\mu$ is upper Ahlfors regular.

  * •

For any $q>0$,

$\limsup_{N}\sum_{n=\lfloor q\log N\rfloor}^{N}M_{n}=\infty$ (3)

Then $\mu(R(\\{M_{n}\\}_{n},f))=1$.

In [KKP], the authors proved Theorem 1 and Theorem 2 for $f$ being the
identity map. We will apply some of the ideas and techniques used in the proof
in [KKP], but the proof is different due to the new setup.

We get an immediate corollary for Ahlfors regular systems.

######  Corollary 3.

Let $p(n)=C\gamma^{n}$ for some $C>0$ and $0<\gamma<1$ and suppose the
correlations for $(X,\mu,T)$ decays as $p$ for $L^{1}$ against BV. Suppose
$\mu$ is $\delta$-Ahlfors regular. Assume $f$ is piecewise Lipschitz and
piecewise monotone. Then

  1. 1.

If $\sum_{n=1}^{\infty}\psi(n)^{\delta}<\infty$, then $R(\psi,f)$ is null.

  2. 2.

If for any $q>0$,

$\limsup_{N}\sum_{n=\lfloor q\log N\rfloor}^{N}M_{n}=\infty,$ (4)

then $R(\psi,f)$ is full.

We shall remark that Corollary 3 is a generalization of the convergence part
of [KZ, Theorem 1.2] in the case $X=[0,1]$. Most notably, the expanding,
bounded distortion and conformality assumptions are omitted. Corollary 3
partially generalizes the divergence case of [KZ, Theorem 1.2], with a
stronger summability assumption of the measures of the targets (4).

The structure of the paper is as follows. In §2, we reduce the proofs of
Theorem 1 and Theorem 2 to the case where $f:X\to X$ is Lipschitz and it only
changes monotonicity at finitely many points. In §3 we construct a sequence of
measurable sets whose limsup set is $R(\\{M_{n}\\}_{n},f)$, and we estimate
the measure of each set and conclude Theorem 1. In §4, we study quasi-
independence properties of this sequence and prove Theorem 2.

## Acknowledgements

The author would like to thank Tomas Persson for bringing this problem to his
attention and discussing possible generalizations. The author is grateful to
Dmitry Kleinbock for his wonderful advice and guidance throughout this
project.

##  2 Piecewise Lipschitz and piecewise monotone twists

By the properties of $R(\psi,f)$, to prove Theorem 1 and Theorem 2, it
suffices to show the statements for $f$ Lipschitz and monotone.

######  Lemma 4.

Theorem 1 and Theorem 2 hold for $f$ Lipschitz and monotone.

We will prove Lemma 4 in §§3-4. Here we first conclude Theorem 1 and Theorem 2
from this lemma.

###### Proofs of Theorem 1 and Theorem 2.

Suppose there exists a countable collection of disjoint open intervals
$\\{X_{i}=(a_{i},b_{i})\\}_{i\in\mathcal{I}}$ so that
$\bigcup_{i\in\mathcal{I}}X_{i}$ is full and $f$ is Lipschitz and monotone on
each $X_{i}$. Then for each $i\in\mathcal{I}$, $f$ is bounded on $X_{i}$ and
hence we can define

$f_{i}(x)=\begin{cases}f(x)&\text{if }x\in(a_{i},b_{i})\\\ \lim_{x\to
a_{i}^{-}}f(x)&\text{if }x\leq a_{i}\\\ \lim_{x\to b_{i}^{+}}f(x)&\text{if
}x\geq b_{i}.\end{cases}$

Then $f_{i}$ is Lipschitz and monotone. By Lemma 4, Theorem 1 and Theorem 2
hold for $f_{i}$ for each $i\in\mathcal{I}$. When $\mu(R(\psi,f_{i}))=0$ for
all $i\in\mathcal{I}$,

$\mu(R(\psi,f))=\sum_{i\in\mathcal{I}}\mu(R(\psi,f)\cap
X_{i})=\sum_{i\in\mathcal{I}}\mu(R(\psi,f_{i})\cap X_{i})=0;$

when $\mu(R(\psi,f_{i}))=1$ for all $i\in\mathcal{I}$,

$\mu(R(\psi,f))=\sum_{i\in\mathcal{I}}\mu(R(\psi,f)\cap
X_{i})=\sum_{i\in\mathcal{I}}\mu(R(\psi,f_{i})\cap
X_{i})=\sum_{i\in\mathcal{I}}\mu(X_{i})=1,$

so the theorems are proved once we prove Lemma 4 in the following sections. ∎

##  3 The convergence part

In this section we prove the convergence part of Lemma 4, thereby fixing
$p:\mathbb{N}\to\mathbb{R}^{+}$, $(X,\mu,T)$, $\\{M_{n}\\}_{n}$ and $f:X\to X$
to be Lipschitz and monotone. Let us define

$R_{n}(\\{M_{n}\\}_{n},f):=\\{x\in X:T^{n}x\in B(f(x),r_{n}(x))\\}.$

Without ambiguity, we shall denote $R_{n}(\\{M_{n}\\}_{n},f)$ simply by
$R_{n}$. Clearly $R(\\{M_{n}\\}_{n},f)=\limsup_{n\to\infty}R_{n}$.

We first prove a fact about the functions $r_{n}$.

######  Lemma 5.

For all $n\in\mathbb{N}$, $r_{n}$ is $1$-Lipschitz.

