# Quantization for a set of discrete distributions on the set of natural
numbers

1Juan Gomez  ,  2Haily Martinez  ,  3Mrinal K. Roychowdhury  ,  4Alexis
Salazar  and  5Daniel J. Vallez

###### Abstract.

The quantization scheme in probability theory deals with finding a best
approximation of a given probability distribution by a probability
distribution that is supported on finitely many points. In this paper, first
we state and prove a theorem, and then give a conjecture. We verify the
conjecture by a few examples. Assuming that the conjecture is true, for a set
of discrete distributions on the set of natural numbers we have calculated the
optimal sets of $n$-means and the $n$th quantization errors for all positive
integers $n$. In addition, the quantization dimension is also calculated.

###### Key words and phrases:

Discrete distribution, optimal sets, quantization error

###### 2010 Mathematics Subject Classification:

60Exx, 94A34.

##  1\. Introduction

The most common form of quantization is rounding-off. Its purpose is to reduce
the cardinality of the representation space, in particular, when the input
data is real-valued. It has broad applications in communications, information
theory, signal processing and data compression (see [GG, GL1, GL2, GN, P, Z1,
Z2]). Let $\mathbb{R}^{d}$ denote the $d$-dimensional Euclidean space equipped
with the Euclidean norm $\|\cdot\|$, and let $P$ be a Borel probability
measure on $\mathbb{R}^{d}$. Then, the $n$th quantization error for $P$, with
respect to the squared Euclidean distance, is defined by

$V_{n}:=V_{n}(P)=\inf\Big{\\{}V(P;\alpha):\alpha\subset\mathbb{R}^{d},1\leq\text{
card}(\alpha)\leq n\Big{\\}},$

where $V(P;\alpha)=\int\mathop{\min}\limits_{a\in\alpha}\|x-a\|^{2}dP(x)$
represents the distortion error due to the set $\alpha$ with respect to the
probability distribution $P$, and for a set $A$, $\text{card}(A)$ represents
the cardinality of the set $A$. A set $\alpha$ for which the infimum occurs
and contains no more than $n$ points is called an optimal set of $n$-means,
and is denoted by $\alpha_{n}:=\alpha_{n}(P)$. The elements of an optimal set
are also called as optimal quantizers. It is known that for a Borel
probability measure $P$ if its support contains infinitely many elements and
$\int\|x\|^{2}dP(x)$ is finite, then an optimal set of $n$-means always exists
and has exactly $n$-elements [AW, GKL, GL1, GL2]. The number

(1) $D(P):=\lim_{n\to\infty}\frac{2\log n}{-\log V_{n}(P)},$

if it exists, is called the quantization dimension of $P$. Quantization
dimension measures the speed at which the specified measure of the error goes
to zero as $n$ tends to infinity. For a finite set
$\alpha\subset\mathbb{R}^{d}$ and $a\in\alpha$, by $M(a|\alpha)$ we denote the
set of all elements in $\mathbb{R}^{d}$ which are the nearest to $a$ among all
the elements in $\alpha$, i.e.,
$M(a|\alpha)=\\{x\in\mathbb{R}^{d}:\|x-a\|=\min_{b\in\alpha}\|x-b\|\\}.$
$M(a|\alpha)$ is called the Voronoi region generated by $a\in\alpha$. On the
other hand, the set $\\{M(a|\alpha):a\in\alpha\\}$ is called the Voronoi
diagram or Voronoi tessellation of $\mathbb{R}^{d}$ with respect to the set
$\alpha$. The following proposition provides further information on the
Voronoi regions generated by an optimal set of $n$-means (see [GG, GL2]).

######  Proposition 1.1.

Let $\alpha$ be an optimal set of $n$-means, $a\in\alpha$, and $M(a|\alpha)$
be the Voronoi region generated by $a\in\alpha$, i.e.,

$M(a|\alpha)=\\{x\in\mathbb{R}^{d}:\|x-a\|=\min_{b\in\alpha}\|x-b\|\\}.$

Then, for every $a\in\alpha$,

  * (i)

$P(M(a|\alpha))>0$,

  * (ii)

$P(\partial M(a|\alpha))=0$,

  * (iii)

$a=E(X:X\in M(a|\alpha))$, and

  * (iv)

$P$-almost surely the set $\\{M(a|\alpha):a\in\alpha\\}$ forms a Voronoi
partition of $\mathbb{R}^{d}$.

