# Weak Poissonian Correlations

Manuel Hauke  TU Graz, Austria<EMAIL_ADDRESS>and  Agamemnon
Zafeiropoulos  NTNU Trondheim, Norway<EMAIL_ADDRESS>
###### Abstract.

We examine a property of sequences called Poissonian pair correlations with
parameter $0\leqslant\beta\leqslant 1$ (abbreviated as $\beta$ – PPC). We
prove that when $\beta<1,$ the property of $\beta$ – PPC, also known as weak
Poissonian correlations, can be detected at the behaviour of sequences at
small scales, and show that this does not happen for the classical notion of
PPC, that is, when $\beta=1$. Furthermore, we show that whenever
$0\leqslant\alpha<\beta\leqslant 1$, $\beta$ – PPC is stronger than $\alpha$ –
PPC. We also include a discussion on weak Poissonian correlations of higher
orders, showing that for $\beta<1$, Poissonian $\beta$ – correlations of order
$k+1$ imply Poissonian $\beta$ – correlations of $k$ – th order with the same
parameter $\beta$.

###### 2010 Mathematics Subject Classification:

Primary 11K06, 11J71; Secondary 11K99

AZ is supported by a postdoctoral fellowship funded by Grant 275113 of the
Research Council of Norway

##  1\. Introduction and statement of results

###  1.1. Introduction

Let $(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence of points in the
unit interval. For $s>0$ and for any positive integer $N\geqslant 1,$ we
define the pair correlation function of the sequence
$(x_{n})_{n\in\mathbb{N}}$ to be

$R_{2}(s,N)=\frac{1}{N}\\#\Big{\\{}m,n\leqslant N,m\neq
n\mathrel{\mathop{\ordinarycolon}}\|x_{m}-x_{n}\|\leqslant\frac{s}{N}\Big{\\}},$

where $\|x\|$ denotes the distance of $x\in\mathbb{R}$ to the nearest integer
(see Section 1.4 for a proper definition). We say that the sequence
$(x_{n})_{n\in\mathbb{N}}$ has Poissonian pair correlations  (from now on
abbreviated as PPC) if

(1) $\lim_{N\to\infty}R_{2}(s,N)=2s\qquad\text{ for all }s>0.$

The notion of pair correlations of sequences has been studied in various
contexts. Its natural connection with mathematical physics is exhibited by the
famous Berry – Tabor conjecture [3]. A series of more recent papers have
studied pair correlations from a purely theoretical point of view. To mention
an example, it was an open problem within the theoretical setup to determine
the relation of Poissonian pair correlations with uniform distribution. It has
been recently shown that any sequence $(x_{n})_{n\in\mathbb{N}}$ with PPC is
also uniformly distributed mod $1,$ that is, we have

$\lim_{N\to\infty}\frac{1}{N}\\#\\{i\leqslant
N\mathrel{\mathop{\ordinarycolon}}x_{i}\in[a,b]\\}=b-a\quad\text{ for all
}\,0\leqslant a<b\leqslant 1.$

This result was established by Aistleiter, Larcher and Lewko [2] and
independently by Grepstad and Larcher [8], while a subsequent proof was also
given by Steinerberger [24] and, in a much more general setup, by Marklof
[16].

In the present paper, we focus our attention on the notion of weak Poissonian
pair correlations with parameter $0\leqslant\beta\leqslant 1$. As we shall
soon explain, this notion forms a weaker variant of the classical property of
PPC, a fact which explains the term “weak”. Our main purpose is to compare it
with the standard property of Poissonian pair correlations and demonstrate
several differences. To the best of our knowledge, the notion of weak
Poissonian pair correlations was first introduced by Nair and Pollicot in
[18].

Let $N\geqslant 1,0\leqslant\beta\leqslant 1$ and $s>0$. We define the pair
correlation function with parameter $\beta$ of a sequence
$(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ to be

$R_{2}(\beta;s,N)=\frac{1}{N^{2-\beta}}\\#\Big{\\{}m,n\leqslant N,m\neq
n\mathrel{\mathop{\ordinarycolon}}\|x_{m}-x_{n}\|\leqslant\frac{s}{N^{\beta}}\Big{\\}}.$

From a statistical perspective, the variants of the pair correlation function
(with a different scaling factor) which are considered in the present paper
fall into the framework of Ripley’s K-function [19].

We say that the sequence $(x_{n})_{n\in\mathbb{N}}$ has _weak Poissonian pair
correlations with parameter $0<\beta<1$_ (also called _Poissonian $\beta$ –
pair correlations_, or, for abbreviation, $\beta$ – PPC) if

(2) $\lim_{N\to\infty}R_{2}(\beta;s,N)=2s\qquad\text{ for all }\,s>0.$

For the value $\beta=0$, we say that the sequence $(x_{n})_{n\in\mathbb{N}}$
has $0$ – PPC if

$\lim_{N\to\infty}R_{2}(\beta;s,N)=2s\qquad\text{ for all
}\,0<s\leqslant\frac{1}{2}\cdot$

The reason why in the definition of $0$ – PPC the values of the scale $s$ are
restricted to the range $0<s\leqslant\frac{1}{2}$ is quite simple: when
$\beta=0$ we trivially have $R_{2}(0;s,N)\leqslant 1$ and thus (2) cannot hold
for $s>\frac{1}{2}$.

It is clear that for $\beta=1$ the notion of $\beta$ – PPC coincides with the
classical Poissonian pair correlations property. For $\beta=0$, the property
of $0$ – PPC appears to be known as being equivalent to uniform distribution,
see e.g. [24, Section 2.2]. Since we have not managed to find a rigorous
explanation in the literature, we provide a proof of this fact.

######  Theorem 1.

Let $(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence. The following are
equivalent.

  1. (i)

The sequence $(x_{n})_{n\in\mathbb{N}}$ is uniformly distributed mod $1$.

  2. (ii)

The sequence $(x_{n})_{n\in\mathbb{N}}$ has $0$ – PPC.

We now turn our attention to a rather striking aspect of weak Poissonian
correlations which, in our opinion, constitutes the most remarkable difference
of $\beta$ – PPC on the one hand, and the classical property of PPC, on the
other. We show that in order for a given sequence $(x_{n})_{n\in\mathbb{N}}$
to have $\beta$ – PPC for some $0\leqslant\beta<1$, it is sufficient that the
defining condition $\lim\limits_{N\to\infty}R_{2}(\beta;s,N)=2s$ holds for all
values of $s$ that lie within an interval of the form $(0,s_{0}),$ where
$s_{0}>0$ can be chosen to be arbitrarily small.

######  Theorem 2.

Let $0\leqslant\beta<1$. Assume there exists some constant $s_{0}>0$ such that

(3) $\lim_{N\to\infty}R_{2}(\beta;s,N)=2s\qquad\text{ for all }s<s_{0}.$

Then the sequence $(x_{n})_{n\in\mathbb{N}}$ has $\beta$ – PPC.

The key that served as a hint for this surprising fact was the proof of
Theorem 1. As the reader will be able to see, to deduce uniform distribution
from the hypothesis of 0 – PPC in the proof of Theorem 1, we only employ the
fact that $R_{2}(0;s,N)\to 2s$ for all values of $s>0$ that are sufficiently
small.

As already alluded to, the assumption that $\beta$ is strictly less than $1$
in Theorem 2 turns out to be essential. In other words, it is not possible to
extend Theorem 2 to the setup of PPC. We actually prove a rather stronger
statement: for any choice of the number $S>0$, we can find a sequence
$(x_{n})_{n\in\mathbb{N}}$ with $R_{2}(s,N)\to 2s$ for all $s<S$ which is not
even uniformly distributed, whence it does not have PPC.

######  Theorem 3.

For every $S>0$, there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ such that

$\lim_{N\to\infty}R_{2}(s,N)=2s\qquad\text{ for all }s<S$

but the sequence $(x_{n})_{n\in\mathbb{N}}$ is not uniformly distributed, and
hence does not have PPC.

The statement of Theorem 3 is in agreement with a phenomenon that enthusiasts
of the theory of Poissonian correlations might have observed: all existing
proofs of the fact that PPC implies uniform distribution [1, 8, 24] at some
point use the fact that the convergence $R_{2}(s,N)\to 2s$ holds for
arbitrarily large values of $s>0.$

In a remark after the proof of Theorem 1, we give a short, elementary proof of
the fact that sequences with $R_{2}(s,N)\to 2s$ for all $s>s_{0}$ are
uniformly distributed modulo $1$. This assumption can actually be relaxed: in
order for a sequence to be uniformly distributed mod $1$, it suffices that
$R_{2}(s,N)\to 2s$ holds for all positive integers $s\in\mathbb{N}$. This was
proved in [11] in a multidimensional setup, and later in [24], where it was
further pointed out that a sequence is equidistributed as long as
$R_{2}(s,N)\to 2s$ holds for all $s$ in a discrete set satisfying some
“maximum gap” condition. The fact that sequences with $R_{2}(s,N)\to 2s$ for
all $s>s_{0}$ are uniformly distributed follows directly from this observation
in [24], but we nevertheless decided to include one more proof because of its
simplicity and its underlying connection with the arguments in [1].

Concerning the relation of weak Poissonian pair correlations with uniform
distribution, Steinerberger [23, 24] proved that if the sequence
$(x_{n})_{n\in\mathbb{N}}$ has $\beta$ – PPC for some $0<\beta\leqslant 1$
then it is uniformly distributed. This showed that the property of $\beta$ –
PPC is stronger than $0$ – PPC for any $0<\beta\leqslant 1,$ extending the
already known fact that $1$ – PPC is a stronger property than $0$ – PPC
(uniform distribution).

Furthermore, Steinerberger refers in [24] to $\beta$ – PPC (for a given
$0<\beta<1)$ as a property that interpolates between uniform distribution and
PPC. The following result describes this phenomenon in a more precise fashion:
whenever $0\leqslant\alpha<\beta\leqslant 1,$ the property of $\beta$ – PPC is
stronger than $\alpha$ – PPC.

######  Theorem 4.

Let $0\leqslant\alpha<\beta\leqslant 1.$ If a sequence
$(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ has weak Poissonian pair correlations
with parameter $\beta$, then it also has weak Poissonian pair correlations
with parameter $\alpha$.

In turn, Theorem 4 leads naturally to the following question: are the notions
of $\beta$ – PPC for the various values of $0<\beta<1$ essentially different
or do they actually coincide, possibly for certain values of $\beta$? We prove
that the former is the case.

######  Theorem 5.

For any $0<\beta<1$, there exists a sequence $(y_{n})_{n\in\mathbb{N}}$ that
has Poissonian $\alpha$-pair correlations for any $0\leqslant\alpha<\beta$ but
does not have Poissonian $\alpha$ – pair correlations for any
$\beta\leqslant\alpha\leqslant 1$.

Therefore for the different values of the parameter $0\leqslant\beta\leqslant
1$ the property of $\beta\text{\,--\,PPC}$ can be seen as covering a spectrum
ranging from PPC to uniform distribution, with $\beta_{1}$ – PPC being
genuinely stronger than $\beta_{2}$ – PPC whenever $\beta_{1}>\beta_{2}.$ We
note that an analogue of Theorem 5 is already known for the value $\beta=1$:
the van der Corput sequence has $\alpha$ – PPC for any $\alpha<1$ but does not
have PPC, see e.g. [27]. The same paper [27] also contains an alternative
proof of Theorem 4; in our opinion, though, this proof contains an oversight
since the range $s$ is allowed to vary depending on $N$.

###  1.2. A metric consideration

Another difference of the weak Poissonian correlations compared to the
standard PPC occurs in the metric setup. From the metric point of view, an
increasing sequence $(a_{n})_{n\in\mathbb{N}}$ of positive integers is
considered fixed, and we examine the Lebesgue measure of the set of those
$x\in[0,1]$ for which the sequence $(a_{n}x)_{n\in\mathbb{N}}$ has Poissonian
pair correlations. Towards this direction, several results have been proved
for specific choices of the sequence $(a_{n})_{n\in\mathbb{N}}$. To name a few
examples, we mention that for any exponent $k\geqslant 2$ the sequence
$(n^{k}x)_{n\in\mathbb{N}}$ has Poissonian pair correlations for almost all
$x\in[0,1]$, see [10, 20]. On the other hand, writing $p_{n}$ for the $n$-th
prime number, the sequence $(p_{n}x)_{n\in\mathbb{N}}$ does not have
Poissonian pair correlations for almost all $x\in[0,1]$, see [26]. For more
such results we refer to [2, 21, 22].

A fundamental question in the theory of metric Poissonian pair correlations is
whether a zero – one law holds in the setup described above. That is, given an
increasing sequence $(a_{n})_{n\in\mathbb{N}}\subseteq\mathbb{N}$, does the
set of $x\in[0,1]$ such that $(a_{n}x)_{n\in\mathbb{N}}$ has PPC have Lebesgue
measure either $0$ or $1$? Although all results so far suggest that the answer
is positive, this question still remains unanswered.

We now briefly examine $\beta$ – pair correlations from the metric point of
view. Since $\beta-$PPC is equivalent to uniform distribution, for the $0$ –
PPC a zero-one law is true in a trivial sense: for any choice of the sequence
$(a_{n})_{n\in\mathbb{N}}$ the sequence $(a_{n}x)$ is uniformly distributed
and hence has $0-$PPC for almost all $x$ (see [12, Theorem 4.1]). We show that
this is also the case with weak pair correlations for all ranges $0<\beta<1.$

In view of the multiple examples of sequences
$(a_{n})_{n\in\mathbb{N}}\subseteq\mathbb{N}$ for which
$(a_{n}x)_{n\in\mathbb{N}}$ fails to have PPC almost surely (see references
above), the following theorem exhibits another difference between the
properties of weak Poissonian correlations with the standard notion of PPC.

######  Theorem 6.

Let $(a_{n})_{n\in\mathbb{N}}$ be an increasing sequence of positive integers
and $0<\beta<1.$ Then for almost all $x\in[0,1]$ the sequence
$(a_{n}x)_{n\in\mathbb{N}}$ has Poissonian $\beta$ – pair correlations.