###### Proof.

Let $x,y\in[0,1]$. Without the loss of generality, suppose $r_{n}(x)\leq
r_{n}(y)$. Then

$\displaystyle B(x,r_{n}(x))\subset B(y,r_{n}(x)+d(x,y)),$

so $\mu(B(y,r_{n}(x)+d(x,y)))\geq M_{n}$ and hence

$\displaystyle r_{n}(y)\leq r_{n}(x)+d(x,y),$

i.e., $|r_{n}(y)-r_{n}(x)|\leq|x-y|$. ∎

For each $n$, we define $Y_{n}$ to be a subset of $[0,1]^{2}$ such that

$Y_{n}=\\{(x,y):y\in B(f(x),r_{n}(f(x)))\\}.$

Then we have

######  Lemma 6.

For each $n\in\mathbb{N}$, $Y_{n}$ is an open subset of $[0,1]^{2}$.

###### Proof.

Fix $n\in\mathbb{N}$. We prove that $Y_{n}$ has closed complement in
$[0,1]^{2}$. Let $\\{(x_{m},y_{m})\\}_{m}$ be a Cauchy sequence in the
complement of $Y_{n}$ and we denote its limit in $[0,1]^{2}$ by $(x,y)$. We
show that $(x,y)\not\in Y_{n}$. Let $\varepsilon>0$. Since $f$ is continuous,
there exists $k\in\mathbb{N}$ so that

$\displaystyle|f(x_{k})-f(x)|<\frac{\varepsilon}{L}\quad\text{and}\quad|y_{k}-y|<\varepsilon.$

Then

$\displaystyle|f(x)-y|\geq$
$\displaystyle|f(x_{k})-y_{k}|-|f(x_{k})-f(x)|-|y_{k}-y|$ $\displaystyle\geq$
$\displaystyle r_{n}(f(x_{k}))-2\varepsilon$
$\displaystyle\underset{\text{Lemma }\ref{lemma3}}{\geq}$ $\displaystyle
r_{n}(f(x))-3\varepsilon.$

Since $\varepsilon$ is chosen arbitrarily, we must have $|f(x)-y|\geq
r_{n}(f(x))$ as desired. ∎

Now we are ready to estimate the measure of $R_{n}$ for each $n$.

######  Lemma 7.

For each $n\in\mathbb{N}$,

$|\mu(R_{n})-M_{n}|\leq 3p(n).$

###### Proof.

Define $F_{n}:[0,1]^{2}\to\mathbb{R}$ to be the characteristic function of
$Y_{n}$. Since $Y_{n}$ is open, we can approximate $F_{n}$ by the following a
sequence of uniformly continuous functions $\\{F_{n,k}\\}_{k}$, where

$\displaystyle F_{n,k}(x,y):=\begin{cases}0&\text{if }(x,y)\not\in Y_{n}\\\
\min\\{1,kd((x,y),\partial Y_{n})&\text{if }(x,y)\in Y_{n}\end{cases}$

and $\partial Y_{n}$ denotes the boundary of $Y_{n}$. Note that
$\\{F_{n,k}\\}_{k}$ is increasing in $k$ and it converges pointwise to
$F_{n}$, so by the monotone convergence theorem, for each $\varepsilon>0$,
there exists some $k$ so that

$\left|\int
F_{n}(x,T^{n}x)\,d\mu(x)-F_{n,k}(x,T^{n}x)\,d\mu(x)\right|<\varepsilon$

and

$\left|\int F_{n}-\int F_{n,k}\right|<\varepsilon.$

Since $F_{n,k}$ is $2k$-Lipschitz, we can choose a partition by intervals
$\\{I_{h}\\}_{h=0}^{m-1}$ of $[0,1]$ so that

$\left|F_{n,k}(x,y)-\sum_{h=0}^{m-1}F_{n,k}(x,y_{h})\chi_{I_{h}}(y)\right|<\varepsilon$

for all $x,y\in[0,1]$, where $y_{h}$ is the middle point of $I_{h}$. Then
consider the integral

$\displaystyle\int\sum_{h=0}^{m-1}F_{n,k}(x,y_{h})\chi_{I_{h}}(T^{n}x)$ (5)

For each summand in (5), apply the decay of correlations to get

$\displaystyle\left|\int F_{n,k}(x,y_{h})\chi_{I_{h}}(T^{n}x)\,d\mu(x)-\int
F_{n,k}(x,y_{h})\,d\mu(x)\int\chi_{I_{h}}(x)\,d\mu(x)\right|$
$\displaystyle\leq\mu(I_{h})||F_{n,k}(x,y_{h})||_{BV}p(n)$

Note that for each $y_{h}$,

$F_{n,k}(x,y_{h})=\begin{cases}1&\text{if }d(f(x),y_{h})<r_{n}(f(x))\\\
0&\text{else}\end{cases}=\chi_{\\{x:d(f(x),y_{h})<r_{n}(f(x))\\}},$

so $||F_{n,k}(x,y_{h})||_{BV}\leq 3$. Then summing over $h$ we get

$\left|\int\sum_{h=0}^{m-1}F_{n,k}(x,y_{h})\chi_{I_{h}}(T^{n}x)\,d\mu(x)-\sum_{h=0}^{m-1}\int
F_{n,k}(x,y_{h})\,d\mu(x)\cdot\mu(I_{h})\right|\leq 3p(n).$