From the above proposition, we can say that if $\alpha$ is an optimal set of
$n$-means for $P$, then each $a\in\alpha$ is the conditional expectation of
the random variable $X$ given that $X$ takes values on the Voronoi region of
$a$. Sometimes, we also refer to such an $a\in\alpha$ as the centroid of its
own Voronoi region. In this regard, interested readers can see [DFG, DR, R1].

A vector $(p_{1},p_{2},p_{3},\cdots)$ is called a probability distribution if
$0<p_{j}<1$ for all $j\in\mathbb{N}$ and $j\geq 0$ such that $\sum_{j\geq
1}p_{j}=1$. Notice that $(p_{1},p_{2},p_{3},\cdots)$ can be a finite, or an
infinite vector, where by a finite vector it is meant that the number of
coordinates in the vector is a finite number, otherwise it is called an
infinite vector.

For a Borel probability measure $P$ on the set $\mathbb{R}$ of real numbers
let $U$ be the largest open subset of $\mathbb{R}$ such that $P(U)=0$, then
$\mathbb{R}\setminus U$ is called the support of the probability measure $P$.
For example, when an unbiased die is thrown one time, then $P$ is a Borel
probability measure on the real line with

$\text{support}(P)=\\{1,2,3,4,5,6\\},$

and the associated probability distribution
$(\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6})$.
When an unbiased coin is tossed two times, then $P$ is a Borel probability
measure on $\mathbb{R}^{2}$ with

$\text{support}(P)=\\{(1,1),(1,2),(2,1),(2,2)\\},$

and the associated probability distribution
$(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4})$, where $1$ stands for
‘Head’ and $2$ stands for ‘Tail’.

######  Definition 1.2.

Let $(p_{1},p_{2},\cdots,p_{k-1})$ be a permutation of the set
$\\{\frac{1}{2},\frac{1}{2^{2}},\cdots,\frac{1}{2^{k-1}}\\}$, where
$k\in\mathbb{N}:=\\{1,2,\cdots\\}$ with $k\geq 2$. Define a probability
measure $P$ on the set $\mathbb{R}$ of real numbers with the support the set
$\mathbb{N}$ of natural numbers as follows:

$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j},$

where for $x\in\mathbb{R}$ the function $\delta_{x}$ represents the dirac
measure, i.e., for any subset $A\subseteq\mathbb{R}$, we have
$\delta_{x}(A)=1$ if $x\in A$, and zero otherwise.

Let us now state the following theorem and the conjecture.

######  Theorem 1.3.

Let
$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j}$
be the probability measure as defined by Definition 1.2. Let
$\\{a_{1},a_{2},\cdots,a_{n-3},a_{n-2},a_{n-1},a_{n}\\}$ be an optimal set of
$n$-means with $n\geq k+2$. Suppose that $a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$.
Then, either $a_{n-2}=n-2,a_{n-1}=Av[n-1,n],a_{n}=Av[n+1,\infty)$, or
$a_{n-2}=Av[n-2,n-1],a_{n-1}=Av[n,n+1],a_{n}=Av[n+2,\infty)$ with quantization
error $V_{n}=\frac{2^{3-n}}{3}$, where for any $k,\ell\in\mathbb{N}$,
$Av[k,\ell]$ and $Av[k,\infty)$ are defined in the next section.