###  1.3. Weak correlations of higher orders

As a last part of this paper, we extend the previous discussion on weaker
variants of Poissonian correlations to orders greater than $2$. We shall study
the $k$ – th order correlations of sequences rescaled by a factor of
$N^{\beta},$ which for convenience we name $(k,\beta)$ – correlations. Given
any integer $k\geqslant 2$, a parameter $0\leqslant\beta\leqslant 1$ and a
closed rectangle

(4)
$\mathcal{R}=[a_{1},b_{1}]\times[a_{2},b_{2}]\times\ldots\times[a_{k-1},b_{k-1}]\subseteq\mathbb{R}^{k-1},$

we define the $(k,\beta)$-correlation function of a sequence
$(x_{n})_{n\in\mathbb{N}}$ as

$\displaystyle
R_{k}(\beta;\mathcal{R},N)\\!=\\!\frac{1}{N^{k-(k-1)\beta}}\\#\\!\\!\left\\{\parbox{70.0001pt}{
$i_{1},\ldots,i_{k}\leqslant N$ \\\ \text{ \, distinct
}}\mathrel{\mathop{\ordinarycolon}}N^{\beta}((\\!(x_{i_{1}}-x_{i_{2}})\\!),\ldots,(\\!(x_{i_{1}}-x_{i_{k-1}})\\!))\in\mathcal{R}\right\\}\\!.$

(Here and in what follows, we say that the indices $i_{1},\ldots,i_{k}$ are
distinct when $i_{m}\neq i_{n}$ for $m\neq n.$) As one might expect, when
$0<\beta\leqslant 1$ we define a sequence $(x_{n})_{n\in\mathbb{N}}$ to have
Poissonian $(k,\beta)$ – correlations if for all rectangles
$\mathcal{R}\subseteq\mathbb{R}^{k-1}$ as in (4) it holds that

$\lim_{N\to\infty}R_{k}(\beta;\mathcal{R},N)=\lambda(\mathcal{R}),$

where $\lambda$ denotes the $(k-1)$ – dimensional Lebesgue measure. Evidently,
when $\beta=1$ in the preceding definition, one obtains the usual definition
of Poissonian $k$ – th order correlations [13].

For the specific value $\beta=0$, we say that a sequence has Poissonian
$(k,0)$ – correlations if
$\lim_{N\to\infty}R_{k}(0;\mathcal{R},N)=\lambda(\mathcal{R})$ holds for any
rectangle $\mathcal{R}\subseteq[-\frac{1}{2},\frac{1}{2}]^{k-1}.$

We note that the previous definitions are generalizations of just one of the
equivalent ways to define Poissonian $k$ – th order correlations. We refer the
reader to [9, Appendix A] for a relevant discussion, which can be easily
adapted to the context of weak correlations.

As a first result in this section, we prove that for any order $k\geqslant 2$,
Poissonian $(k,0)$ – correlations are equivalent to the property of uniform
distribution, thus generalising Theorem 1.

######  Theorem 7.

Let $k\geqslant 2$ be an integer. Then a sequence
$(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ is uniformly distributed if and only
if it has Poissonian $(k,0)$ – correlations.

As a corollary, Theorem 7 implies that Poissonian $(k+1,0)$ – correlations are
equivalent to Poissonian $(k,0)$ – correlations for any $k\geqslant 2$. This
brings us to the main motivating point for studying weak correlations of
higher orders.

It is an interesting open problem to determine whether the property of
Poissonian correlations of some given order $k\geqslant 3$ is a property
stronger than Poissonian correlations of lower orders. In the opposite
direction, Lutsko, Sourmelidis and Technau recently [15] gave explicit
examples of sequences with Poissonian pair correlations, but without
Poissonian triple correlations. Their argument can be easily modified to yield
the existence of sequences with Poissonian pair correlations but without
Poissonian $k$ – th order correlations for $k>3.$ In addition, the sequence
$(\sqrt{n})_{n^{2}\notin\mathbb{N}}$ is known to have Poissonian pair
correlations [5] but does not have Poissonian correlations of $k$ – th order
for at least one value of $k\geqslant 3$; if the correlations of all orders
followed the Poisson model, the gap distribution of this sequence would be the
Poisson distribution [13, Appendix]. However, this is not the case for
$(\sqrt{n})_{n^{2}\notin\mathbb{N}}$, as was proved by Elkies and McMullen in
[6].

It still remains unknown whether the other implication holds, i.e. whether
Poissonian $k+1$ – correlations imply Poissonian $k$ – correlations. In the
setup of weak correlations, it turns out that we are able to answer such a
question positively.

######  Theorem 8.

Let $k>2,0\leqslant\beta<1$. If a sequence $(x_{n})_{n\in\mathbb{N}}$ has
Poisonnian $(k,\beta)$ – correlations, then it also has Poissonian
$(k-1,\beta)$ – correlations. In particular, Poissonian ($k,\beta$) –
correlations imply weak Poissonian pair correlations with parameter
$\alpha\leqslant\beta$ and therefore, uniform distribution.

We end the introductory part of the paper with some directions for further
research. Concerning the relation between the properties of Poissonian
correlations of different orders, in view of Theorem 8 one might expect that
any sequence with Poissonian $(k+1)$ – order correlations also has Poissonian
correlations of order $k.$ However, we are hesitant to conjecture that this is
indeed the case. Attempting to adapt the proof of Theorem 8 to the case
$\beta=1,$ one can notice that the impasse arises from inequality (51)
appearing therein. More precisely, when $\beta=1$ the terms in (51) that
involve correlation functions of orders $2\leqslant m\leqslant k-1$ are not
negligible when $N\to\infty,$ and this in turn does not allow for a
characterisation of Poissonian correlations of order $k$ in terms of the
functions $R_{k}^{*}$ introduced later in the paper.

Further, we are confident that $(k,\beta)$ – Poissonian correlations of any
order $k\geqslant 3$ is a property that can be detected at small scales, in
other words an analogue of Theorem 2 holds for weak correlations of higher
orders. We believe that in order to prove this fact it would suffice to
establish a condition equivalent to Poissonian $(k,\beta)$ – correlations in
terms of the function $F_{\beta}$ defined later in (5).

###  1.4. Notation

Given two functions
$f,g\\!\mathrel{\mathop{\ordinarycolon}}\\!(0,\infty)\to\mathbb{R},$ we shall
write $f(t)=\mathcal{O}(g(t))$, $t\to\infty$ and $f(t)=o(g(t)),\,t\to\infty$
when

$\limsup_{t\to\infty}\frac{|f(t)|}{|g(t)|}<\infty\quad\text{ or
}\quad\lim_{t\to\infty}\frac{f(t)}{g(t)}=0$

respectively. For a real number $x\in\mathbb{R},$ we write $\\{x\\}$ for the
fractional part of $x$,
$\|x\|=\min\\{|x-k|\mathrel{\mathop{\ordinarycolon}}k\in\mathbb{N}\\}$ for the
distance of $x$ from its nearest integer, and

$(\\!(x)\\!)=\begin{cases}\\{x\\},&\text{ if }0\leqslant
x\leqslant\tfrac{1}{2}\\\ 1-\\{x\\},&\text{ if
}\tfrac{1}{2}<\\{x\\}<1\end{cases}$

for the signed distance of $x$ from the origin modulo $1$. Further, we use the
symbol $\\{\,\cdot\,\\}^{+}$ for the function

$\\{x\\}^{+}=\begin{cases}x,&\text{ if }x\geqslant 0\\\ 0,&\text{ if
}x<0.\end{cases}$

We use the standard notation $e(x)=e^{2\pi ix}$ and also write
$B(x_{0},r)=\\{x\in[0,1]\mathrel{\mathop{\ordinarycolon}}\|x-x_{0}\|\leqslant
r\\}$ for the closed interval with center $x_{0}$ and length $2r$ modulo $1$.
Given a positive integer $m\geqslant 1$ we write $[m]=\\{1,\ldots,m\\}$ for
the set of positive integers which are less than or equal to $m.$

##  2\. Preliminary Results

For any value of $0\leqslant\beta\leqslant 1$ we define the function

(5) $F_{\beta}(t,s,N)=\frac{1}{N^{1-\beta}}\\#\Big{\\{}1\leqslant n\leqslant
N\mathrel{\mathop{\ordinarycolon}}\mathinner{\\!\left\lVert
x_{n}-t\right\rVert}\leqslant\frac{s}{2N^{\beta}}\Big{\\}},\quad 0\leqslant
t\leqslant 1.$

Heuristically, $F_{\beta}$ can be thought of as counting the number of points
among the first $N$ terms of $(x_{n})_{n\in\mathbb{N}}$ lying within an
interval of length $s/N^{\beta}$ and random center $0\leqslant t\leqslant 1,$
uniformly distributed in $[0,1]$. We also define the integral

(6) $I_{\beta}(s,N)=\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t.$

The importance of $I_{\beta}(s,N)$ in the proofs of our main results is clear
from the following lemma, which is an analogue of [9, Proposition 9] in the
context of weak correlations.

######  Lemma 9.

Let $(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence,
$0\leqslant\beta\leqslant 1$ and $s_{0}>0$ be a constant. The following are
equivalent.
(i) The $\beta$ – pair correlation function satisfies

$\lim_{N\to\infty}R_{2}(\beta;s,N)=2s\qquad\text{ for all }s<s_{0}.$

(ii) We have

$\lim_{N\to\infty}\frac{1}{s}\int_{0}^{s}\\!R_{2}(\beta;\sigma,N)\,\mathrm{d}\sigma=s\qquad\text{
for all }s<s_{0}.$

(iii) The integral $I_{\beta}(s,N)$ defined in (6) satisfies

$\lim_{N\to\infty}I_{\beta}(s,N)=\begin{cases}s^{2},&\text{ if }\beta<1\\\
s^{2}+s,&\text{ if }\beta=1\end{cases}\qquad\text{ for all }s<s_{0}.$

###### Proof.

We first prove the equivalence of (i) with (ii). Assuming (i) is true, let
$s<s_{0}.$ Then by the monotonicity of $R_{2}(\beta;\sigma,N)$ as a function
of $\sigma$, for any integer $M\geqslant 1$ and for any $0\leqslant j\leqslant
M-1$ we get

$\frac{s}{M}R_{2}\Big{(}\beta;\frac{js}{M},N\Big{)}\leqslant\int_{[\frac{js}{M},\frac{(j+1)s}{M}]}R_{2}\big{(}\beta;\sigma,N\big{)}\,\mathrm{d}\sigma\leqslant\frac{s}{M}R_{2}\Big{(}\beta;\frac{(j+1)s}{M},N\Big{)}.$

Summing over all $0\leqslant j\leqslant M-1$ we get

$\frac{s}{M}\sum_{0\leqslant j\leqslant
M-1}\\!R_{2}\Big{(}\beta;\frac{js}{M},N\Big{)}\leqslant\int_{0}^{s}R_{2}(\beta;\sigma,N)\,\mathrm{d}\sigma\leqslant\frac{s}{M}\sum_{1\leqslant
j\leqslant M}R_{2}\Big{(}\beta;\frac{js}{M},N\Big{)}.$

Letting $N\to\infty$ and using the hypothesis in (i) we deduce that (ii)
holds.

Conversely, when (ii) is true, using again the monotonicity of
$R(\beta;\sigma,N)$ in $\sigma$ gives that

$\frac{1}{\varepsilon}\int_{s-\varepsilon}^{s}R_{2}(\beta;\sigma,N)\,\mathrm{d}\sigma\leqslant
R_{2}(\beta;s,N)\leqslant\frac{1}{\varepsilon}\int_{s}^{s+\varepsilon}R_{2}(\beta;\sigma,N)\,\mathrm{d}\sigma$

for $s<s_{0}$ fixed and for all $\varepsilon>0$ sufficiently small. Letting
first $N\to\infty$ and then $\varepsilon\to 0$ we conclude that
$\lim\limits_{N\to\infty}R_{2}(\beta;s,N)=2s.$
We now proceed to prove that (ii) is equivalent to (iii). Observe that

$\displaystyle F_{\beta}(t,s,N)^{2}$
$\displaystyle=\frac{1}{N^{2-2\beta}}\sum_{m,n\leqslant
N}\mathds{1}_{B\left(x_{m},\frac{s}{2N^{\beta}}\right)\cap
B\left(x_{n},\frac{s}{2N^{\beta}}\right)}(t).$

Denoting by $\lambda$ the 1 – dimensional Lebesgue measure, we can write

$\displaystyle I_{\beta}(s,N)$
$\displaystyle=\frac{1}{N^{2-2\beta}}\sum_{m,n\leqslant
N}\lambda\Big{(}B\Big{(}x_{m},\frac{s}{2N^{\beta}}\Big{)}\cap
B\Big{(}x_{n},\frac{s}{2N^{\beta}}\Big{)}\Big{)}$
$\displaystyle=\frac{1}{N^{2-2\beta}}\sum_{\begin{subarray}{c}m,n\leqslant
N\\\ m\neq
n\end{subarray}}\lambda\Big{(}B\Big{(}x_{m},\frac{s}{2N^{\beta}}\Big{)}\cap
B\Big{(}x_{n},\frac{s}{2N^{\beta}}\Big{)}\Big{)}+\frac{1}{N^{2-2\beta}}\cdot\frac{Ns}{N^{\beta}}$
$\displaystyle=\frac{1}{N^{2-2\beta}}\sum_{\begin{subarray}{c}m,n\leqslant
N\\\ m\neq
n\end{subarray}}\left\\{\frac{s}{N^{\beta}}-\mathinner{\\!\left\lVert
x_{n}-x_{m}\right\rVert}\right\\}^{+}+\frac{s}{N^{1-\beta}}$
$\displaystyle=\frac{s}{N^{2-\beta}}\sum_{\begin{subarray}{c}m,n\leqslant N\\\
m\neq n\end{subarray}}\left\\{1-\frac{\mathinner{\\!\left\lVert
x_{n}-x_{m}\right\rVert}}{s/N^{\beta}}\right\\}^{+}+\frac{s}{N^{1-\beta}}\,\cdot$

On the other hand,

$\displaystyle\frac{1}{s}\int_{0}^{s}R_{2}(\beta;\sigma,N)\,\mathrm{d}\sigma$
$\displaystyle=\frac{1}{s}\int_{0}^{s}\frac{1}{N^{2-\beta}}\sum_{\begin{subarray}{c}m,n\leqslant
N\\\ m\neq
n\end{subarray}}\mathds{1}_{B(0,\frac{\sigma}{N^{\beta}})}\left(x_{m}-x_{n}\right)\,\mathrm{d}\sigma$
$\displaystyle=\frac{1}{sN^{2-\beta}}\sum_{\begin{subarray}{c}m,n\leqslant
N\\\ m\neq
n\end{subarray}}\int_{0}^{s}\mathds{1}_{[\,\mathinner{\\!\left\lVert
x_{m}-x_{n}\right\rVert}N^{\beta},\infty)}(\sigma)\,\mathrm{d}\sigma$
$\displaystyle=\frac{1}{sN^{2-\beta}}\sum_{\begin{subarray}{c}m,n\leqslant
N\\\ m\neq n\end{subarray}}\left\\{s-\frac{\mathinner{\\!\left\lVert
x_{n}-x_{m}\right\rVert}}{1/N^{\beta}}\right\\}^{+}$
$\displaystyle=\frac{1}{N^{2-\beta}}\sum_{\begin{subarray}{c}m,n\leqslant N\\\
m\neq n\end{subarray}}\left\\{1-\frac{\mathinner{\\!\left\lVert
x_{n}-x_{m}\right\rVert}}{s/N^{\beta}}\right\\}^{+},$

and this finishes the proof. ∎

Finally, we mention the following lemma on the size of the integral
$I_{\beta}(s,N)$ defined in (6) that will be used in the proofs of many of the
main results.