Hence

$\displaystyle\left|\mu(R_{n})-M_{n}\right|$ $\displaystyle=$
$\displaystyle\left|\int F_{n}(x,T^{n}x)\,d\mu(x)-\int
F_{n}\,d\mu\,d\mu\right|$ $\displaystyle\leq$ $\displaystyle\left|\int
F_{n}(x,T^{n}x)\,d\mu(x)-\int F_{n,k}(x,T^{n}x)\,d\mu(x)\right|$
$\displaystyle+\left|\int F_{n,k}(x,T^{n}x)\,d\mu(x)-\sum_{h=0}^{m-1}\int
F_{n,k}(x,y_{h})\,d\mu(x)\cdot\mu(I_{h})\right|$
$\displaystyle+\left|\sum_{h=0}^{m-1}\int
F_{n,k}(x,y_{h})\,d\mu(x)\cdot\mu(I_{h})-\int F_{n,k}\,d\mu\,d\mu\right|$
$\displaystyle+\left|\int F_{n,k}\,d\mu\,d\mu-\int F_{n}\,d\mu\,d\mu\right|$
$\displaystyle\leq$
$\displaystyle\varepsilon+(\varepsilon+3p(n))+\varepsilon+\varepsilon.$

As we let $k\to\infty$, the lemma is proved. ∎

The convergence/zero measure case will follow immediately from Lemma 7.

###### Proof of Theorem 1.

By Lemma 7,
$\sum_{n=1}^{\infty}\mu(R_{n})\leq\sum_{n=1}^{\infty}(M_{n}+3p(n))<\infty$, so
$\mu(R(\\{M_{n}\\}_{n},f))=0$ by the Borel-Cantelli lemma. ∎

##  4 The divergence part

In this section, we prove the divergence part of Lemma LABEL:lemm2, so
throughout the section we will assume that $(X,\mu,T)$, $\\{M_{n}\\}_{n}$ and
$f:X\to X$ satisfy the conditions stated in Theorem 2 and that
$\lim_{n\to\infty}p(n)=0$. The key ingredients for the proof of the divergence
case are the estimate of the measure of each $R_{n}$, which was proved in
Lemma 7, and a quasi-independence property of $\\{R_{n}\\}_{n}$, which we will
prove next.

######  Lemma 8.

There exist positive constants $K_{1},K_{2}$ and $K_{3}$ such that for all
$n,m\in\mathbb{N}$,

$\mu(R_{n}\cap E_{R+m})\leq
M_{n}M_{n+m}(1+K_{1}\sqrt{p(n)})+K_{2}(M_{n}p(n)^{s/2}+M_{n+m}(p(n)^{s/2}+p(m)))+K_{3}p(n)^{s}$

for some positive constants $K_{1},K_{2}$ and $K_{3}$.

###### Proof.

Let $n,m\in\mathbb{N}$. Define

$E_{n,m}(x,y,z)=\begin{cases}1&\text{if }y\in B(f(x),r_{n}(f(x))),z\in
B(f(x),r_{n+m}(f(x)))\\\ 0&\text{otherwise}\end{cases}$

Note that

$\mu(R_{n}\cap R_{n+m})=\int F_{n,m}(x,T^{n}x,T^{n+m}x)\,d\mu(x)$

We will approximate $E_{n,m}$ by a sum of products of characteristic functions
of intervals. For each $\ell\in\mathbb{N}$, we first partition $X$ evenly into
$\ell$ subintervals $\\{I_{h}\\}_{h=0}^{\ell-1}$ and denote the middle point
of $I_{h}$ by $x_{h}$ for each $h=0,\ldots,\ell-1$. Define

$F_{n,m,\ell}(x,y,z):=\sum_{h=0}^{\ell-1}\chi_{I_{k}}(x)\cdot\chi_{B(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell})}(y)\cdot\chi_{B(f(x_{h}),r_{n+m}(f(x_{h}))+\frac{L}{\ell})}(z)$

Note that for each $x\in I_{h}$ and $w\in B(f(x),r_{n}(f(x)))$,
$|x-x_{h}|<\frac{1}{2\ell}$, $|f(x)-f(x_{h})|<\frac{L}{2\ell}$ and
$|r_{n}(f(x))-r_{n}(f(x_{h}))|<\frac{L}{2\ell}$, so $w\in
B(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell})$ and hence $F_{n,m,\ell}\geq
F_{n,m}$. Then

$\displaystyle\int F_{n,m}(x,T^{n}x,T^{n+m}x)\,d\mu(x)$ $\displaystyle\leq$
$\displaystyle\int F_{n,m,\ell}(x,T^{n}x,T^{n+m}x)\,d\mu(x)$ $\displaystyle=$
$\displaystyle\sum_{h=0}^{\ell-1}\int\chi_{I_{k}}(x)\cdot\chi_{B(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell})}(T^{n}x)\cdot\chi_{B(f(x_{h}),r_{n+m}(f(x_{h}))+\frac{L}{\ell})}(T^{n+m}x)\,d\mu(x)$
$\displaystyle\underset{\text{BV}}{\leq}$
$\displaystyle\sum_{h=0}^{\ell-1}\int\chi_{B(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell})}(x)\cdot\chi_{B(f(x_{h}),r_{n+m}(f(x_{h}))+\frac{L}{\ell})}(T^{m}x)\,d\mu(x)\cdot\left(\mu(I_{h})+3p(n)\right)$
$\displaystyle\underset{\text{BV}}{\leq}$
$\displaystyle\sum_{h=0}^{\ell-1}\left(\mu\left(B\left(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell}\right)\right)+3p(m)\right)\left(\mu\left(B\left(f(x_{h}),r_{n+m}(f(x_{h}))+\frac{L}{\ell}\right)\right)\right)$
$\displaystyle\cdot(\mu(I_{k})+3p(n))$