######  Example 1.4.

Let $(\frac{1}{2^{3}},\frac{1}{2^{2}},\frac{1}{2})$ be a permutation of the
set $\\{\frac{1}{2},\frac{1}{2^{2}},\frac{1}{2^{3}}\\}$. Write

$P:=\frac{1}{2^{3}}\delta_{1}+\frac{1}{2^{2}}\delta_{2}+\frac{1}{2}\delta_{3}+\sum_{j=4}^{\infty}\frac{1}{2^{j}}\delta_{j}.$

Then, $P$ is a Borel probability measure on $\mathbb{R}$ with support the set
$\mathbb{N}$ of natural numbers. Let us assume that
$\\{a_{1},a_{2},\cdots,a_{n}\\}$ is an optimal set of $n$-means for $n=6$. If
$a_{1}=1$, $a_{2}=2$, and $a_{3}=3$, then by Theorem 1.3, we must have the set
$\\{a_{4},a_{5},a_{6}\\}$ equals either the set
$\\{4,Av[5,6],Av[7,\infty)\\}$, or the set
$\\{Av[4,5],Av[6,7],Av[8,\infty)\\}$ with quantization error
$V_{6}=\frac{1}{24}$.

######  Conjecture 1.5.

Let
$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j}$
be the probability measure as defined by Definition 1.2. Let
$\\{a_{1},a_{2},a_{3},\cdots,a_{n}\\}$ be an optimal set of $n$-means with
$n\geq k+2$ such that $a_{1}<a_{2}<\cdots<a_{n}$. Then,
$a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$.

In this paper, first we give a complete proof of Theorem 1.3. Then, we verify
the conjecture by two discrete distributions as mentioned in Remark 3.2.2 and
Remark 3.3.2. Under the assumption that the conjecture is true, we calculate
the optimal sets of $n$-means and the $n$th quantization errors for the two
discrete distributions for all $n\in\mathbb{N}$. Once the quantization error
is known, the quantization dimension can easily be calculated, see Proposition
3.3.3. In addition, in the last section, we give a proposition Proposition
5.1. By this proposition, we deduce that if $(p_{1},p_{2},\cdots,p_{k-1})$ is
not a permutation of the set
$\\{\frac{1}{2},\frac{1}{2^{2}},\cdots,\frac{1}{2^{k-1}}\\}$, where
$k\in\mathbb{N}:=\\{1,2,\cdots\\}$ with $k\geq 2$, then the conjecture is not
true. The general proof of the conjecture is not known yet. Such a problem
still remains open.

##  2\. Preliminaries

Let $\mathbb{N}:=\\{1,2,3,\cdots\\}$ be the set of natural numbers. Let
$(p_{1},p_{2},p_{3},\cdots)$, where $0<p_{j}<1$ for all $j\in\mathbb{N}$ and
$\sum_{j=1}^{\infty}p_{j}=1$, be a probability distribution. Let

$P=\sum_{j=1}^{\infty}p_{j}\delta_{j},$

where $\delta_{j}$ is the dirac measure as given in Definition 1.2. Then, $P$
is a discrete probability measure on the set $\mathbb{R}$ of real numbers with
the support the set of natural numbers $\mathbb{N}$ associated with the
probability distribution $(p_{1},p_{2},p_{3},\cdots)$. In fact, if $X$ is a
random variable associated with the probability measure $P$, and $f$ is the
probability mass function, then we have

$P(X=j)=f(j)=p_{j}.$

Define the following notations: For $k,\ell\in\mathbb{N}$, where $k\leq\ell$,
write

$[k,\ell]:=\\{n:n\in\mathbb{N}\text{ and }k\leq n\leq\ell\\},\text{ and
}[k,\infty):=\\{n:n\in\mathbb{N}\text{ and }n\geq k\\}.$

Further, write

$Av[k,\ell]:=E\Big{(}X:X\in[k,\ell]\Big{)}=\frac{\sum_{n=k}^{\ell}p_{n}n}{\sum_{n=k}^{\ell}p_{n}},\
Av[k,\infty):=E\Big{(}X:X\in[k,\infty)\Big{)}=\frac{\sum_{n=k}^{\infty}p_{n}n}{\sum_{n=k}^{\infty}p_{n}},$
$Er[k,\ell]:=\sum_{n=k}^{\ell}p_{n}\Big{(}n-Av[k,\ell]\Big{)}^{2},\text{ and
}Er[k,\infty):=\sum_{n=k}^{\infty}p_{n}\Big{(}n-Av[k,\infty)\Big{)}^{2}.$