######  Lemma 10.

Let $0\leqslant\beta\leqslant 1$. For any $s>0$ and $N\geqslant 1$, we have
$I_{\beta}(s,N)\geqslant s^{2}.$

###### Proof.

By the Cauchy – Schwarz inequality,

$I_{\beta}(s,N)=\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t\geqslant\left(\int_{0}^{1}\\!\\!F_{\beta}(t,s,N)\,\mathrm{d}t\right)^{2}=s^{2}.$

∎

##  3\. Proof of Theorem 1

Assume the sequence $(x_{n})_{n\in\mathbb{N}}$ is uniformly distributed mod
$1$. For any $0<s\leqslant\frac{1}{2}$ we have

$\displaystyle\left\lvert R_{2}(0;s,N)-2s\right\rvert$
$\displaystyle=\Big{\lvert}\frac{1}{N^{2}}\sum_{\begin{subarray}{c}k,\ell\leqslant
N\\\
k\neq\ell\end{subarray}}\mathds{1}_{[-s,s]}\left(x_{k}-x_{\ell}\right)-2s\Big{\rvert}$
$\displaystyle\leqslant\frac{1}{N}\sum_{k=1}^{N}\Big{\lvert}\frac{1}{N}\sum_{\begin{subarray}{c}\ell\leqslant
N\\\ \ell\neq
k\end{subarray}}\mathds{1}_{[-s,s]}\left(x_{k}-x_{\ell}\right)-2s\Big{\rvert}$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}\Big{\lvert}\frac{1}{N}\sum_{\ell\leqslant
N}\mathds{1}_{B(x_{k},s)}\left(x_{\ell}\right)-2s-\frac{1}{N}\Big{\rvert}$
$\displaystyle\leqslant D_{N}+\frac{1}{N},$

where

$D_{N}=D_{N}(\\{x_{1},\ldots,x_{N}\\})\mathrel{\mathop{\ordinarycolon}}=\sup_{0\leqslant
a<b\leqslant 1}\Big{\lvert}\frac{1}{N}\\#\left\\{i\leqslant
N\mathrel{\mathop{\ordinarycolon}}x_{i}\in[a,b]\right\\}-(b-a)\Big{\rvert}$

denotes the discrepancy of $\\{x_{1},\ldots,x_{N}\\}$. Since
$(x_{n})_{n\in\mathbb{N}}$ is uniformly distributed,
$\lim\limits_{N\to\infty}D_{N}=0$ and hence it follows that
$(x_{n})_{n\in\mathbb{N}}$ has $0$ – PPC.

Conversely, assume $(x_{n})_{n\in\mathbb{N}}$ has $0$ – PPC but is not
uniformly distributed. Then there exists an interval $[0,a]\subseteq[0,1]$ and
a sequence $(N_{k})_{k\in\mathbb{N}}$ such that

(7) $\lim_{k\to\infty}\frac{1}{N_{k}}\\#\\{n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}x_{n}\in[0,a]\\}=b\neq a.$

We now establish a relation between the correlation function
$R_{2}(0;s,N_{k})$ and the function $F_{0}(t,s,N_{k})$ defined in (5). A
counting argument gives

$\displaystyle\frac{1}{N_{k}^{2}}$ $\displaystyle\\#\Big{\\{}1\leqslant n\neq
m\leqslant N_{k}\mathrel{\mathop{\ordinarycolon}}\lVert
x_{n}-x_{m}\rVert\leqslant s\Big{\\}}$ (8)
$\displaystyle\quad\geqslant\frac{1}{N_{k}^{2}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-t\|\leqslant\frac{s}{2}\Big{\\}}^{2}-\frac{1}{N_{k}^{2}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-t\|\leqslant\frac{s}{2}\Big{\\}}$
$\displaystyle\quad\geqslant\Big{(}\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-t\|\leqslant\frac{s}{2}\Big{\\}}\Big{)}^{2}-\frac{1}{N_{k}},\quad\text{
for any }0\leqslant t\leqslant 1.$

By the assumption that $(x_{n})_{n\in\mathbb{N}}$ has $0$ – PPC, letting
$k\to\infty$ in (8) we get

$\displaystyle 2s$
$\displaystyle=\limsup_{k\to\infty}\frac{1}{N_{k}^{2}}\\#\Big{\\{}1\leqslant
n\neq m\leqslant N_{k}\mathrel{\mathop{\ordinarycolon}}\lVert
x_{n}-x_{m}\rVert\leqslant s\Big{\\}}$
$\displaystyle\geqslant\limsup_{k\to\infty}\Big{(}\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-t\|\leqslant\frac{s}{2}\Big{\\}}\Big{)}^{2}$

for all $0\leqslant t\leqslant 1$ and all $0<s\leqslant\frac{1}{2}$. Thus if
we fix some $\varepsilon>0,$ there exist some $s>0$ and $K\in\mathbb{N}$ such
that for all $k\geqslant K,$

$\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}\|\leqslant\frac{s}{2}\Big{\\}}\,<\,\frac{\varepsilon}{3}$

and

$\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-a\|\leqslant\frac{s}{2}\Big{\\}}\,<\,\frac{\varepsilon}{3}\cdot$

In view of (7), we can additionally assume that for all $k\geqslant K$ we have

$\frac{1}{N_{k}}\\#\big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}x_{n}\in[0,a]\big{\\}}>b-\frac{\varepsilon}{3}\cdot$

Observe that the function $F_{0}(t,s,N)$ defined in (5) satisfies

$\displaystyle\int_{0}^{a}F_{0}(t,s,N_{k})\,\mathrm{d}t$ $\displaystyle=$
$\displaystyle\frac{1}{N_{k}}\sum_{n\leqslant
N_{k}}\int_{0}^{1}\mathds{1}_{B\left(x_{n},\frac{s}{2}\right)}(t)\mathds{1}_{[0,a]}(t)\,\mathrm{d}t$
$\displaystyle=$ $\displaystyle\frac{1}{N_{k}}\sum_{n\leqslant
N_{k}}\lambda\Big{(}B(x_{n},\tfrac{s}{2})\cap[0,a]\Big{)}$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{N_{k}}\sum_{\begin{subarray}{c}n\leqslant N_{k}\\\
\frac{s}{2}\leqslant x_{n}\leqslant
a-\frac{s}{2}\end{subarray}}\lambda\Big{(}B\big{(}x_{n},\tfrac{s}{2}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{s}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\frac{s}{2}\leqslant x_{n}\leqslant
a-\frac{s}{2}\Big{\\}}.$

Thus for all $k\geqslant K$,

$\displaystyle\frac{1}{s}\int_{0}^{a}F_{0}(t,s,N_{k})\,\mathrm{d}t$
$\displaystyle\geqslant\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\frac{s}{2}\leqslant x_{n}\leqslant
a-\frac{s}{2}\Big{\\}}$ $\displaystyle\geqslant\frac{1}{N_{k}}\\#\\{n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}x_{n}\in[0,a]\\}$ $\displaystyle\hskip
11.38109pt-\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}\|\leqslant\frac{s}{2}\Big{\\}}-\frac{1}{N_{k}}\\#\Big{\\{}n\leqslant
N_{k}\mathrel{\mathop{\ordinarycolon}}\|x_{n}-a\|\leqslant\frac{s}{2}\Big{\\}}$
$\displaystyle\geqslant b-\varepsilon$

and similarly, we can show that

$\frac{1}{s}\int_{a}^{1}F_{0}(t,s,N_{k})\,\mathrm{d}t\geqslant
1-b-\varepsilon.$

By applying the Cauchy-Schwarz inequality, we obtain

$\displaystyle\int_{0}^{1}F_{0}(t,s,N_{k})^{2}\,\mathrm{d}t$ $\displaystyle=$
$\displaystyle\int_{0}^{a}F_{0}(t,s,N_{k})^{2}\,\mathrm{d}t+\int_{a}^{1}F_{0}(t,s,N_{k})^{2}\,\mathrm{d}t$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{a}\left(\int_{0}^{a}F_{0}(t,s,N_{k})\,\mathrm{d}t\right)^{2}+\frac{1}{1-a}\left(\int_{a}^{1}F_{0}(t,s,N_{k})\,\mathrm{d}t\right)^{2}$
$\displaystyle\geqslant$
$\displaystyle\frac{s^{2}}{a}(b-\varepsilon)^{2}+\frac{s^{2}}{1-a}(1-b-\varepsilon)^{2}.$

Letting $k\to\infty,$ the left hand side converges to $s^{2}$ by the
assumption of $0$ – PPC (this follows by Lemma 9), therefore

$1>\frac{(b-\varepsilon)^{2}}{a}+\frac{(1-b-\varepsilon)^{2}}{1-a}\cdot$

This is a contradiction for $\varepsilon$ sufficiently small because $b\neq
a.$

######  Remark.

The arguments of the second part of the previous proof, i.e. the proof of the
fact that $0$ – PPC implies uniform distribution, can be used to deduce that
$\beta$ – PPC implies uniform distribution for any $\beta<1.$ When $\beta=1$,
the same method can actually yield something more: any sequence for which

$\lim_{N\to\infty}R_{2}(s,N)=2s\qquad\text{ for all }s\geqslant s_{0}$

is uniformly distributed. Indeed, under this hypothesis a modification of the
proof of Lemma 9 gives that for all $s\geqslant s_{0},$

$\limsup_{N\to\infty}\int_{0}^{1}F_{1}(t,s,N)^{2}\,\mathrm{d}t\leqslant
s^{2}+s+s_{0}^{2}.$

The implicit constant here does not depend on the value of $s$. Assuming such
a sequence is not uniformly distributed, we obtain (7) and thus for all $k$
sufficiently large we get

$\int_{0}^{a}F_{1}(t,s,N_{k})\,\mathrm{d}t\geqslant(b-\varepsilon)s\quad\text{
and
}\quad\int_{a}^{1}F_{1}(t,s,N_{k})\,\mathrm{d}t\geqslant(1-b-\varepsilon)s.$

With an application of the Cauchy–Schwarz inequality this gives

(9)
$\displaystyle\int_{0}^{1}F_{1}(t,s,N_{k})^{2}\,\mathrm{d}t\geqslant\frac{s^{2}}{a}(b-\varepsilon)^{2}+\frac{s^{2}}{1-a}(1-b-\varepsilon)^{2}.$

Letting $k\to\infty$ we obtain

$\frac{s^{2}}{a}(b-\varepsilon)^{2}+\frac{s^{2}}{1-a}(1-b-\varepsilon)^{2}\leqslant
s^{2}+s+s_{0}^{2}\quad\text{for all }s\geqslant s_{0}$

and if we choose $s>0$ sufficiently large, we arrive at a contradiction. We
note that this proof can be seen as a “continuous analogue” of the proof of
Aistleitner, Lachmann and Pausinger [1] that avoids the need to use the
positivity of the Fejér kernel.

##  4\. Proof of Theorem 2

In this section we shall prove that whenever $0<\beta<1$ and

(10) $\lim_{N\to\infty}R_{2}(\beta;s,N)=2s\qquad\text{ for all }s<s_{0}.$

holds for some value of $s_{0}>0,$ the sequence $(x_{n})_{n\in\mathbb{N}}$ has
$\beta$ – PPC. In view of Lemmas 9 and 10, it suffices to show that for all
$s>0$ we have

(11) $\limsup_{N\to\infty}I_{\beta}(s,N)\leqslant s^{2}.$

To do so, note that for any fixed $s>0$, there exists an
$N_{0}=N_{0}(s,s_{0})$ such that for any even integer $K\geqslant N_{0}$ we
have $s/K<s_{0}$. Note that when $N$ is sufficiently large, for any
$t\in[0,1]$ we have

$B\Big{(}t,\frac{s}{2N^{\beta}}\Big{)}\subseteq\bigcup_{|\ell|\leqslant\frac{K}{2}}B\Big{(}t+\frac{\ell
s}{KN^{\beta}},\frac{s}{2KN^{\beta}}\Big{)}$

and therefore,

$F_{\beta}(t,s,N)\leqslant\sum_{|\ell|\leqslant\frac{K}{2}}F_{\beta}\Big{(}t+\frac{\ell
s}{KN^{\beta}},\frac{s}{K},N\Big{)}.$

The Cauchy – Schwarz inequality gives

$\displaystyle\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\Big{(}\sum_{|\ell|\leqslant\frac{K}{2}}F_{\beta}\Big{(}t+\frac{\ell
s}{KN^{\beta}},\frac{s}{K},N\Big{)}\Big{)}^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}(K+1)\sum_{|\ell|\leqslant\frac{K}{2}}F_{\beta}\Big{(}t+\frac{\ell
s}{KN^{\beta}},\frac{s}{K},N\Big{)}^{2}\,\mathrm{d}t$
$\displaystyle=(K+1)^{2}\int_{0}^{1}F_{\beta}\Big{(}t,\frac{s}{K},N\Big{)}^{2}\,\mathrm{d}t.$

By the assumption of (10) and using Lemma 9,

$\lim_{N\to\infty}\int_{0}^{1}F_{\beta}\Big{(}t,\frac{s}{K},N\Big{)}^{2}\,\mathrm{d}t=\frac{s^{2}}{K^{2}},$

so we obtain

$\limsup_{N\to\infty}\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t\leqslant\frac{(K+1)^{2}}{K^{2}}s^{2}.$

Letting $K\to\infty$, the result follows.