By upper Ahlfors regularity (LABEL:ar),

$\mu\left(B\left(f(x_{h}),r_{n}(f(x_{h}))+\frac{L}{\ell}\right)\right)\leq
M_{n}+2c\left(\frac{L}{\ell}\right)^{s}$

and

$\mu\left(B\left(f(x_{h}),r_{n+m}(f(x_{h}))+\frac{L}{\ell}\right)\right)\leq
M_{n+m}+2c\left(\frac{L}{\ell}\right)^{s}.$

Now we pick $\ell$ so that $\sqrt{p(n)}/2<\frac{L}{\ell}<\sqrt{p(n)}$. Then

$\displaystyle\int F_{n,m}(x,T^{n}x,T^{n+m}x)\,d\mu(x)$ $\displaystyle\leq$
$\displaystyle(M_{n}+2cp(n)^{s/2}+3p(m))(M_{n+m}+2cp(n)^{s/2})\cdot(1+6L\sqrt{p(n)})$
$\displaystyle\leq$ $\displaystyle
M_{n}M_{n+m}(1+K_{1}\sqrt{p(n)})+K_{2}(M_{n}p(n)^{s/2}+M_{n+m}(p(n)^{s/2}+p(m)))+K_{3}p(n)^{s}$

where $K_{1}=6L$, $K_{2}=(1+6L\sqrt{\sup_{n}p(n)})(2c+3)$ and
$K_{3}=4c^{2}(1+6L\sqrt{\sup_{n}p(n)})$. ∎

To prove the divergence case, we will need the Chung-Erdös inequality.

######  Lemma 9.

In a probability space, for measurable sets $A_{1},\ldots,A_{n}$,

$\mu(A_{1}\cup\cdots\cup
A_{n})\geq\frac{\left(\sum_{j=1}^{n}\mu(A_{j})\right)^{2}}{\sum_{j,k=1}^{n}\mu(A_{j}\cup
A_{k})}$

Now we finish the proof of Theorem 2.

###### Proof of Theorem 2.

In addition to the assumptions at the beginning of this section, we assume
that $p(n)=C\gamma^{n}$ for some $C>0$ and $0<\gamma<1$. Let

$I_{N}=\left\\{j:-\frac{2}{s}\log_{\gamma}N\leq j\leq N\right\\}$

and

$U_{N}=\bigcup_{j\in I_{N}}R_{j}.$

Note that $\limsup_{n}R_{n}=\limsup_{N}U_{N}$.

Let

$S_{N}=\sum_{j\in I_{N}}\mu(R_{j})$

and

$\sigma_{N}=\sum_{j\in I_{N}}M_{j}$

By Lemma 7, we have

$\sigma_{N}-c_{1}\leq S_{N}\leq\sigma_{N}+c_{1}$

where the constant $c_{1}=C\gamma^{-1}$. On the other hand, let

$C_{N}=\sum_{j,k\in I_{N}}\mu(R_{j}\cap R_{k})$

and by Lemma 8, we have

$\displaystyle C_{N}=$ $\displaystyle S_{N}+2\sum_{j,k\in
I_{N},j>k}\mu(E_{j}\cap E_{k})$ $\displaystyle\leq$ $\displaystyle
S_{N}+(1+K_{1}CN^{-1/s})\sigma_{N}^{2}+2K\sum_{j,k\in
I_{N},j>k}(M_{k}\gamma^{ks/2}+M_{j}(\gamma^{ks/2}+\gamma^{j-k})+\gamma^{ks/2})$

where $K=K_{2}+K_{3}$. Denote

$D_{N}:=\sum_{j,k\in
I_{N},j>k}(M_{k}\gamma^{ks/2}+M_{j}(\gamma^{ks/2}+\gamma^{j-k})+\gamma^{ks/2})$

We show that $R_{N}$ is bounded, proceeding term by term. Each of the first
two and last sums is less than or equal to

$\displaystyle\sum_{j=-\frac{2}{s}\log_{\gamma}N}^{N}\sum_{k=-\frac{2}{s}\log_{\gamma}N}^{j-1}\gamma^{ks/2}\leq\sum_{j=-\frac{2}{s}\log_{\gamma}N}^{N}c_{2}e^{-\log
N}\leq c_{2}$

for some constant $c_{2}$ dependent on the bound in (3). For the third sum,

$\sum_{j,k\in
I_{N},j>k}M_{j}\gamma^{j-k}=\sum_{j=-\frac{2}{s}\log_{\gamma}N}M_{j}\sum_{k=-\frac{2}{s}\log_{\gamma}N}^{j-1}\gamma^{j-k}\leq
c_{3}\sigma_{N}$

Hence

$\displaystyle C_{N}\leq$ $\displaystyle
S_{N}+(1+K_{1}CN^{-1/2})\sigma_{N}^{2}+2K(3c_{2}+c_{3}\sigma_{N})$
$\displaystyle\leq$
$\displaystyle\sigma_{N}+c_{1}+(1+K_{1}CN^{-1/2})\sigma_{N}^{2}+2K(3c_{2}+c_{3}\sigma_{N})$

Now we can use the Chung-Erdös inequality to conclude that

$\mu(U_{N})\geq\frac{S_{N}^{2}}{C_{N}}\geq\frac{(\sigma_{N}-c_{1})^{2}}{(1+K_{1}CN^{-1/2})\sigma_{N}^{2}+c_{4}\sigma_{N}+c_{5}}$
(6)

for some constants $c_{4},c_{5}$; as we take $\limsup_{N\to\infty}$ in (6), we
have that