Notice that $E(X):=E(X:X\in\text{supp}(P))=\sum_{n=1}^{\infty}p_{n}n$, and so
the optimal set of one-mean is the set $\\{\sum_{n=1}^{\infty}p_{n}n\\}$ with
quantization error

$V(P)=\sum_{n=1}^{\infty}p_{n}(n-E(X))^{2}.$

In the following sections, we give the main results of the paper.

##  3\. Proof of Theorem 1.3 and verifications of Conjecture 1.5

In this section, in the following subsections first we prove Theorem 1.3, and
then by two different examples, we verify that Conjecture 1.5 is true. Then,
we state and prove Proposition 3.3.3, which gives the quantization dimension
of the probability measure $P$.

###  3.1. Proof of Theorem 1.3

Let $n\geq k+2$, where $k\geq 2$. The distortion error due to the set
$\beta:=\\{1,2,\cdots,n-3,n-2,Av[n-1,n],Av[n+1,\infty)\\}$ is given by

$V(P;\beta)=Er[n-1,n]+Er[n+1,\infty)=\frac{2^{3-n}}{3}.$

Since $V_{n}$ is the quantization error for $n$-means, we have
$V_{n}\leq\frac{2^{3-n}}{3}$. Let
$\alpha:=\\{a_{1},a_{2},a_{3},\cdots,a_{n}\\}$ be an optimal set of $n$-means,
where $1\leq a_{1}<a_{2}<a_{3}<\cdots<a_{n}<\infty$. Assume that
$a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$. Then, the Voronoi region of $a_{n-2}$
must contain the element $n-2$. Suppose that the Voronoi region of $a_{n-2}$
contains the set $\\{n-2,n-1,n\\}$. Then,

$V_{n}\geq Er[n-2,n]=\frac{13}{7}2^{1-n}>\frac{2^{3-n}}{3}\geq V_{n},$

which is a contradiction. Hence, we can assume that the Voronoi region of
$a_{n-2}$ contains only the set $\\{n-2\\}$ or the set $\\{n-2,n-1\\}$. Let us
consider the following two cases:

Case 1. The Voronoi region of $a_{n-2}$ contains only the set $\\{n-2\\}$.

Then, the Voronoi region of $a_{n-1}$ must contain the element $n-1$. Suppose
that the Voronoi region of $a_{n-1}$ contains the set $\\{n-1,n,n+1,n+2\\}$.
Then,

$V_{n}\geq\frac{97}{15}\ 2^{-n-1}>V_{n},$

which is a contradiction. Assume that the Voronoi region of $a_{n-1}$ contains
only the set $\\{n-1,n,n+1\\}$. Then, as the Voronoi region of $a_{n}$
contains the set $\\{k:k\geq n+2\\}$,

$V_{n}\geq Er[n-1,n+1]+Er[n+2,\infty)=\frac{5}{7}2^{2-n}>V_{n},$

which leads to a contradiction. Next, assume that the Voronoi region of
$a_{n-1}$ contains only the set $\\{n-1\\}$. Then, the Voronoi region of
$a_{n}$ contains the set $\\{k:k\geq n\\}$ yielding

$V_{n}=Er[n,\infty)=2^{2-n}>V_{n},$

which gives a contradiction. This yields the fact that the Voronoi region of
$a_{n-1}$ contains only the set $\\{n-1,n\\}$, and hence, the Voronoi region
of $a_{n}$ contains only the set $\\{k:k\geq n+1\\}$. Thus, in this case we
have $a_{n-2}=n-2,a_{n-1}=Av[n-1,n],a_{n}=Av[n+1,\infty)$ with quantization
error $V_{n}=\frac{2^{3-n}}{3}.$

Case 2. The Voronoi region of $a_{n-2}$ contains only the set $\\{n-2,n-1\\}$.