##  5\. Proof of Theorem 3

We prove that given any $S>0,$ we can find a sequence
$(x_{n})_{n\in\mathbb{N}}\subseteq[0,1]$ such that on the one hand
$R_{2}(s,N)\to 2s$ for all $s<S$ but on the other hand,
$(x_{n})_{n\in\mathbb{N}}$ is not uniformly distributed in $[0,1].$ During the
course of the proof we shall deal with the correlation functions of different
sequences at the same time, and thus for convenience we stress out the
dependence of the correlation function on the sequence.

We begin with a brief description of the main heuristic idea of the proof.
Given the value $S>0$, we first want to define a sequence
$(x_{n})_{n\in\mathbb{N}}$ whose pair correlation function is asymptotically
smaller than the value that corresponds to the Poisson model for all $s<S.$
More specifically, we want to find a sequence such that $R_{2}(s,N)\to 2cs$
for all $s<\frac{1}{c}S$, where $0<c<1$ is some constant. We then expect that
if we contract the sequence by a factor $c,$ on the one hand the “contracted”
sequence will have pair correlations asymptotically equal to $2s$ for all
$s<S,$ on the other hand it cannot be uniformly distributed modulo $1,$ since
it will be contained within the interval $[0,c].$

We thus need to take a careful look on how contracting a sequence affects its
pair correlation function. Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence in
$[0,1]$ and $0<c<1$ be an arbitrary constant. For any two distinct terms
$x_{n},x_{m}\in[0,1]$, we have

(12) $\displaystyle\|c(x_{n}-x_{m})\|\leqslant\frac{s}{N}$
$\displaystyle\Leftrightarrow\quad\exists
k\in\mathbb{Z}\mathrel{\mathop{\ordinarycolon}}\lvert
c(x_{n}-x_{m})-k\rvert\leqslant\frac{s}{N}$
$\displaystyle\Leftrightarrow\quad\exists
k\in\mathbb{Z}\mathrel{\mathop{\ordinarycolon}}\big{\lvert}x_{n}-x_{m}-k/c\big{\rvert}\leqslant\frac{s/c}{N}\cdot$

We now observe that for all $N\geqslant 1$ sufficiently large, the only
candidate for $k$ is $0$: If $k\neq 0$, then $\lvert k/c\rvert\geqslant
1+\delta$ for some fixed $\delta>0$. Now if $N\geqslant 1$ is large enough
such that $\dfrac{s/c}{N}<\delta$, since $x_{n}-x_{m}\in[-1,1]$ we see that
$\lvert x_{n}-x_{m}-\frac{k}{c}\rvert>\dfrac{s/c}{N}$ and the inequalities in
(12) will have to fail. Hence,

(13) $\displaystyle\|c(x_{n}-x_{m})\|\leqslant\frac{s}{N}\quad$
$\displaystyle\Leftrightarrow\quad\lvert
x_{n}-x_{m}\rvert\leqslant\frac{s/c}{N}$ (14)
$\displaystyle\Leftrightarrow\quad\|x_{n}-x_{m}\|\leqslant\frac{s/c}{N}\hskip
8.53581pt\text{ and }\hskip 8.53581pt\lvert
x_{n}-x_{m}\rvert<1-\frac{s/c}{N}\cdot$

Therefore writing $R_{2}^{\mathfrak{X}}(s,N)$ and $R_{2}^{c\mathfrak{X}}(s,N)$
for the correlation functions of the sequences $(x_{n})_{n\in\mathbb{N}}$ and
$(cx_{n})_{n\in\mathbb{N}},$ respectively, we have

(15) $\displaystyle R_{2}^{c\mathfrak{X}}(s,N)$
$\displaystyle=\frac{1}{N}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|x_{n}-x_{m}\|\leqslant\frac{s/c}{N}\,\,\&\,\,\lvert
x_{n}-x_{m}\rvert<1-\frac{s/c}{N}\Big{\\}}$ (16)
$\displaystyle=\frac{1}{N}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|x_{n}-x_{m}\|\leqslant\frac{s/c}{N}\Big{\\}}$
(17) $\displaystyle\quad-\frac{1}{N}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|x_{n}-x_{m}\|\leqslant\frac{s/c}{N}\,\,\&\,\,\lvert
x_{n}-x_{m}\rvert\geqslant 1-\frac{s/c}{N}\Big{\\}}$ (18)
$\displaystyle=R_{2}^{\mathfrak{X}}(s/c,N)-E^{\mathfrak{X}}(s/c,N),$

where $E^{\mathfrak{X}}(s,N)$ is an error term, by definition equal to

(19) $E^{\mathfrak{X}}(s,N)=\frac{1}{N}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|x_{n}-x_{m}\|\leqslant\frac{s}{N}\,\&\,\lvert
x_{n}-x_{m}\rvert\geqslant 1-\frac{s}{N}\Big{\\}}.$

The upshot is that for sequences $(x_{n})_{n\in\mathbb{N}}$ for which the
error term $E^{\mathfrak{X}}(s/c,N)$ tends to $0$ as $N\to\infty,$ we will
have

(20)
$\lim_{N\to\infty}R_{2}^{c\mathfrak{X}}(s,N)=\lim_{N\to\infty}R_{2}^{\mathfrak{X}}(s/c,N),$

provided, of course, that the limit on the right-hand side exists. So as long
as we find a sequence $(x_{n})_{n\in\mathbb{N}}$ with
$E^{\mathfrak{X}}(s,N)\to 0$ for any $s>0$ and a number $0<c<1$ such that the
correlation function $R_{2}^{\mathfrak{X}}(s,N)$ of $(x_{n})_{n\in\mathbb{N}}$
satisfies

(21) $\lim_{N\to\infty}R_{2}^{\mathfrak{X}}(s,N)=2cs\qquad\text{for any
}s<\frac{1}{c}S,$

the proof of Theorem 3 can be finished as already mentioned: the sequence
$(cx_{n})_{n\in\mathbb{N}}$ satisfies
$\lim_{N\to\infty}R_{2}^{c\mathfrak{X}}(s,N)=2s$ for every $s<S$, but is only
supported in $[0,c]$, which makes it impossible to be uniformly distributed.

We now move our discussion to how we can construct a sequence that satisfies
(21). The idea is to start with some sequence $(y_{n})_{n\in\mathbb{N}}$ that
has PPC and “dilute” its pair correlations by inserting periodically terms
from some other sequence $(z_{n})_{n\in\mathbb{N}}$, for which the pair
correlation function is $0$ for all sufficiently small scales $s>0.$

More precisely, we consider a positive integer $M\geqslant 2S$, a sequence
$(y_{n})_{n\in\mathbb{N}}$ that will be specified later, as well as the binary
van der Corput sequence $(z_{n})_{n\in\mathbb{N}}$. By the definition of
$(z_{n})_{n\in\mathbb{N}}$ (see e.g. [12, p. 127]) one can deduce that for any
$N\geqslant 1,$ all gaps between elements of the set
$\\{z_{1},\ldots,z_{N}\\}$ are at least $1/(2N).$ It follows immediately that
the correlation function $R_{2}^{\mathcal{Z}}(s,N)$ of
$(z_{n})_{n\in\mathbb{N}}$ satisfies

(22) $\lim_{N\to\infty}R_{2}^{\mathcal{Z}}(s,N)=0\qquad\text{ for any
}\,0<s<\frac{1}{2}\cdot$

We now build $(x_{n})_{n\in\mathbb{N}}$ by inserting an element of
$(z_{n})_{n\in\mathbb{N}}$ after every $M-1$ elements of
$(y_{n})_{n\in\mathbb{N}}$; that is, writing $n=kM+r$ where $k\in\mathbb{N}$
and $0\leqslant r<M$, we define

(23) $x_{n}=\begin{cases}z_{k},&\text{ if $r=0$}\\\ y_{k(M-1)+r},&\text{ if
$r\neq 0$}.\end{cases}$

We then compute

(24) $\displaystyle R_{2}^{\mathfrak{X}}(s,MN)$
$\displaystyle=\frac{1}{MN}\\#\Big{\\{}n\neq
m\leqslant(M-1)N\mathrel{\mathop{\ordinarycolon}}\|y_{n}-y_{m}\|\leqslant\frac{s/M}{N}\Big{\\}}$
(25) $\displaystyle\quad+\frac{1}{MN}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|z_{n}-z_{m}\|\leqslant\frac{s/M}{N}\Big{\\}}$
(26) $\displaystyle\quad+\frac{2}{MN}\\#\Big{\\{}n\leqslant(M-1)N,m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|y_{n}-z_{m}\|\leqslant\frac{s/M}{N}\Big{\\}}$
(27)
$\displaystyle=\frac{M-1}{M}R_{2}^{\Upsilon}\left(s(M-1)/M,(M-1)N\right)+\frac{1}{M}R_{2}^{\mathcal{Z}}(s/M,N)$
(28) $\displaystyle\quad+\frac{2}{MN}\\#\Big{\\{}n\leqslant(M-1)N,m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|y_{n}-z_{m}\|\leqslant\frac{s/M}{N}\Big{\\}}.$

In view of (22) and the assumption that $M\geqslant 2S$, we see that
$\lim_{N\to\infty}R_{2}^{\mathcal{Z}}(s/M,N)=0$ for any $s<S$. So we are close
to the end of the proof as long as we find a sequence
$(y_{n})_{n\in\mathbb{N}}$ fulfilling the following properties:
${\rm(i)}$ $(y_{n})_{n\in\mathbb{N}}$ has PPC,
${\rm(ii)}$ for every $s>0,$

(29) $\lim_{N\to\infty}\frac{1}{N}\\#\Big{\\{}n\leqslant(M-1)N,m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|y_{n}-z_{m}\|\leqslant\frac{s/M}{N}\Big{\\}}=\frac{2s(M-1)}{M},$

${\rm(iii)}$ for every $s>0,$ the error term $E^{\Upsilon}(s,N)$ defined in
(19) corresponding to the sequence $(y_{n})_{n\in\mathbb{N}}$ satisfies
$\lim\limits_{N\to\infty}E^{\Upsilon}(s,N)=0$.

The existence of such a sequence $(y_{n})_{n\in\mathbb{N}}$ is guaranteed by
the following lemma.

######  Lemma 11.

Let $M\geqslant 1$ be an integer and let $(z_{n})_{n\in\mathbb{N}}$ denote the
binary van der Corput sequence. Furthermore let $(Y_{n})_{n\in\mathbb{N}}$ be
a sequence of independent, uniformly distributed random variables in $[0,1]$.
Then almost surely, the sequence $(Y_{n}(\omega))_{n\in\mathbb{N}}$ has PPC,
satisfies property (29) and $\lim\limits_{N\to\infty}E(s,N)=0$.

###### Proof.

It is a well – known fact that any sequence $(Y_{n})_{n\in\mathbb{N}}$ of
independent, uniformly distributed random variables in $[0,1]$ has PPC almost
surely. For a proof, we refer to [9, Appendix B] in the setup of higher order
correlations, or also to [11] for sequences in higher dimensions.

We now prove the second property, namely that $(Y_{n})_{n\in\mathbb{N}}$
satisfies (29) almost surely. Choose an arbitrary $s>0$. Writing

$Y_{n,N}(s)=\\#\Big{\\{}1\leqslant m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\|Y_{n}-z_{m}\|\leqslant\frac{s}{MN}\Big{\\}},\quad
n=1,\ldots,(M-1)N,$

the quantity in question is equal to

$I_{N}(s)=\frac{1}{N}\sum_{n\leqslant(M-1)N}Y_{n,N}(s).$

It is easy to see that

$\mathbb{E}[I_{N}(s)]=\frac{2s(M-1)}{M},\qquad N\geqslant 1$

and we proceed to compute the variance of $I_{N}.$ We first observe that for
any $N\geqslant 1$, the functions $(Y_{n,N})_{n=1}^{N(M-1)}$ form a family of
independent random variables. Furthermore, for $2^{k}\leqslant N<2^{k+1}$ we
have
$(z_{m})_{m=1}^{N}\subseteq\\{\frac{\ell}{2^{k+1}},\ell=1,\ldots,2^{k+1}\\}$,
so we can bound

$Y_{n,N}(s)\leqslant\\#\Big{\\{}1\leqslant m\leqslant
2^{k+1}\mathrel{\mathop{\ordinarycolon}}\frac{m}{2^{k+1}}\in\Big{[}y_{n}-\frac{2s/M}{2^{k+1}},y_{n}+\frac{2s/M}{2^{k+1}}\Big{]}\Big{\\}}\leqslant\frac{4s}{M}+2.$

This implies that

(30)
$\displaystyle\text{Var}[I_{N}]\\!=\\!\frac{1}{N^{2}}\sum_{n=1}^{N(M-1)}\text{Var}[Y_{n,N}]=\frac{1}{N^{2}}\sum_{n=1}^{N(M-1)}\Big{(}\mathbb{E}[Y_{n,N}^{2}]-\mathbb{E}[Y_{n,N}]^{2}\Big{)}\\!=\\!\mathcal{O}_{M,s}\Big{(}\frac{1}{N}\Big{)}.$

To finish the proof we use a standard approximation argument (for more details
see e.g. [22]). Let $\gamma>0$ and consider the subsequence

$B_{N}\mathrel{\mathop{\ordinarycolon}}=\left\lceil{N^{\gamma}}\right\rceil,\qquad
N\geqslant 1.$

For $s>0$ fixed, we use Chebyshev’s inequality, the first Borel–Cantelli Lemma
and the variance estimate from (30) to see that

(31) $\lim_{N\to\infty}I_{B_{N}}(s)=\frac{2s(M-1)}{M}\qquad\text{ almost
surely }$

(where the zero – measure set depends on $s$). Repeating the argument for all
$s$ lying in a dense, countable subset of $\mathbb{R}_{+}$ and employing the
monotonicity of $I_{N}(s)$ as a function of $s$, we see that (31) actually
holds for all $s>0$. Next, if $N\geqslant 1$ is an arbitrary integer, we let
$K\geqslant 1$ be such that $B_{K}\leqslant N<B_{K+1}$ and observe that for
any $s>0,$

(32)
$\displaystyle\frac{B_{K}}{B_{K+1}}I_{B_{K}}\Big{(}\frac{B_{K}s}{B_{K+1}}\Big{)}\leqslant
I_{N}(s)\leqslant\frac{B_{K+1}}{B_{K}}I_{B_{K+1}}\Big{(}\frac{B_{K+1}s}{B_{K}}\Big{)}.$

Since $\lim\limits_{N\to\infty}\dfrac{B_{N}}{B_{N+1}}=1$, we deduce from (31)
that (29) holds almost surely for any fixed $s>0$.