$\limsup_{N}\mu(U_{N})\geq 1$

Hence we proved that $\limsup_{n}R_{n}=\limsup_{N}U_{N}=1$. ∎

## References

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[a]DPNC, Section de Physique, Université de Genève, Geneva, Switzerland
[b]Laboratory of Instrumentation and Experimental Particles Physics, Lisboa
and Coimbra, Portugal [c]Laboratoire de L’accélérateur Linéaire from
CNRS/IN2P3, Orsay, France [d]Institute for Nuclear Research of the Russian
Academy of Sciences, Moscow, Russia [e]Moscow Institute of Physics and
Technology (MIPT), Moscow, Russia [f]National Research Nuclear University
(MEPhI), Moscow, Russia [g]Institut de Física d’Altes Energies (IFAE) - The
Barcelona Institute of Science and Technology (BIST), Campus UAB, Bellaterra
(Barcelona), Spain<EMAIL_ADDRESS>
# A $4\pi$ time-of-flight detector for the ND280/T2K upgrade

A. KorzenevCurrent address: Joint Institute for Nuclear Research, Dubna,
Russia  F. Barao  S. Bordoni  D. Breton  F. Cadoux  Y. Favre  M. Khabibullin
A. Khotyantsev  Y. Kudenko  T. Lux  J. Maalmi  P. MermodDeceased in the summer
of 2020  O. Mineev  F. Sanchez

###### Abstract

ND280 is a near detector of the T2K experiment which is located in the J-PARC
accelerator complex in Japan. After a decade of fruitful data-taking, ND280 is
scheduled for upgrade. The time-of-flight (ToF) detector, which is described
in this article, is one of three new detectors that will be installed in the
basket of ND280. The ToF detector has a modular structure. Each module
represents an array of 20 plastic scintillator bars which are stacked in a
plane of $2.4\times 2.2$ m2 area. Six modules of similar construction will be
assembled in a cube, thus providing an almost $4\pi$ enclosure for an active
neutrino target and two TPCs. The light emitted by scintillator is absorbed by
arrays of large-area silicon photo-multipliers (SiPMs) which are attached to
both ends of every bar. The readout of SiPMs, shaping and analog sum of
individual SiPM signals within the array are performed by a discrete circuit
amplifier. An average time resolution of about 0.14 ns is achieved for a
single bar when measured with cosmic muons. The detector will be installed in
the basket of ND280, where it will be used to veto particle originating
outside the neutrino target, improve the particle identification and provide a
cosmic trigger for calibration of detectors which are enclosed inside it.

Detector design and construction technologies and materials, Instrumentation
and methods for time-of-flight (TOF) spectroscopy, Scintillators and
scintillating fibres and light guides, Photon detectors for UV, visible and IR
photons (solid-state)

##  1 Introduction

An improvement of characteristics of the ND280 detector is one of the goals of
the ongoing T2K upgrade which is a part of the second phase of the T2K
experiment [1]. For this purpose, an active neutrino target (sFGD), consisting
of small scintillator cubes [2, 3], and two TPCs, which will cover large
angles [4], will be installed in the basket of ND280. The time-of-flight
system (ToF) will surround these detectors providing a close to 4$\pi$
coverage as shown in Fig. 1 (left). ToF will serve to determine the direction
of charged particles (inward or outward with respect to the target), thereby
rejecting those originating from interactions in materials outside the target
volume. The corresponding uncertainty is currently one of the main
contributors to the systematic uncertainty the ND280 physical results [5]. A
time resolution of 0.5 ns is required to unambiguously determine the direction
of particle flight. ToF can also be used to improve particle identification if
one can achieve the time-of-flight resolution better than 0.2 ns. Although
this identification is mostly ensured by $dE/dx$ measurements in the TPCs,
discrimination between muons and electrons or between protons and positrons is
not possible in some momentum ranges due to similar energy loss. In addition,
the fact that ToF encloses the sFGD and TPC detectors makes it convenient for
triggering on cosmic muons. The ToF will also ensure a precision timing
reference for detectors enclosed inside it.

The paper is organized as follows. In this section we introduce the ToF
detector concept. Details on the amplifier board are given in Sec. 2. Results
of the timing characteristic measurements for a single ToF bar are presented
in Sec. 3. The paper concludes with Sec. 4.

[width=0.6]Fig1_1.png [width=0.36]Fig1_2.jpg

Figure 1:  Left: six modules of ToF assembled in a cube covering the sFGD and
two TPC detectors. The front ToF module is not drawn in order to see the
interior. Right: photo of four ToF modules during the test with cosmics in the
CERN neutrino platform building.

###  1.1 Detector concept

Plastic scintillators, due to their fast photon emission properties, are
widely employed for neutrino detectors that are located on the surface of the
earth. Neutrino beams are bunched with a duration of the order of
microseconds, therefore the cosmic ray background is filtered out during
analysis using timing information.

There are two types of plastic scintillators commonly used in experiments: an
extruded plastic with wavelength shifting (WLS) for readout and a cast
scintillator with direct light readout. We briefly compare their relevant
parameters. A typical number of optical photons released isotropically by a
minimum-ionising particle crossing a 1 cm plastic scintillator is $\sim$104.
The resulting number of detected photons depends on a distance of propagation
and photo-detection efficiencies. In the case of extruded plastic, the
scintillator is usually shaped as a bar with a central hole [6] or a surface
groove [7] for a WLS fiber. A small trapping angle for transmitted photons in
the fiber strongly limits the absolute amount of light collected. Typically,
several 10’s of photons are detected. The time spread due to re-emission in
the WLS material introduces an extra uncertainty for the time measurement.
That makes the overall time resolution to be around 1 ns.