Then, the Voronoi region of $a_{n-1}$ must contain the element $n$. Suppose
that the Voronoi region of $a_{n-1}$ contains the set $\\{n,n+1,n+2,n+3\\}$.
Then,

$V_{n}\geq Er[n-2,n-1]+Er[n,n+3]=\frac{59}{5}2^{-n-2}>\frac{2^{3-n}}{3}\geq
V_{n},$

which leads to a contradiction. Assume that the Voronoi region of $a_{n-1}$
contains only the set $\\{n,n+1,n+2\\}$. Then, as the Voronoi region of
$a_{n}$ contains the set $\\{k:k\geq n+3\\}$,

$V_{n}\geq Er[n-2,n-1]+Er[n,n+2]+Er[n+3,\infty)=\frac{29}{21}2^{1-n}>V_{n},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{n-1}$ contains only the set $\\{n\\}$, or the set $\\{n,n+1\\}$. If the
Voronoi region of $a_{n-1}$ contains only the set $\\{n\\}$, then

$V_{n}\geq Er[n-2,n-1]+Er[n+1,\infty)=\frac{5}{3}2^{1-n}>V_{n},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{n-1}$ contains only the set $\\{n,n+1\\}$, and the Voronoi region of
$a_{n}$ contains only the set $\\{k:k\geq n+2\\}$ yielding
$a_{n-2}=Av[n-2,n-1],a_{n-1}=Av[n,n+1],a_{n}=Av[n+2,\infty)$ with quantization
error $V_{n}=\frac{2^{3-n}}{3}$.

Case 1 and Case 2 together give the optimal sets of $n$-means and the $n$th
quantization errors for all positive integers $n$. Thus, the proof of the
theorem Theorem 1.3 is completed. ∎

###  3.2. Verification of Conjecture 1.5 when
$(p_{1},p_{2},p_{3},p_{4},\cdots)=(\frac{1}{2^{2}},\frac{1}{2},\frac{1}{2^{3}},\frac{1}{2^{4}},\frac{1}{2^{5}},\cdots)$

In this case the probability mass function $f$ for the probability measure $P$
on the set of real numbers $\mathbb{R}$ is given by

$f(j)=\left\\{\begin{array}[]{ll}\frac{1}{2^{2}}&\text{ if }j=1,\\\
\frac{1}{2}&\text{ if }j=2,\\\ \frac{1}{2^{n}}&\text{ if }j=n\text{ for
}n\in\mathbb{N}\text{ and }n\neq 1,2,\\\ 0&\text{
otherwise}.\end{array}\right.$

Notice that here $k=3$ and
$(p_{1},\cdots,p_{k-1})=(p_{1},p_{2})=(\frac{1}{2^{2}},\frac{1}{2})$, where
$k\in\mathbb{N}$ as defined by Definition 1.2.

Let us now prove the following proposition.

######  Proposition 3.2.1.

Let $n\geq 5$, and let $\alpha_{n}$ be an optimal set of $n$-means for the
probability measure $P$ given by

$P=\frac{1}{2^{2}}\delta_{1}+\frac{1}{2}\delta_{2}+\sum_{j=3}^{\infty}\frac{1}{2^{j}}\delta_{j}.$

Then, $\alpha_{n}$ must contain the set $\\{1,2,\cdots,(n-3)\\}$.

###### Proof.

The distortion error due to the set
$\beta:=\\{1,2,\cdots,(n-2),Av[n-1,n],Av[n+1,\infty)\\}$ is given by

$V(P;\beta)=Er[n-1,n]+Er[n+1,\infty)=\frac{2^{3-n}}{3}.$

Since $V_{n}$ is the quantization error for $n$-means, we have
$V_{n}\leq\frac{2^{3-n}}{3}$. Let $\alpha_{n}:=\\{a_{1},a_{2},\cdots,a_{n}\\}$
be an optimal set of $n$-means such that $1\leq
a_{1}<a_{2}<\cdots<a_{n}<\infty$. We show that
$a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$. We prove it by induction. Notice that
the Voronoi region of $a_{1}$ must contain the element $1$. Suppose that the
Voronoi region of $a_{1}$ also contains the element $2$. Then,