Finally, it remains to prove that the third property, namely
$\lim_{N\to\infty}E(s,N)=0,$ is satisfied almost surely. For the error term
$E^{\Upsilon}(s,N)$ corresponding to $(y_{n})_{n\in\mathbb{N}}$, we provide
the upper bound

(33) $\displaystyle E^{\Upsilon}(s,N)$
$\displaystyle\leqslant\frac{1}{N}\\#\Big{\\{}n\neq m\leqslant
N\mathrel{\mathop{\ordinarycolon}}\lvert y_{n}-y_{m}\rvert\geqslant
1-\frac{s}{N}\Big{\\}}$
$\displaystyle\leqslant\frac{2}{N}\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}y_{n}\geqslant
1-\frac{s}{N}\Big{\\}}\cdot\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}y_{n}\leqslant\frac{s}{N}\Big{\\}}.$

It is therefore sufficient to prove that almost surely, we have

(34) $\lim_{N\to\infty}\frac{1}{N}\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}Y_{n}\geqslant
1-\frac{s}{N}\Big{\\}}\cdot\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}Y_{n}\leqslant\frac{s}{N}\Big{\\}}=0.$

This can be proved using a mean – variance argument, completely analogous to
the one used to prove (29). We leave the details to the interested reader. ∎

Having proved the existence of a sequence $(y_{n})_{n\in\mathbb{N}}$ with the
three desired properties as in Lemma 11, we define $(x_{n})_{n\in\mathbb{N}}$
as in (23). In view of these properties, (27) implies that for any $s>0,$ the
pair correlation $R_{2}^{\mathfrak{X}}(s,N)$ of $(x_{n})_{n\in\mathbb{N}}$
satisfies

(35)
$R_{2}^{\mathfrak{X}}(s,MN)=\Big{(}1-\frac{1}{M^{2}}\Big{)}\,2s+\frac{1}{M}R_{2}^{\mathcal{Z}}(s/M,N)+o(1),\quad
N\to\infty.$

From now on, we focus our attention on values $s<M/2.$ By the assumption that
$M\geqslant 2S,$ these values include all scales $s<S.$ Since for $s<M/2$ we
have $R_{2}^{\mathcal{Z}}(\frac{s}{M},MN\big{)}\to 0$, we deduce that for
$s<M/2,$

$R_{2}^{\mathfrak{X}}(s,MN)=\Big{(}1-\frac{1}{M^{2}}\Big{)}\,2s+o(1),\qquad
N\to\infty.$

Then a standard approximation argument, similar to the one we used to derive
(32), gives

(36)
$\lim_{N\to\infty}R_{2}^{\mathfrak{X}}(s,N)=\Big{(}1-\frac{1}{M^{2}}\Big{)}\,2s\qquad\text{
for all }s<\frac{M}{2}\cdot$

Finally, we set $c=1-\dfrac{1}{M^{2}}\in(0,1)$ and consider the sequence
$(cx_{n})_{n\in\mathbb{N}}.$ In order to prove that this sequence satisfies
the statement of Theorem 2, it remains to verify that the error term
$E^{\mathfrak{X}}(s,N)$ corresponding to $(x_{n})_{n\in\mathbb{N}}$ tends to
$0$ for any $s>0.$ As in (33), we find

(37) $\displaystyle E^{\mathfrak{X}}(s,MN)$
$\displaystyle=\frac{1}{MN}\\#\Big{\\{}n\neq m\leqslant
MN\mathrel{\mathop{\ordinarycolon}}\|x_{n}-x_{m}\|\leqslant\frac{s}{MN}\,\&\,\lvert
x_{n}-x_{m}\rvert\geqslant 1-\frac{s}{MN}\Big{\\}}$ (38)
$\displaystyle\leqslant\frac{M-1}{M}E^{\Upsilon}(s(M-1)/M,(M-1)N)+\frac{1}{M}E^{\mathcal{Z}}(s/M,N)$
(39)
$\displaystyle\,\,+\frac{1}{MN}\\#\Big{\\{}n\leqslant(M-1)N\mathrel{\mathop{\ordinarycolon}}y_{n}\leqslant\frac{s/M}{N}\Big{\\}}\cdot\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}z_{n}\geqslant 1-\frac{s/M}{N}\Big{\\}}$
(40)
$\displaystyle\,\,+\frac{1}{MN}\\#\Big{\\{}n\leqslant(M-1)N\mathrel{\mathop{\ordinarycolon}}y_{n}\geqslant
1-\frac{s/M}{N}\Big{\\}}\cdot\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}z_{n}\leqslant\frac{s/M}{N}\Big{\\}}.$

Using that $s/M<1/2$ and the property $\\{z_{1},\ldots
z_{N}\\}\subseteq[1/(2N),1-1/(2N)]$ that follows straightforwardly from the
definition of the binary van der Corput sequence, we see that

$\\#\Big{\\{}n\leqslant N\mathrel{\mathop{\ordinarycolon}}z_{n}\geqslant
1-\frac{s/M}{N}\Big{\\}}=\\#\Big{\\{}n\leqslant
N\mathrel{\mathop{\ordinarycolon}}z_{n}\leqslant\frac{s/M}{N}\Big{\\}}=0,$

whence $E^{\mathcal{Z}}(s/M,N)=0$ and

$E^{\mathfrak{X}}(s,MN)\leqslant\frac{M-1}{M}E^{\Upsilon}(s(M-1)/M,(M-1)N).$

By the construction of $(y_{n})_{n\in\mathbb{N}}$,
$E^{\Upsilon}(s(M-1)/M,(M-1)N)\to 0$ as $N\to\infty$, so
$\lim\limits_{N\to\infty}E^{\mathfrak{X}}(s,MN)=0$ and subsequently
$\lim\limits_{N\to\infty}E^{\mathfrak{X}}(s,N)=0$ follows. According to the
discussion in the beginning of the section, this allows us to deduce that for
the rescaled sequence $(cx_{n})_{n\in\mathbb{N}}$ we have

$R_{2}^{c\mathfrak{X}}(s,N)=R_{2}^{\mathfrak{X}}(s/c,N)+o(1),\quad
N\to\infty.$

Combined with (36), this implies that

$\lim_{N\to\infty}R_{2}^{c\mathfrak{X}}(s,N)=2s\quad\text{ for all }s<S$

which concludes the proof of Theorem 3 by the discussion above.

##  6\.  Proof of Theorem 4

We now proceed to the proof of Theorem 4: we prove that when $\alpha<\beta$,
the property of $\beta$ – PPC is stronger than that of $\alpha$ – PPC. We
shall distinguish two different cases according to whether $\beta=1$ or
$\beta<1.$ In both cases, by Lemma 9 and Lemma 10, it suffices to show that
for all $s>0$

$\limsup_{N\to\infty}I_{\alpha}(s,N)\leqslant s^{2}.$

###  6.1. The case $0<\beta<1$

Let $0\leqslant\alpha<\beta<1.$ We bound the function $F_{\alpha}(t,s,N)$ from
above in terms of $F_{\beta}(t,s,N)$. Writing

$M=\Big{\lceil}\frac{N^{\beta-\alpha}}{2}\Big{\rceil},$

we use the same reasoning as in the proof of Theorem 2: we note that

$B\Big{(}t,\frac{s}{2N^{\alpha}}\Big{)}\subseteq\bigcup_{|\ell|\leqslant
M}B\Big{(}t+\frac{\ell s}{N^{\beta}},\frac{s}{2N^{\beta}}\Big{)},$

whence

(41)
$F_{\alpha}(t,s,N)\leqslant\frac{1}{N^{\beta-\alpha}}\sum_{|\ell|\leqslant
M}F_{\beta}\Big{(}t+\frac{\ell s}{N^{\beta}},s,N\Big{)}.$

Another application of the Cauchy-Schwarz inequality gives

$\displaystyle\int_{0}^{1}F_{\alpha}(t,s,N)^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\frac{1}{N^{2(\beta-\alpha)}}\Bigg{(}\sum_{|\ell|\leqslant
M}F_{\beta}\Big{(}t+\frac{\ell
s}{N^{\beta}},s,N\Big{)}\Bigg{)}^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\frac{N^{\beta-\alpha}+1}{N^{2(\beta-\alpha)}}\sum_{|\ell|\leqslant
M}F_{\beta}\Big{(}t+\frac{\ell s}{N^{\beta}},s,N\Big{)}^{2}\,\mathrm{d}t$
$\displaystyle=\frac{N^{\beta-\alpha}+1}{N^{2(\beta-\alpha)}}\sum_{|\ell|\leqslant
M}\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\Big{(}1+\frac{3}{N^{\beta-\alpha}}\Big{)}\int_{0}^{1}F_{\beta}(t,s,N)^{2}\,\mathrm{d}t.$

By the assumption of $\beta$ – PPC and Lemma 9, we have

$\lim_{N\to\infty}\int_{0}^{1}F_{\beta}(t,s,N)^{2}\mathrm{d}t=s^{2},$

so the result follows.

###  6.2. The case $\beta=1$

Assume the sequence $(x_{n})_{n\in\mathbb{N}}$ has PPC and let $s>0$. For all
integers $K,N\geqslant 1$ we set

$M=M(N,K)=\Bigg{\lceil}\frac{N^{1-\alpha}}{2K}\Bigg{\rceil}.$

As in the previous case, we have

$B\Big{(}t,\frac{s}{2N^{\alpha}}\Big{)}\subseteq\bigcup_{|\ell|\leqslant
M}B\Big{(}t+\frac{\ell sK}{N},\frac{sK}{2N}\Big{)}$

and therefore

$F_{\alpha}(t,s,N)\leqslant\frac{1}{N^{1-\alpha}}\sum_{|\ell|\leqslant
M}F_{1}\Big{(}t+\frac{\ell sK}{N},sK,N\Big{)}.$

By the Cauchy-Schwarz inequality and argumentations as previously, it follows
that

$\displaystyle\int_{0}^{1}F_{\alpha}(t,s,N)^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\frac{1}{N^{2(1-\alpha)}}\Big{(}\sum_{|\ell|\leqslant
M}F_{1}\Big{(}t+\frac{\ell sK}{N},sK,N\Big{)}\Big{)}^{2}\,\mathrm{d}t$
$\displaystyle\leqslant\int_{0}^{1}\frac{2M+1}{N^{2(1-\alpha)}}\sum_{|\ell|\leqslant
M}F_{1}\Big{(}t+\frac{\ell sK}{N},sK,N\Big{)}^{2}\,\mathrm{d}t$
$\displaystyle=\frac{(2M+1)^{2}}{N^{2(1-\alpha)}}\int_{0}^{1}F_{1}(t,sK,N)^{2}\,\mathrm{d}t.$

Clearly,

$\lim_{N\to\infty}\frac{(2M+1)^{2}}{N^{2(1-\alpha)}}=\frac{1}{K^{2}}$

and by the assumption of PPC we have

$\lim_{N\to\infty}\int_{0}^{1}F_{1}(t,sK,N)^{2}\,\mathrm{d}t=(sK)^{2}+sK.$

Therefore, we can deduce that

$\limsup_{N\to\infty}\int_{0}^{1}F_{\alpha}(t,s,N)^{2}\,\mathrm{d}t\leqslant\frac{s^{2}K^{2}+sK}{K^{2}}$

for any $K\in\mathbb{N}$. Letting $K\to\infty$, the statement follows.

##  7\. Proof of Theorem 5

In the current section we fix a value of $0<\beta<1$ and construct a sequence
$(x_{n})_{n\in\mathbb{N}}$ that has $\alpha$ – PPC for all $\alpha<\beta$ but
does not have $\alpha$ – PPC for any $\alpha\geqslant\beta.$ The construction
is based on the following idea: we start with an arbitrary sequence
$(y_{n})_{n\in\mathbb{N}}$ that has PPC. The required sequence
$(x_{n})_{n\in\mathbb{N}}$ is then defined in a way that for each $1\leqslant
n\leqslant N$, the number $y_{n}$ occurs asymptotically
$N^{\tfrac{1}{\beta}-1}$ times in the first $N^{\tfrac{1}{\beta}}$ elements of
$(x_{n})_{n\in\mathbb{N}}$. In that way, the contribution to
$R_{2}(\alpha;s,N)$ of terms $x_{i},x_{j}$ with $1\leqslant i\neq j\leqslant
N$ and $x_{i}=x_{j}$ will be of order $N^{2-\beta}$. Looking at the factor
$\frac{1}{N^{2-\alpha}}$ in the definition of $R_{2}(\alpha;s,N)$, we see that
the contribution of these pairs is negligible when $\alpha<\beta$, whereas for
$\alpha>\beta$, it is impossible to have the Poissonian correlation property
since the correlation function will diverge.

To achieve the aforementioned construction, we employ an “expansion” of
positive integers with respect to a sequence that grows like $\frac{1}{\beta}$
– th powers of integers.