The time resolution of 0.5 ns which is necessary for determining the direction
of flight in ND280, would require several layers of plastic scintillator
described above. A better value of the resolution assumes an even larger
number of layers which makes this approach inconvenient. Given a very limited
space inside the ND280 basket, the solution with a cast plastic scintillator
that propagates near-UV photons of the primary fluor has been taken as a
baseline for the ToF detector. The bulk attenuation length of several meters
and very low light losses in successive internal reflections at the surface of
the bar eliminate the need for WLS and allow the number of photons detected at
the bar ends to be of the order of hundreds. The combination of high light
yield and fast response time of the scintillator makes possible a reduction in
the uncertainty typically down to the 0.1 ns level for bars with a length of 2
m. This improvement of the time resolution by an order of magnitude compared
to the case of extruded plastic would require, however, a photo-sensor with a
much larger surface coverage.

Traditionally, vacuum photomultiplier tubes (PMTs) are utilised to convert the
light to an electronic signal. However, due to the limited space and 0.2 T
magnetic field inside the ND280, this option was not possible. A readout by
arrays of large-area silicon photomultipliers (SiPMs) was chosen as a
baseline. This approach was already tested earlier for the readout of a single
plastic scintillator bar [8, 9] and WLS optical modules [10]. Nowadays, large-
area SiPMs have appeared on the market at relatively low cost and offer
several advantages over PMTs: magnetic field tolerance, a much smaller volume
and footprint allowing a compact design for bars without light-guides, low
operation voltage and high photon detection efficiency. An increased
sensitivity of SiPMs to the longer-$\lambda$ of the optical spectrum, which is
least attenuated inside the plastic, favours the use of longer bars.

###  1.2 Assembly of the ToF system

The ToF system consists of six modules of similar construction, which are
assembled in a cube as shown in Fig. 1 (left). Each module is a set of 20
plastic scintillator bars which are arranged in a plane with a total active
area of $5.4$ m2. In view of the limited space inside the basket, demands for
robustness of the construction111 The module must have a shock resistance of
$0.5g$ in the transverse direction to the plane, according to the J-PARC
safety regulations.  and similar dimensions for all modules, the configuration
with staggered bars was disfavoured. The planar configuration makes the
detector cheaper and easier to assemble. Bars are therefore positioned in a
plane with a gap of 1.5 mm between the side faces. The gap is necessary to
include steel brackets which keep the bars attached to the outer aluminum
frame. Two bars of the bottom module are removed to leave space for trays
which will guide service cables of sFGD and TPCs outside the basket.

The size of each bar is $220\times 12\times 1$ cm3. The material is a plastic
scintillator EJ-200 with a scintillation rise time of 0.9 ns, a decay time of
2.1 ns and an attenuation length of 380 cm [11]. The wavelength emission
spectrum peaks at 425 nm perfectly matching the SiPM spectral response. The
bars were wrapped in aluminium foils and a black plastic stretch film on top
to ensure opacity.

[width=0.49]Fig2_1 [width=0.49]Fig2_2

Figure 2: Left: an amplifier PCB with 8 SiPMs (top) and its location inside
the end-cap when mounted on the bar (bottom). Right: a simplified circuit
diagram of the 8 SiPM amplifier board.

Each scintillator bar is read out on both ends with an array of 8 SiPMs as
shown in Fig. 2 (left), for a total of 236 readout channels. A plastic end-cap
was designed in order to provide stable mounting, easy replacement and light-
tightness for the SiPM array. The inner volume of the end-cap fits the cross-
sectional dimension of the SiPM board and it bonds the bar as shown in Fig. 2
(left).

Once the assembly of one module was completed, the module was placed upright,
as shown in the photo of Fig. 1 (right), in order to take data with cosmic
rays. The data acquisition module WaveCatcher [12], two power supplies and the
DAQ notebook are also shown in the photo.

##  2 8 SiPM amplifier board

The jitter of a timing measurement is inversely proportional to the slope of
the pulse leading edge [13]. Therefore, to avoid additional dilution of the
time resolution, it is preferable to keep the rise time of the photosensor
response function shorter than the rise time of the photon emission pulse in
the scintillator. A large terminal capacitance of SiPM (1.3 nF for a $6\times
6$ mm2 sensor used in this work), however, increases the charge depletion time
that makes the front edge longer. It is the main reason why a parallel
connection of many sensors can not be accepted for precision timing
measurements. This issue can be resolved by applying an independent readout
for each sensor in such a way as to isolate the SiPM capacitances from each
other. The signals are then summed up at the end. Another option is to reduce
the capacitance by connecting SiPMs in series which decreases the rise time of
the leading edge but worsens the signal-to-noise ratio [14]. There is an
additional time dilution in this case associated with the dependence of
transit time on the position of SiPM in the series. Given the above
considerations, a combination of parallel and serial connection has been
chosen for the amplifier presented here.

Eight surface-mount devices S13360-6050PE222Specifications: area $6\times 6$
mm2, pixel pitch 50 $\mu$m, number of pixels 14400, PDE($\lambda=450$ nm)=40%,
gain $1.7\cdot 10^{6}$, dark counts 2 Mcpc, crosstalk probability 3% at
$T=25\,^{\circ}$C and $V_{\rm over}=3V$ [15]. from Hamamatsu [15] have been
soldered in an array to a custom-made PCB. A schematic of the 8 SiPM amplifier
circuit is shown in Fig. 2 (right). The eight SiPMs at each bar end are
grouped in four pairs. A series connection is used for SiPMs in a single pair,
while the current of each pair is amplified and shaped independently and added
to the summing node only at the end. A positive bias voltage is applied to the
SiPM cathode and the readout line is connected to the anode, which is close to
ground potential.