$V_{n}>\sum_{j=1}^{2}f(j)(j-Av[1,2])^{2}=\frac{1}{6}\geq\frac{2^{3-n}}{3}\geq
V_{n},$

which is a contradiction. Hence, we can conclude that the Voronoi region of
$a_{1}$ contains only the element $1$ yielding $a_{1}=1$. Thus, we can deduce
that there exists a positive integer $\ell$, where $1\leq\ell<n-3$, such that
$a_{1}=1,a_{2}=2,\cdots,a_{\ell}=\ell$. We now show that $a_{\ell+1}=\ell+1$.
Notice that the Voronoi region of $a_{\ell+1}$ must contain $\ell+1$. Suppose
that the Voronoi region of $a_{\ell+1}$ also contains the element $\ell+2$.
Then, we have

$V_{n}>\sum_{j=\ell+1}^{\ell+2}\frac{1}{2^{j}}(j-Av[\ell+1,\ell+2])^{2}=Er[\ell+1,\ell+2]=\frac{2^{-\ell-1}}{3}\geq\frac{2^{3-n}}{3}\geq
V_{n},$

which is a contradiction. Hence, we can conclude that the Voronoi region of
$a_{\ell+1}$ contains only the element $\ell+1$ yielding $a_{\ell+1}=\ell+1$.
Notice that $2\leq\ell+1\leq n-3$. Thus, by the Principle of Mathematical
Induction, we deduce that $a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$. Thus, the
proof of the proposition is complete. ∎

######  Remark 3.2.2.

Proposition 3.2.1 verifies that the conjecture Conjecture 1.5 is true.

###  3.3. Verification of Conjecture 1.5 when
$(p_{1},p_{2},p_{3},p_{4},\cdots)=(\frac{1}{2^{3}},\frac{1}{2^{2}},\frac{1}{2},\frac{1}{2^{4}},\frac{1}{2^{5}},\cdots)$

In this case the probability mass function $f$ for the probability measure $P$
on the set of real numbers $\mathbb{R}$ is given by

$f(j)=\left\\{\begin{array}[]{ll}\frac{1}{2^{3}}&\text{ if }j=1,\\\
\frac{1}{2}&\text{ if }j=3,\\\ \frac{1}{2^{n}}&\text{ if }j=n\text{ for
}n\in\mathbb{N}\text{ and }n\neq 1,3,\\\ 0&\text{
otherwise}.\end{array}\right.$

Notice that here $k=4$ and
$(p_{1},\cdots,p_{k-1})=(p_{1},p_{2},p_{3})=(\frac{1}{2^{3}},\frac{1}{2^{2}},\frac{1}{2})$,
where $k\in\mathbb{N}$ as defined by Definition 1.2.

Let us now prove the following proposition.

######  Proposition 3.3.1.

Let $n\geq 6$, and let $\alpha_{n}$ be an optimal set of $n$-means for the
probability measure $P$ given by

$P=\frac{1}{2^{3}}\delta_{1}+\frac{1}{2^{2}}\delta_{2}+\frac{1}{2}\delta_{3}+\sum_{j=4}^{\infty}\frac{1}{2^{j}}\delta_{j}.$

Then, $\alpha_{n}$ must contain the set $\\{1,2,\cdots,(n-3)\\}$.