Fix $0<\beta<1$ and let $(y_{n})_{n\in\mathbb{N}}$ be a sequence with PPC such
that $y_{m}\neq y_{n}$ whenever $m\neq n$. (The existence of such a sequence
is justified, for example, by [20, Theorem 1].) Also, consider the sequence of
indices

$A_{N}=N\lfloor N^{\frac{1}{\beta}-1}\rfloor,\qquad N\geqslant 1.$

Every $N\geqslant 1$ can be written uniquely as

(42) $N=A_{M}+\varepsilon_{N}\lfloor
M^{\frac{1}{\beta}-1}\rfloor+q_{N}(M+1)+r_{N},$

where

  1. (i)

$M=M_{N}\geqslant 1$ is the unique integer such that $A_{M}<N\leqslant
A_{M+1},$

  2. (ii)

$\varepsilon_{N}\in\\{0,1\\},\quad q_{N}\geqslant 0,\quad 1\leqslant
r_{N}\leqslant M+1,\quad$ and

  3. (iii)

$\varepsilon_{N}=0$ if and only if  $A_{M}<N\leqslant A_{M}+\lfloor
M^{\frac{1}{\beta}-1}\rfloor.$

We now define a new sequence $(x_{n})_{n\in\mathbb{N}}$ by letting
$x_{1}=y_{1}$ and

$x_{N}=\begin{cases}y_{M+1},&\text{ if }\varepsilon_{N}=0,\\\
y_{r_{N}},&\text{ otherwise.}\end{cases}$

Here $M\geqslant 1,\varepsilon_{N}\in\\{0,1\\}$ and $r_{N}\geqslant 1$ are as
in (42).

We aim to show that $(x_{n})_{n\in\mathbb{N}}$ has the required property, that
is, it does not have $\beta$ – PPC but it has $\alpha$ – PPC for any
$\alpha<\beta.$ We first need the following lemma.

######  Lemma 12.

For any $M\geqslant 1,$ we have

(43) $\\#\\{1\leqslant n\leqslant
A_{M}\mathrel{\mathop{\ordinarycolon}}x_{n}=y_{k}\\}=\lfloor
M^{\frac{1}{\beta}-1}\rfloor,\quad k=1,2,\ldots,M.$

###### Proof.

We prove this statement by induction on $M$. Trivially, the statement holds
for $M=1$, so we assume (43) is true for some fixed $M\geqslant 1$. By
construction of the sequence, $x_{N}=y_{m+1}$ for $A_{M}<N\leqslant
A_{M}+\lfloor M^{\frac{1}{\beta}-1}\rfloor=(M+1)\lfloor
M^{\frac{1}{\beta}-1}\rfloor.$ This implies that for all $\ell,k\in[M+1]$,

(44) $\\#\\{n\leqslant(M+1)\lfloor
M^{\frac{1}{\beta}-1}\rfloor\mathrel{\mathop{\ordinarycolon}}x_{n}=y_{k}\\}=\\#\\{n\leqslant(M+1)\lfloor
M^{\frac{1}{\beta}-1}\rfloor\mathrel{\mathop{\ordinarycolon}}x_{n}=y_{\ell}\\}.$

Since both $A_{M+1}$ and $A_{M}+\lfloor
M^{\frac{1}{\beta}-1}\rfloor=(M+1)\lfloor M^{\frac{1}{\beta}-1}\rfloor$ are
divisible by $M+1$, so is their difference. Thus, as $N$ runs through the
values $(M+1)\lfloor M^{\frac{1}{\beta}-1}\rfloor<N\leqslant A_{M+1}$, it
attains each residue class mod ${M+1}$ the same number of times. This means
that $r_{N}$ attains each value between $1$ and $M+1$ the same number of
times, and finally $x_{N}$ assumes the values $y_{1},\ldots,y_{M+1}$ equally
often. Hence, we can extend (44) to $1\leqslant n\leqslant A_{M+1}$. Note that
no element $y_{k}$ with $k\geqslant M+2$ appears in
$\\{x_{1},\ldots,x_{A_{M+1}}\\}$, so the statement follows. ∎

We can now show that $(x_{n})_{n\in\mathbb{N}}$ does not have $\beta$ – PPC.
Let $s>0$. Whenever $m,n\leqslant A_{M},$ the inequality
$\|x_{m}-x_{n}\|\leqslant s/A_{M}^{\,\beta}$ is satisfied:

  1. $\bullet$

either when $x_{m}=y_{k}$ and $x_{n}=y_{\ell}$ for some indices $1\leqslant
k,\ell\leqslant M$ with $k\neq\ell$ such that $\|y_{k}-y_{\ell}\|\leqslant
s/A_{M}^{\,\beta},$ or

  2. $\bullet$

when $x_{n}=x_{m}=y_{k}$ for some $1\leqslant k\leqslant M.$

Therefore by Lemma 12, as $M\to\infty$ we have

$\displaystyle R_{2}(\beta;s,A_{M})$
$\displaystyle=\frac{1}{A_{M}^{2-\beta}}\\#\Big{\\{}1\leqslant m\neq
n\leqslant
A_{M}\mathrel{\mathop{\ordinarycolon}}\|x_{m}-x_{n}\|\leqslant\frac{s}{A_{M}^{\,\beta}}\Big{\\}}$
$\displaystyle=\frac{M\lfloor
M^{\frac{1}{\beta}-1}\rfloor^{2}}{A_{M}^{2-\beta}}\cdot\frac{1}{M}\\#\Big{\\{}1\leqslant
k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\beta}}\Big{\\}}$
$\displaystyle\qquad+\frac{1}{A_{M}^{2-\beta}}\lfloor
M^{\frac{1}{\beta}-1}\rfloor\big{(}\lfloor
M^{\frac{1}{\beta}-1}\rfloor-1\big{)}M$
$\displaystyle=\big{(}1+o(1)\big{)}\cdot\frac{1}{M}\\#\Big{\\{}1\leqslant
k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\beta}}\Big{\\}}+1+o(1).$

Let $\varepsilon>0.$ Since
$\lim\limits_{M\to\infty}\dfrac{A_{M}^{\beta}}{M}=1,$ for all $M\geqslant 1$
large enough we have

(45) $\displaystyle R_{2}^{\Upsilon}((1-\varepsilon)s,M)$
$\displaystyle=\frac{1}{M}\\#\Big{\\{}1\leqslant k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant(1-\varepsilon)\frac{s}{M}\Big{\\}}$
$\displaystyle\leqslant\frac{1}{M}\\#\Big{\\{}1\leqslant k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\beta}}\Big{\\}}$
$\displaystyle\leqslant\frac{1}{M}\\#\Big{\\{}1\leqslant k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant(1+\varepsilon)\frac{s}{M}\Big{\\}}$
$\displaystyle=R_{2}^{\Upsilon}((1+\varepsilon)s,M),$

where we write $R_{2}^{\Upsilon}(s,N)$ for the pair correlation function of
the sequence $(y_{n})_{n\in\mathbb{N}}$ By the hypothesis that
$(y_{n})_{n\in\mathbb{N}}$ has PPC, we deduce that

$\lim_{M\to\infty}\frac{1}{M}\\#\Big{\\{}1\leqslant k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\beta}}\Big{\\}}=2s.$

Therefore,

$\lim_{M\to\infty}R_{2}(\beta;s,A_{M})=2s+1,$

which implies that $(x_{n})_{n\in\mathbb{N}}$ does not have $\beta$ – PPC. By
Theorem 4, it does not have $\alpha$ – PPC for any $\alpha>\beta.$ (We leave
it to the interested reader to verify that actually for $\alpha>\beta,$
$\lim\limits_{N\to\infty}R_{2}(\alpha;s,N)=\infty$ for any $s>0.)$

We now fix $0\leqslant\alpha<\beta.$ Arguing as previously we obtain

$\displaystyle R_{2}(\alpha;s,A_{M})$
$\displaystyle=\frac{1}{A_{M}^{2-\alpha}}\\#\Big{\\{}1\leqslant m\neq
n\leqslant
A_{M}\mathrel{\mathop{\ordinarycolon}}\|x_{m}-x_{n}\|\leqslant\frac{s}{A_{M}^{\alpha}}\Big{\\}}$
$\displaystyle=\frac{\lfloor
M^{\frac{1}{\beta}-1}\rfloor^{2}}{A_{M}^{2-\alpha}}\cdot\\#\Big{\\{}1\leqslant
k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\alpha}}\Big{\\}}$
$\displaystyle\qquad+\frac{1}{A_{M}^{2-\alpha}}\lfloor
M^{\frac{1}{\beta}-1}\rfloor\big{(}\lfloor
M^{\frac{1}{\beta}-1}\rfloor-1\big{)}M.$

We first notice that since $\alpha<\beta,$

$\frac{M}{A_{M}^{2-\alpha}}\lfloor M^{\frac{1}{\beta}-1}\rfloor\big{(}\lfloor
M^{\frac{1}{\beta}-1}\rfloor-1\big{)}=\frac{1}{M^{1-\frac{\alpha}{\beta}}}+o(1)=o(1),\qquad
M\to\infty.$

It remains to understand the asymptotic behaviour of the factor

$\\#\Big{\\{}1\leqslant k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\alpha}}\Big{\\}}.$

By the definition of $(A_{M})_{M\in\mathbb{N}},$ we have
$\lim\limits_{M\to\infty}\dfrac{A_{M}^{\alpha}}{M^{\frac{\alpha}{\beta}}}=1.$
The sequence $(y_{n})_{n\in\mathbb{N}}$ has PPC, so by Theorem 4 it also has
$(\alpha/\beta)$ – PPC. Thus arguing as in (45) we deduce that

$\lim_{M\to\infty}\frac{1}{M^{2-\frac{\alpha}{\beta}}}\\#\Big{\\{}1\leqslant
k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\alpha}}\Big{\\}}=2s.$

Therefore,

$\displaystyle R_{2}(\alpha;s,A_{M})$ $\displaystyle=\frac{\lfloor
M^{\frac{1}{\beta}-1}\rfloor^{2}}{A_{M}^{2-\alpha}}\\#\Big{\\{}1\leqslant
k\neq\ell\leqslant
M\mathrel{\mathop{\ordinarycolon}}\|y_{k}-y_{\ell}\|\leqslant\frac{s}{A_{M}^{\alpha}}\Big{\\}}+o(1)$
$\displaystyle=\frac{\lfloor
M^{\frac{1}{\beta}-1}\rfloor^{2}}{A_{M}^{2-\alpha}}M^{2-\frac{\alpha}{\beta}}\big{(}2s+o(1)\big{)}+o(1)$
$\displaystyle=2s+o(1),\qquad M\to\infty.$

To complete the proof that $(x_{n})_{n\in\mathbb{N}}$ has $\alpha$ – PPC, we
observe that $\lim\limits_{M\to\infty}\dfrac{A_{M+1}}{A_{M}}=1$ and by an
approximation argument analogous to (32), we deduce that

$\lim\limits_{N\to\infty}R_{2}(\alpha;s,N)=2s.$

##  8\. Proof of Theorem 6

Theorem 6 will follow from a standard argument that, to the best of our
knowledge, was employed for the first time in the context of pair correlations
in [20] and subsequently has been used quite often in the relevant literature,
see e.g. [2] and the references therein. We have decided to provide only the
basic elements of the proof.

We stress out the dependence of the pair correlation function with parameter
$\beta$ of the sequence $(a_{n}x)_{n\in\mathbb{N}}$ on $x\in[0,1)$ by writing

$R_{2}(\beta;s,N,x)\mathrel{\mathop{\ordinarycolon}}=\frac{1}{N^{2-\beta}}\left\\{1\leqslant
i\neq j\leqslant N\mathrel{\mathop{\ordinarycolon}}\mathinner{\\!\left\lVert
a_{i}x-a_{j}x\right\rVert}\leqslant\frac{s}{N^{\beta}}\right\\}.$

It then suffices to show that for Lebesgue – almost all $x\in[0,1),$ we have

$\lim_{N\to\infty}R_{2}(\beta;s,N,x)=2s\qquad\text{ for any }s>0.$

First we fix a value of $s>0$. It follows by the definition of
$R_{2}(\beta;s,N,x)$ that

$\int_{0}^{1}R_{2}(\beta;s,N,x)\,\mathrm{d}x=\frac{2(N-1)s}{N}\cdot$

A sufficient upper bound on the variance is given by the following lemma.