Taking advantage of the fact that the company provided values of the breakdown
voltage for every individual SiPM, we could sort them in groups by eight
having the same properties. Thus, each SiPM array operates using a single bias
voltage, which reduces significantly the complexity and cost of the board,
since fine tuning of individual SiPM is not required.

The input stage of the amplifier is based on the BJT transistor BFP420
operating in the common-emitter configuration. A photocurrent generated in the
SiPM discharge modifies the base-to-emitter current thereby regulating the
collector current which, in turn, is converted into voltage before entering
the RRC shaper. The transistor is equipped with negative feedback which makes
the circuit more stable, reduces distortions of the input waveform and widens
the bandwidth which justifies a compromise of the gain lowering at this stage.

At the next step, the signal passes the RRC filter. The filter reduces the
time constant of the signal trailing edge which helps to quickly restore a
stable baseline. The signal width is decreased by about 4 times and has a
typical value of 10 ns. The rising edge of the single photon signal was
measured to be 2.5 ns. The power consumption of a single input stage with RRC
filter is rather low, 28 mW. It is important in view of the fact that this
part of the circuit is placed right on the back side of the PCB with respect
to SiPMs. The heat dissipation is carried out by two inner conductive layers
of PCB which are grounded and are not used for the circuit.

In order to obtain a combined timing signal, four outputs of the shapers are
connected together and fed into the output stage which is characterised by a
low input impedance. The trace length from each transistor to the summing node
is made equal to avoid time smearing due to different transit times. The
output stage of the circuit is represented by a transimpedance amplifier based
on a current feedback operational amplifier (OPA) AD8001 which is suitable to
this high speed application. A 50 Ohm resistor at the end makes the output
impedance compatible with the input of the digitizer. The single-ended output
signal has a positive polarity. The RRC shaper and OPA form a high-pass filter
with a 2.9 MHz low frequency cutoff having a 0.18 gain and a high frequency
cutoff of 29 MHz having a gain of 1.81. The dynamic range of the amplifier has
been adjusted to 1 V to be close to the range of the DAQ digitizer. The
overall power consumption of the circuit is 0.2 W, which increases the
operating board temperature by almost 4∘C over ambient temperature.

The back side of the board was additionally screened with an aluminium plate
which was electrically connected to the circuit ground. To avoid short-
circuits, the plate and board were interleaved with a soft thermally
conductive interface pad.

[width=0.47]Fig3_1 [width=0.49]Fig3_2

Figure 3: Left: photo of the experimental setup for the cosmic ray
measurements showing two reference counters with the bar between them. Right:
averaged waveforms recorded by a single SiPM array. The waveforms correspond
to different positions of the trigger counters along the bar. Markers indicate
the time corresponding to 10% of amplitude as used in the dCFD analysis. The
corresponding position of trigger counters is also indicated.

##  3 Time resolution of a single bar

Characterisation of the time response for a single bar was done using a cosmic
test-bench. The setup is shown in Fig. 3 (left). The trigger was formed by the
coincidence of signals of two identical counters located at the top and bottom
of the bar under test. The counters were made of a fast scintillator, EJ-228,
and had a cubic shape with sides of 2 cm. They were coupled to 1” PMTs
(Hamamatsu R4998) from two sides via 5 cm long light-guides. The distance
between the centers of the two scintillators was 4 cm. The trigger time was
calculated as the average time registered by the four PMTs.

The HV of the trigger PMTs was adjusted in oder to accommodate the signal
amplitudes within a 1 V range. Only events with amplitudes of all four PMTs
larger than 0.5 V were used in the analysis. The resolution of the trigger
system was obtained from the distribution of the time difference measured by
top and bottom counters. The standard deviation $\sigma$ of a Gaussian fit to
this distribution was found to be 0.02 ns. Since this value is much smaller
than the time resolution of the bar, this uncertainty associated with the
trigger counters was neglected in the analysis.

Pulses from four trigger PMTs and two SiPM arrays of the test bar were sent to
a WaveCatcher digitizer module [12] for data acquisition. WaveCatcher provides
amplitude information from 1024 sampling cells, thus covering a 320 ns time
interval when operating at a sampling rate of 3.2 GS/s. Thus, it gives
complete information about the waveform of all pulses.

The measurements were done at 11 positions along the 2.2 m axis of the bar
with an exposure time of 24 hours for each position. Approximately six hundred
events were taken per exposure. For signals from the SiPM arrays, no minimum
amplitude cutoff was applied so as not to bias the time resolution study. Not
a single event was observed with the trigger fired and not having a signal in
the test bar, which indicates the full effectiveness of the counter.

Pulse waveforms averaged over all events taken for a single exposure and
corresponding to different locations of the trigger counters along the bar are
shown in Fig. 3 (right). Due to the photon absorption in plastic, the signal
amplitude decreases from 220 mV to 95 mV for signals corresponding to trigger
positions at 5 cm and 170 cm from the photosensors, respectively. One can also
observe a slight increase in the amplitude for trigger positions near the far
end of the bar. It can be explained by backwards reflected light which can
provide some improvement in the measurement precision. A similar effect was
observed in Ref. [16]. Moreover, the rise time333Time corresponding to the
amplitude interval from 10% to 90%. increases from 3.5 ns to 5 ns which is a
consequence of the length dispersion of photon paths during their propagation
in the bar. Both effects, attenuation and dispersion, lead to a deterioration
in the accuracy of measurements.