######  Lemma 13.

For $\varepsilon\mathrel{\mathop{\ordinarycolon}}=\frac{1}{2}(1-\beta)>0$, we
have

$\int_{0}^{1}\left(R_{2}(s,N,\beta,x)-\frac{2(N-1)s}{N}\right)^{2}\mathrm{d}x\ll
N^{-\varepsilon},\quad N\to\infty.$

###### Proof.

The proof goes along the lines of [2]. We use the Fourier series expansion

$\mathds{1}_{[-\tfrac{s}{N^{\beta}},\tfrac{s}{N^{\beta}}]}(x)\sim\sum_{n=-\infty}^{+\infty}c_{n}e(nx),$

where

$c_{0}=\frac{2s}{N^{\beta}}\qquad\text{ and }\qquad c_{n}=\frac{1}{\pi
n}\sin\Big{(}\frac{2\pi ns}{N^{\beta}}\Big{)},\quad n\neq 0.$

Using Parseval’s identity we get

(46)
$\int_{0}^{1}\\!\left(R_{2}(\beta;s,N,x)-\frac{2(N-1)s}{N}\right)^{2}\\!\\!\mathrm{d}x=\frac{1}{N^{4-2\beta}}\sum_{m,n\in\mathbb{Z}\setminus\\{0\\}}r_{N}(m)r_{N}(n)\sum_{\begin{subarray}{c}i,j\in\mathbb{Z}\setminus\\{0\\}\\\
im=jn\end{subarray}}c_{i}c_{j}$

where

$r_{N}(m)=\\#\\{1\leqslant i,j\leqslant N,a_{i}-a_{j}=m\\},\quad m\geqslant 1$

denotes the number of representations of $m\geqslant 1$ as a difference of two
elements of $(a_{i})_{i\leqslant N}$. It can now be shown as in [2] that for
some absolute constant $C>0,$

$\sum_{\begin{subarray}{c}i,j\in\mathbb{Z}\setminus\\{0\\}\\\
im=jn\end{subarray}}\lvert c_{i}c_{j}\rvert\leqslant C(\log
N)\frac{s}{N^{\beta}}\frac{\gcd(m,n)}{\sqrt{\lvert mn\rvert}},\quad N\geqslant
1.$

Since

$\sum_{m\geqslant 1}r_{N}(m)^{2}=\\#\\{i,j,k,\ell\leqslant
N\mathrel{\mathop{\ordinarycolon}}a_{i}-a_{j}=a_{k}-a_{\ell}\\}\leqslant
N^{3},$

we can use the gcd – sum estimates from [2, Lemma 1] (in fact, the result from
[4] suffices) to show that

$\frac{1}{N^{3}}\sum_{m,n\geqslant
1}r_{N}(m)r_{N}(n)\frac{\text{gcd}(m,n)}{\sqrt{mn}}\ll\exp\left(\frac{c\sqrt{(\log
N)\log\log\log N}}{\sqrt{\log\log N}}\right)\leqslant N^{\varepsilon}$

which proves the desired statement. ∎

The proof of Theorem 6 can be completed by arguments identical to the ones
used at the end of Theorem 3. We let $\gamma>1/\varepsilon$ and consider the
sequence
$B_{M}\mathrel{\mathop{\ordinarycolon}}=\left\lceil{M^{\gamma}}\right\rceil,\,M\geqslant
1.$ We use Chebyshev’s inequality and the first Borel – Cantelli Lemma to
deduce that

$\lim_{M\to\infty}R_{2}(\beta;s,B_{M},x)=2s$

for almost all $x$ (where the zero – measure set depends on $s$). For an
arbitrary integer $N\geqslant 1$, let $M=M_{N}$ be the unique index such that
$B_{M}\leqslant N<B_{M+1}.$ Since

$\frac{B_{M}}{B_{M+1}}R_{2}\Big{(}\beta;\frac{B_{M}s}{B_{M+1}},B_{M},x\Big{)}\leqslant
R_{2}(\beta;s,N,x)\leqslant\frac{B_{M+1}}{B_{M}}R_{2}\Big{(}\beta;\frac{B_{M+1}s}{B_{M}},B_{M+1},x\Big{)}$

holds for any value of $x$ and also
$\lim\limits_{M\to\infty}\dfrac{B_{M}}{B_{M+1}}=1,$ we get

$\lim_{N\to\infty}R_{2}(\beta;s,N,x)=2s$

for all $x$ in a set of full Lebesgue measure. It remains to show that the
almost sure convergence is actually true for all $s>0$; this follows if we
consider a dense countable set in $\mathbb{R}_{+}$ and employ the monotonicity
of $R_{2}(\beta;s,N,x)$ as a function of $s>0$.

##  9\. Proof of Theorems 7 and 8

In this last section we prove the results relevant to weak correlations of
higher orders. We start with the proof Theorem 8 and then use some of the
arguments involved to deduce Theorem 7.

Throughout this section, we shall use the following notation. Given
$k\geqslant 2$, a rectangle $\mathcal{R}\subseteq\mathbb{R}^{k-1}$ and
$0\leqslant\beta\leqslant 1$, we define a new correlation counting function by

(47)
$R_{k}^{*}(\beta;\mathcal{R},N)\\!=\\!\frac{1}{N^{k-(k-1)\beta}}\\#\\!\\!\left\\{i_{1},\ldots,i_{k}\leqslant
N\\!\\!\mathrel{\mathop{\ordinarycolon}}N^{\beta}((\\!(x_{i_{1}}-x_{i_{2}})\\!),\ldots,(\\!(x_{i_{1}}-x_{i_{k-1}})\\!))\\!\in\\!\mathcal{R}\right\\}\\!.$

That is, in the definition of $R_{k}^{*}$ we allow indices to be equal, in
contrast to the definition of $R_{k}$ where all indices have to be pairwise
distinct. Also whenever $a<b$ we write

$y_{i}(a,b)=y_{i}(a,b,N,\beta)=\\#\big{\\{}j\leqslant
N\mathrel{\mathop{\ordinarycolon}}a\leqslant
N^{\beta}(\\!(x_{i}-x_{j})\\!)\leqslant b\big{\\}},\quad i\leqslant N.$

Under this notation, we observe that for the rectangle
$\mathcal{R}=[a_{1},b_{1}]\times\ldots\times[a_{k-1},b_{k-1}]$ we have

(48)
$R_{k}^{*}(\beta;\mathcal{R},N)=\frac{1}{N^{k-(k-1)\beta}}\sum_{i\leqslant
N}y_{i}(a_{1},b_{1})\cdot\ldots\cdot y_{i}(a_{k-1},b_{k-1}).$

As we shall soon see, this form makes $R_{k}^{*}$ much easier to handle than
$R_{k}$.

The following lemma states that for $\beta<1$, under the assumption of
Poissonian $(k,\beta)$ – correlations, the asymptotic size of $R_{k}$ is the
same as that of $R_{k}^{*}$ as $N\to\infty$. In the proof of the lemma we make
use of certain facts on $R_{k}$ and $R_{k}^{*}$ that appear in [9]. There are
only minor modifications, and these are due to the fact that here we deal with
weak correlations; we have chosen to refer the interested reader to the proofs
in [9] and explain here briefly where the minor differences come from.

######  Lemma 14.

Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence and $k\geqslant 2$. The following
are equivalent.
(i) $(x_{n})_{n\in\mathbb{N}}$ has Poissonian $(k,\beta)$ – correlations.
(ii) For all closed rectangles $\mathcal{R}\subseteq\mathbb{R}^{k-1}$ of the
form $[a_{1},b_{1}]\times\ldots\times[a_{k-1},b_{k-1}]$ we have

(49) $\lim_{N\to\infty}R_{k}^{*}(\beta;\mathcal{R},N)=\lambda(\mathcal{R}).$

(iii) For all cubes
$\mathcal{C}\mathrel{\mathop{\ordinarycolon}}=[a,b]^{k-1},a,b\in\mathbb{R}$ we
have

(50)
$\limsup_{N\to\infty}R_{k}^{*}(\beta;\mathcal{C},N)\leqslant\lambda(\mathcal{C}).$

###### Proof.

We start by proving the equivalence of (i) and (ii). We observe that for any
rectangle $\mathcal{R}\subseteq\mathbb{R}^{k-1},$ we have

(51) $R_{k}(\beta;\mathcal{R},N)\leqslant
R_{k}^{*}(\beta;\mathcal{R},N)\leqslant
R_{k}(\beta;\mathcal{R},N)+\sum_{m=1}^{k-1}\Big{(}\frac{1}{N^{1-\beta}}\Big{)}^{k-m}b_{m}R_{m}(\beta;\mathcal{R}_{m},N)$

where $R_{1}(s;N)\mathrel{\mathop{\ordinarycolon}}=1$, for each $1\leqslant
m\leqslant k-1,\mathcal{R}_{m}\subseteq\mathbb{R}^{m-1}$ is an $(m-1)$ –
dimensional rectangle depending on $\mathcal{R},$ and
$b_{1},\ldots,b_{k-1}\in\mathbb{N}$ are constants depending only on $k$. The
proof of (51) is essentially the same as that of [9, Proposition 7], which
deals with standard correlation functions (i.e. when $\beta=1$). The only
differences are the appearance of the coefficients $(N^{-(1-\beta)})^{k-m}$,
that come from the different scaling factor for weak correlations, and the
fact that (51) is about arbitrary rectangles $\mathcal{R}$ rather than
rectangles symmetric with respect to the axes.

Next, we observe that whenever ${\rm(i)}$ or ${\rm(ii)}$ holds, arguing
analogously to [9, Theorem 3], we can show that for $m\leqslant k-1$ we have

(52) $\limsup_{N\to\infty}R_{m}(\beta;\mathcal{S},N)<\infty\quad\text{ for any
rectangle }\mathcal{S}\subset\mathbb{R}^{m-1}.$

Combining (51) and (52) completes the proof of this equivalence.

Since (ii) $\Rightarrow$ (iii) is trival, it remains to show (iii)
$\Rightarrow$ (ii). So let us assume that (50) holds and let
$\mathcal{R}=[a_{1},b_{1}]\times\ldots\times[a_{k-1},b_{k-1}]$ be an arbitrary
rectangle in $\mathbb{R}^{k-1}$. Applying the Hölder inequality with exponents
$p_{i}=1/(k-1),(1\leqslant i\leqslant k-1)$ to (48) we obtain

(53) $\displaystyle R_{k}^{*}(\beta;\mathcal{R},N)$
$\displaystyle=\frac{1}{N^{k-(k-1)\beta}}\sum_{i\leqslant
N}y_{i}(a_{1},b_{1})\cdot\ldots\cdot y_{i}(a_{k-1},b_{k-1})$ (54)
$\displaystyle\leqslant\frac{1}{N^{k-(k-1)\beta}}\Bigg{(}\sum_{i\leqslant
N}y_{i}(a_{1},b_{1})^{k-1}\Bigg{)}^{\frac{1}{k-1}}\cdot\ldots\cdot\Bigg{(}\sum_{i\leqslant
N}y_{i}(a_{k-1},b_{k-1})^{k-1}\Bigg{)}^{\frac{1}{k-1}}$ (55)
$\displaystyle=R_{k}^{*}(\beta;\mathcal{C}_{a_{1},b_{1}},N)^{\frac{1}{k-1}}\cdot\ldots\cdot
R_{k}^{*}(\beta;\mathcal{C}_{a_{k-1},b_{k-1}},N)^{\frac{1}{k-1}},$

where we set $\mathcal{C}_{a_{i},b_{i}}=[a_{i},b_{i}]^{k-1}$ for each
$1\leqslant i\leqslant k-1.$ Using (50) we are able to deduce that

(56)
$\limsup_{N\to\infty}R_{k}^{*}(\beta;\mathcal{R},N)\leqslant\lambda(\mathcal{R})\quad\text{
for any rectangle }\mathcal{R}\subset\mathbb{R}^{k-1}.$

For the remaining part of the proof, it will be convenient to adopt the
following notation: given any
$\mathbf{t}=(t_{1},\ldots,t_{k-1})\in\mathbb{R}^{k-1}$ with
$t_{i}>0,i=1,\ldots,k-1$ we shall write

$\mathcal{R}_{\mathbf{t}}=[-t_{1},t_{1}]\times\ldots\times[-t_{k-1},t_{k-1}]\subseteq\mathbb{R}^{k-1}$

for the rectangle in $\mathbb{R}^{k-1}$ that is symmetric with respect to each
of the axes and has a vertex at the point $\mathbf{t}.$

Continuing with the proof of the implication (iii) $\Rightarrow$ (ii), we
assume for contradiction that (49) does not hold. Then by (56) there must
exist $\mathbf{a}=(a_{1},\ldots,a_{k-1}),$
$\mathbf{b}=(b_{1},\ldots,b_{k-1})\in\mathbb{R}^{k-1}$ such that for the
rectangle
$\mathcal{R}_{[\mathbf{a},\mathbf{b}]}\mathrel{\mathop{\ordinarycolon}}=[a_{1},b_{1}]\times\ldots\times[a_{k-1},b_{k-1}]$
we have

(57)
$\liminf_{N\to\infty}R_{k}^{*}(\beta;\mathcal{R}_{[\mathbf{a},\mathbf{b}]},N)<\lambda(\mathcal{R}_{[\mathbf{a},\mathbf{b}]}).$

Setting
$s=\max\\{|a_{i}|,|b_{i}|\mathrel{\mathop{\ordinarycolon}}i=1,\ldots,k-1\\}$
and $\mathbf{s}=(s,\ldots,s)$, we can find $n\in\mathbb{N}$ and rectangles
$\mathcal{R}_{j},1\leqslant j\leqslant n$ such that the cube
$\mathcal{C}_{s}\mathrel{\mathop{\ordinarycolon}}=[-s,s]^{k-1}$ satisfies

$\mathcal{C}_{s}=\mathcal{R}_{[\mathbf{a},\mathbf{b}]}\cup\bigcup_{j=1}^{n}\mathcal{R}_{j}$

and all rectangles in the right-hand side have disjoint interiors. Using (56)
on every $\mathcal{R}_{j}$ and (57), we obtain

(58) $\displaystyle\liminf_{N\to\infty}R_{k}^{*}(\beta;\mathcal{C}_{s},N)$
$\displaystyle\leqslant\liminf_{N\to\infty}R_{k}^{*}(\beta;\mathcal{R}_{[\mathbf{a},\mathbf{b}]},N)+\sum_{j=1}^{n}\limsup_{N\to\infty}R_{k}^{*}(\beta;\mathcal{R}_{j},N)$
(59) $\displaystyle<(2s)^{k-1}.$

Consequently, there exist an $\eta>0$ and a sequence
$(N_{r})_{r\in\mathbb{N}}\subseteq\mathbb{N}$ such that

(60) $R_{k}^{*}(\beta;\mathcal{C}_{s},N_{r})<(2s)^{k-1}-\eta\qquad\text{for
all }r\geqslant 1.$

Observe that there exists a cube
$\mathcal{C}_{0}=[s-\delta,s]\times\ldots\times[s-\delta,s]\subseteq\mathbb{R}^{k-1}$
with positive $(k-1)$ – dimensional Lebesgue measure such that for any
$r\geqslant 1,$

(61)
$R_{k}^{*}(\beta;\mathcal{R}_{\mathbf{t}},N_{r})<(2t_{1})(2t_{2})\cdots(2t_{k-1})-\frac{\eta}{2}$

for any $\mathbf{t}=(t_{1},\ldots,t_{k-1})\in\mathcal{C}_{0}$.

We shall obtain the desired contradiction by deriving a lower and an upper
bound for the integral

$\int_{[0,s]^{k-1}}R_{k}^{*}(\beta,\mathcal{R}_{\mathbf{t}}\,,N)\,\mathrm{d}\mathbf{t}.$

Arguing as in [9, Proposition 11], we can show that

(62)
$\int_{[0,s]^{k-1}}R_{k}^{*}(\beta,\mathcal{R}_{\mathbf{t}}\,,N)\,\mathrm{d}\mathbf{t}\geqslant
s^{2(k-1)}\quad\text{ for any }N\geqslant 1.$

Let $\varepsilon>0$ be arbitrary. For any integer $M\in\mathbb{N},$ we can
split the integration domain $[0,s]^{k-1}$ in $M^{k-1}$ cubes of the form

$\mathcal{C}_{\mathbf{j}}\mathrel{\mathop{\ordinarycolon}}=\left[\frac{(j_{1}-1)s}{M},\frac{j_{1}s}{M}\right]\times\ldots\times\left[\frac{(j_{k-1}-1)s}{M},\frac{j_{k-1}s}{M}\right],$

where $\mathbf{j}=(j_{1},\ldots,j_{k-1})\in[M]^{k-1}$, and if $M\geqslant 1$
is large enough, then for all indices $\mathbf{j}\in[M]^{k-1}$ we have

(63)
$\lvert\lambda(\mathcal{R}_{\mathbf{t}_{1}})-\lambda(\mathcal{R}_{\mathbf{t}_{2}})\rvert\leqslant\frac{\varepsilon}{2}\qquad\text{
for any }\mathbf{t}_{1},\mathbf{t}_{2}\in\mathcal{C}_{\mathbf{j}}.$

For each of the cubes $\mathcal{C}_{\mathbf{j}}$ defined above, we denote its
top corner by

$\mathbf{c}_{\mathbf{j}}=\left(\frac{j_{1}s}{M},\frac{j_{2}s}{M},\ldots,\frac{j_{k-1}s}{M}\right)\in\mathbb{R}^{k-1}.$

In view of (56), we know that for all $r\geqslant 1$ sufficiently large, we
have

(64)
$R_{k}^{*}(\beta;\mathcal{R}_{\mathbf{c}_{\mathbf{j}}},N_{r})\leqslant\lambda(\mathcal{R}_{\mathbf{c}_{\mathbf{j}}})+\frac{\varepsilon}{2}\qquad\text{for
all }\mathbf{j}\in[M]^{k-1}.$

(Note that the values of $r$ can be chosen independently of $\mathbf{j}$ since
the number of considered $\mathbf{j}$’s is finite.) Given
$\mathbf{t}\in[0,s]^{k-1}$ we write $\mathbf{c}(\mathbf{t})$ for the top
corner $\mathbf{c}_{\mathbf{j}}$ of the cube $\mathcal{C}_{\mathbf{j}}$ that
contains the point $\mathbf{t}.$ Then for all $r\geqslant 1$ large enough, we
have

(65) $\displaystyle R_{k}^{*}(\beta,\mathcal{R}_{\mathbf{t}},N_{r})$
$\displaystyle\leqslant
R_{k}^{*}(\beta,\mathcal{R}_{\mathbf{c}(\mathbf{t})},N_{r})$ (66)
$\displaystyle\leqslant\lambda(\mathcal{R}_{\mathbf{c}_{\mathbf{j}}})+\frac{\varepsilon}{2}\quad\qquad\text{
(by \eqref{corner_bound})}$ (67)
$\displaystyle\leqslant\lambda(\mathcal{R}_{\mathbf{t}})+\varepsilon\quad\qquad\,\,\,\text{
(by \eqref{small_diam})}$

for any $\mathbf{t}\in[0,s]^{k-1}$. Combining (61) and (66), we obtain

(68)
$\displaystyle\int_{[0,s]^{k-1}}R_{k}^{*}(\beta,\mathcal{R}_{\mathbf{t}}\,,N_{r})\,\mathrm{d}\mathbf{t}$
$\displaystyle=\int_{[0,s]^{k-1}\setminus\mathcal{C}_{0}}R_{k}^{*}(\beta;\mathcal{R}_{\mathbf{t}}\,,N_{r})\,\mathrm{d}\mathbf{t}+\int_{\mathcal{C}_{0}}R_{k}^{*}(\beta;\mathcal{R}_{\mathbf{t}}\,,N_{r})\,\mathrm{d}\mathbf{t}$
(69) $\displaystyle\leqslant s^{2(k-1)}+\varepsilon
s^{k-1}-\eta\lambda(\mathcal{C}_{0}),$

a contradiction to (62) when $\varepsilon$ is sufficiently small. ∎

Having established Lemma 14, we can now proceed to the proof of Theorem 8.

###### Proof of Theorem 8.

We first prove that having Poissonian $(k,\beta)$ – correlations is a property
stronger than $\beta$ – PPC. We then use this fact to prove that this is also
stronger than $(k-1,\beta)$ – correlations. We apply the Hölder inequality
with exponents $p=(k-1)/(k-2)$ and $q=k-1$ to obtain

$\displaystyle R_{2}^{*}(\beta;[a,b],N)^{k-1}$
$\displaystyle=\frac{1}{N^{2(k-1)-(k-1)\beta}}\Big{(}\sum_{i\leqslant
N}y_{i}(a,b)\Big{)}^{k-1}$ (70)
$\displaystyle\leqslant\frac{1}{N^{k-(k-1)\beta}}\sum_{i\leqslant
N}y_{i}(a,b)^{k-1}=R_{k}^{*}(\beta;[a,b]^{k-1},N).$

Assuming Poissonian $(k,\beta)$ – correlations, the right-hand side of (70)
tends to
$(b-a)^{k-1}$ as $N\to\infty$. Therefore, we have for all $a<b\in\mathbb{R}$
that

$\limsup_{N\to\infty}R_{2}^{*}(\beta;[a,b],N)\leqslant b-a,$

which by Lemma 14 implies $\beta$ – PPC.

Now we proceed to prove that $(x_{n})_{n\in\mathbb{N}}$ also has Poissonian
$(k-1,\beta)$ – correlations. We make use of the well – known inequality

(71) $N\sum_{i\leqslant N}x_{i}^{k-1}\,\geqslant\,\Big{(}\sum_{i\leqslant
N}x_{i}\Big{)}\Big{(}\sum_{i\leqslant N}x_{i}^{k-2}\Big{)}$

which holds for $N\geqslant 1,x_{i}\geqslant 0$ to show that

(72) $\displaystyle R_{k}^{*}(\beta;[a,b]^{k-1},N)$
$\displaystyle=\frac{1}{N^{k-(k-1)\beta}}\sum_{i\leqslant N}y_{i}(a,b)^{k-1}$
(73) $\displaystyle\geqslant\frac{1}{N^{2-\beta}}\sum_{i\leqslant
N}y_{i}(a,b)\cdot\frac{1}{N^{k-1-(k-2)\beta}}\sum_{i\leqslant
N}y_{i}(a,b)^{k-2}$ (74) $\displaystyle=R_{2}^{*}(\beta;[a,b],N)\cdot
R_{k-1}^{*}(\beta;[a,b]^{k-2},N).$

By Poissonian $(k,\beta)$ – correlations and $\beta$ – PPC, we can deduce that

$\limsup_{N\to\infty}R_{k-1}^{*}(\beta;[a,b]^{k-2},N)\leqslant(b-a)^{k-2},$

and in view of Lemma 14 this completes the proof. ∎

######  Remark.

Note that Theorem 8 holds for $\beta=0$, the proof follows the same lines as
above with the only difference being $s<\frac{1}{2}$. However for $\beta=1$,
this argumentation fails since in case of Poissonian $k$ – correlations,
$R_{k}^{*}(\mathcal{R},N)$ will not tend to $\lambda(\mathcal{R})$.

###### Proof of Theorem 7.

The one direction of the statement, namely that Poissonian $(k,0)$ –
correlations imply uniform distribution, follows from Theorem 8: for
$\beta=0$, we have proven that Poissonian $(k,0)$-correlations imply $0$ –
PPC, which in turn imply uniform distribution by Theorem 1.

We proceed to showing that uniform distribution implies Poissonian $(k,0)$ –
correlations. In view of Lemma 14, it suffices to prove that for any cube
$\mathcal{C}=[a,b]^{k-1}\subseteq[-\tfrac{1}{2},\tfrac{1}{2}]^{k-1}$ we have

(75)
$\limsup_{N\to\infty}R_{k}^{*}(0;\mathcal{C},N)\leqslant\lambda(\mathcal{C}).$

As expected, the proof goes along the lines of Theorem 1. We observe that

$\displaystyle R_{k}^{*}(0;\mathcal{C},N)-\lambda(\mathcal{C})$
$\displaystyle\,=\,\frac{1}{N^{k}}\sum_{i_{1},\ldots,i_{k}\leqslant
N}\mathds{1}_{[a,b]}(x_{i_{1}}-x_{i_{2}})\cdots\mathds{1}_{[a,b]}(x_{i_{1}}-x_{i_{k-1}})-\lambda(\mathcal{C})$
$\displaystyle=\frac{1}{N}\sum_{n\leqslant
N}\Bigg{(}\frac{1}{N}\sum_{m\leqslant
N}\mathds{1}_{[x_{n}-b,x_{n}-a]}(x_{m})\Bigg{)}^{k-1}-(b-a)^{k-1}.$

Since

$\displaystyle\frac{1}{N}\sum_{m\leqslant
N}\mathds{1}_{[x_{n}-b,x_{n}-a]}(x_{m})\leqslant b-a+D_{N},\quad
n=1,\ldots,N,$

with $D_{N}$ denoting the discrepancy of $(x_{n})_{n\in\mathbb{N}}$, we deduce
that

$R_{k}^{*}(0;\mathcal{C},N)-\lambda(\mathcal{C})\leqslant(b-a+D_{N})^{k-1}-(b-a)^{k-1}=\mathcal{O}_{a,b}(D_{N}).$

Assuming that $(x_{n})_{n\in\mathbb{N}}$ is uniformly distributed mod $1$, we
have $D_{N}\to 0$ as $N\to\infty$ and hence (75) follows. This concludes the
proof.∎

### Acknowledgements

We would like to thank Niclas Technau for introducing us to the notion of weak
Poissonian pair correlations. We also thank Christoph Aistleitner for many
valuable discussions.

## References

  * [1] C. Aistleitner, T. Lachmann and F. Pausinger, Pair correlations and equidistribution, J. Number Theory 182, 206–220, (2018).
  * [2] C. Aistleitner, G. Larcher, M. Lewko, Additive Energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. With an appendix by Jean Bourgain, Isr. J. Math. 222, 463 – 485 (2017).
  * [3] M.V. Berry, M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. A 356, 375 – 394 (1977)
  * [4] T. Dyer, G. Harman, Sums Involving Common Divisors, Journal of The London Mathematical Society – second Series, 34, 1 – 11, (1986).
  * [5] D. El-Baz, J.Marklof, I. Vinogradov, The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson, Proceedings of the American Mathematical Society, Vol. 143, No. 7, 2815 – 2828 (2015)
  * [6] N. D. Elkies, C. T. McMullen, Gaps in $\sqrt{n}\bmod 1$ and ergodic theory, Duke Math. J. 123, no. 1, 95 – 139 (2004)
  * [7] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, Seventh Edition, Elsevier Academic Press (2007)
  * [8] S. Grepstad, G. Larcher, On Pair Correlation and Discrepancy, Arch. Math. (Basel) 109, no. 2, 143 – 149 (2017)
  * [9] M. Hauke, A. Zafeiropoulos, Poissonian Correlations of higher orders, pre – print: arXiv:2107.06523
  * [10] D. R. Heath-Brown, Pair correlation for fractional parts of $\alpha n^{2}$, Math. Proc. Cambridge Philos. Soc. 148, 385–407 (2010).
  * [11] A. Hinrichs, L. Kaltenböck, G. Larcher, W. Stockinger, M. Ullrich, On a multi-dimensional Poissonian pair correlation concept and uniform distribution, Monatsh. Math. 190, 333 – 352 (2019).
  * [12] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences, Wiley, 1974.
  * [13] P. Kurlberg, Z. Rudnick, The distribution of spacings between quadratic residues,  Duke J. Math., 100: 211 – 242, (1999).
  * [14] C. Lutsko, Long-Range Correlations of Sequences Modulo 1, pre – print: arXiv:2007.09292
  * [15] C. Lutsko, A. Sourmelidis and N. Technau, Pair Correlation of the Fractional Parts of $\alpha n^{\theta}$, pre – print: arXiv: 2106.09800
  * [16] J. Marklof, _Pair correlation and equidistribution on manifolds,_ Monatsh. Math. 191, 279 – 294 (2020).
  * [17] H. Montgomery, _Ten Lectures at the Interface of Harmonic Analysis and Number Theory,_ American Mathematical Society, 1994.
  * [18] R. Nair, M. Pollicot, Pair Correlations of Sequences in higher dimensions, Israel J. Math. 157, 219 – 238, (2007).
  * [19] B. D. Ripley, Spatial Statistics, Wiley Series on Probability and Statistics, ISBN: 978-0-471-69116-7 (2004)
  * [20] Z. Rudnick, P. Sarnak, The pair correlation function of fractional parts of polynomials, Comm. Math. Phys., 194 (1) 61 – 70, (1998).
  * [21] Z. Rudnick, A. Zaharescu, _The distribution of spacings between fractional parts of lacunary sequences_ , Forum Math., 14(5) 691 – 712, (2002).
  * [22] Z. Rudnick, A. Zaharescu, _A metric result on the pair correlation of fractional parts of sequences,_ Acta Arith., 89(3) 283 – 293, (1999).
  * [23] S. Steinerberger, _Localized Quantitative Criteria for Uniform Distribution_ , Acta Arith., 180, 183 – 199 (2017)
  * [24] S. Steinerberger, Poissonian Pair Correlation in Higher Dimensions, J. Number Theory, 208: 47 – 58, (2020).
  * [25] N. Technau, A. Walker, On the triple correlations of fractional parts of $n^{2}\alpha$, pre-print: arXiv:2005.01490
  * [26] A. Walker, _The primes are not metric poissonian,_ Mathematika, 64(1), 230 – 236, (2018).
  * [27] C. Weiss, T. Skill, Sequences with almost Poissonian Pair Correlations, pre-print: arXiv:1905.02760.