[width=0.5]Fig4_1_mod [width=0.47]Fig4_2

Figure 4: Left: time detected by the array-1 with respect to the trigger time.
Different peaks correspond to different positions of the trigger counters
along the bar. Right: an effective light velocity versus position of the
trigger counters along the bar.

A digital constant fraction discrimination (dCFD) technique was used to define
the arrival time of signals [13]. The time corresponding to 10% of the signal
amplitude was taken in the analysis. A linear interpolation between amplitudes
of adjacent time cells was applied. In the following discussion, the two SiPM
arrays at the two ends of the bar are referred to as array-1 and array-2. The
difference between the time detected by array-1 and the trigger time is shown
in Fig. 4 (left) for different positions of the trigger counters along the
bar. The standard deviation of a Gaussian approximation to this distribution
defines the time resolution of the measurement, while the mean value defines
the arrival time. In general, the dependence of the propagation distance on
the arrival time is close to linear. The ratio of these values represents the
effective light propagation velocity which is shown in Fig. 4 (right). A fit
to the distribution by the sum of a constant and exponent shows that the
velocity increases from 14.5 cm/ns to 15.9 cm/ns with distance. This is
consistent with the fact that photons at larger angles with respect to the
surface of the bar (slower velocity) have to travel longer distances and are
more prone to attenuation. The interaction time $t_{int}$ is calculated using
the time registered by the arrays at both ends of the bar $t_{1}$ and $t_{2}$
and the corresponding velocities $v_{1}$ and $v_{2}$

$t_{int}=\frac{t_{1}v_{1}+t_{2}v_{2}}{v_{1}+v_{2}}-\frac{L}{v_{1}+v_{2}}\stackrel{{\scriptstyle\,v_{1}=v_{2}\,}}{{=}}\frac{t_{1}+t_{2}}{2}-\frac{L}{2v},$
(1)

where $L$ is the bar length. An assumption of a constant propagation velocity,
which is commonly used in analyses, significantly simplifies the formula.
Using the velocity parametrization shown in Fig. 4 (right) one can determine
the bias introduced by this assumption. It does not exceed 0.05 ns which is
smaller by a factor of about four than the required resolution.

The time resolution as a function of position of the interaction point along
the bar obtained with one-side measurements using array-1 or array-2 is shown
in Fig. 5 (left). The graphs were approximated by analytic functions
consisting of a sum of two exponential functions and a constant. The
resolution for each array on either end degrades from 0.13 ns to 0.27 ns with
distance. The resolution of two-side readout was obtained from the gaussian
approximation of the $t_{int}$ distribution calculated with Eq. 1 and it is
also shown in Fig. 5 (left). Even better precision for $t_{int}$ can be
achieved by weighting $t_{1}$ and $t_{2}$ for each event with their respective
resolutions. Unlike the case of one-side readout, the two-side graphs are
almost independent of the interaction position and show a time resolution of
0.14 ns at the center of the bar.

[width=0.49]Fig5_1_mod2 [width=0.49]Fig5_2_mod

Figure 5: Left: time resolution of one-side readout (approximated by solid
curves), the resolution of two-side readout (approximated by broken curves) as
a function of distance along the bar. Right: two-side resolution (left axis of
histogram) and the signal amplitude (right axis of histogram) at the center of
the bar, $x=110$ cm, versus the bias voltage. The voltage used during
datataking is indicated by the arrow.

The dependence of the time resolution and the most probable value of the
amplitude versus the SiPM bias voltage for interactions at the bar center is
shown in Fig. 5 (right). As the photon detection efficiency (PDE) improves
with increasing voltage, the values of amplitude increase and the resolution
improves. The latter however is practically saturated at overvoltage444The
difference between the bias and breakdown voltages. values above 4 V. This
value corresponds to a bias voltage of 111 V applied to a SiPM pair in series.
This voltage was used during the datataking and it is indicated in the figure
by an arrow.

##  4 Conclusions

A time-of-flight system designed for the upgrade of the ND280 detector has
been presented. An array of plastic scintillator bars is used as an active
element of the detector. A new approach for the light readout, which employs
arrays of large-area SiPMs at two ends of each bar, has been dictated by the
limited space inside the basket of ND280 and the requirement of high time
resolution. Six modules of ToF have been successfully assembled, placed
upright in the temporary support frame and tested with cosmics. The time
response of one bar was characterised as a function of position along the bar
and the time resolution was found to be 0.14 ns on average, which meets the
requirements of the ND280 upgrade. The ToF modules will be packed without
disassembly and shipped from CERN to J-PARC. They will be installed in the
basket of ND280 and will provide almost $4\pi$ coverage for the active
neutrino target and two TPCs. The modules will be used to veto charged
particles originated outside the neutrino target and, potentially, for
particle identification starting from the year 2023.

## Acknowledgements

The construction of the ToF detector was made possible thanks to funding from
the Swiss National Science Foundation under Grant FLARE S19516. The R&D part
of the project was covered by the grant PP00P2_150583. The Russian group
acknowledges the support from JSPS-RFBR grant #20-52-50010, Russia, and the
support in the framework of the State project ”Science” by the Ministry of
Science and Higher Education of the Russian Federation under the contract
075-15-2020-778. T. Lux acknowledges funding from the Spanish Ministerio de
Economía y Competitividad (SEIDI - MINECO) under Grants No.
PID2019-107564GB-I00 and SEV-2016-0588. IFAE is partially funded by the CERCA
program of the Generalitat de Catalunya. We also would like to acknowledge the
contribution of FAST (COST action TD1401) which was at the origin of this
project.

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