###### Proof.

The distortion error due to the set
$\beta:=\\{1,2,\cdots,(n-2),Av[n-1,n],Av[n+1,\infty)\\}$ is given by

$V(P;\beta)=Er[n-1,n]+Er[n+1,\infty)=\frac{2^{3-n}}{3}.$

Since $V_{n}$ is the quantization error for $n$-means, we have
$V_{n}\leq\frac{2^{3-n}}{3}$. Let $\alpha_{n}:=\\{a_{1},a_{2},\cdots,a_{n}\\}$
be an optimal set of $n$-means such that $1\leq
a_{1}<a_{2}<\cdots<a_{n}<\infty$. We show that
$a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$. We prove it by induction. The Voronoi
region of $a_{1}$ must contain the element $1$. Suppose that the Voronoi
region of $a_{1}$ also contains the element $2$. Notice that the remaining
elements of the set of natural numbers are contained in the union of the
Voronoi regions of $a_{2},a_{3},\cdots,a_{n}$ with positive distortion error
yielding

$V_{n}>\sum_{j=1}^{2}f(j)(j-Av[1,2])^{2}=\frac{1}{12}\geq\frac{2^{3-n}}{3}\geq
V_{n},$

which is a contradiction. Hence, we can conclude that the Voronoi region of
$a_{1}$ contains only the element $1$, yielding $a_{1}=1$. Thus, we can deduce
that there exists a positive integer $\ell$, where $1\leq\ell<n-3$, such that
$a_{1}=1,a_{2}=2,\cdots,a_{\ell}=\ell$. We now show that $a_{\ell+1}=\ell+1$.
Notice that the Voronoi region of $a_{\ell+1}$ must contain $\ell+1$. Suppose
that the Voronoi region of $a_{\ell+1}$ also contains the element $\ell+2$.
Then, proceeding in the similar lines as given in Proposition 3.2.1, we can
see that a contradiction arises. Hence, we can conclude that the Voronoi
region of $a_{\ell+1}$ contains only the element $\ell+1$ yielding
$a_{\ell+1}=\ell+1$. Notice that $2\leq\ell+2\leq n-3$. Thus, by the Principle
of Mathematical Induction, we deduce that
$a_{1}=1,a_{2}=2,\cdots,a_{n-3}=n-3$. Thus, the proof of the proposition is
complete. ∎

######  Remark 3.3.2.

Proposition 3.3.1 verifies that the conjecture Conjecture 1.5 is true.

######  Proposition 3.3.3.

Let
$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j}$
be the probability measure as defined by Definition 1.2. Assume that
Conjecture 1.5 is true. Then, the quantization dimension $D(P)$ exists and
equals zero.

###### Proof.

By Theorem 1.3 and under the assumption that Conjecture 1.5 is true, the $n$th
quantization error for any positive integer $n\geq k+2$ for the probability
measure $P$, defined by Definition 1.2, is obtained as
$V_{n}(P)=\frac{2^{3-n}}{3}$. Hence, using the formula (1), we have $D(P)=0.$
∎

##  4\. Optimal quantization for the two probability distributions described
in Section 3

In this section, in the following two subsections we determine the optimal
sets of $n$-means and the $n$th quantization errors for all positive integers
$n\geq 2$ for the two probability measures $P$ given in Subsection 3.2 and
Subsection 3.3 under the assumption that Conjecture 1.5 is true.

###  4.1. Optimal quantization for $P$ when
$(p_{1},p_{2},p_{3},p_{4},\cdots)=(\frac{1}{2^{2}},\frac{1}{2},\frac{1}{2^{3}},\frac{1}{2^{4}},\frac{1}{2^{5}},\cdots)$

Let us give the results in the following propositions.

######  Proposition 4.1.1.

The optimal set of two-means is given by $\\{Av[1,3],Av[4,\infty)\\}$ with
quantization error $V_{2}=\frac{17}{28}$.

###### Proof.

We see that $Av[1,3]=\frac{13}{7}$, and $Av[4,\infty)=5$. Since
$3<\frac{1}{2}(\frac{13}{7}+5)=\frac{24}{7}<4$, the distortion error due to
the set $\beta:=\\{\frac{13}{7},5\\}$ is given by

$V(P;\beta)=Er[1,3]+Er[4,\infty)=\frac{17}{28}.$

Since $V_{2}$ is the quantization error for two-means, we have
$V_{2}\leq\frac{17}{28}$. Let $\alpha:=\\{a_{1},a_{2}\\}$ be an optimal set of
two-means such that $a_{1}<a_{2}$. Since the points in an optimal set are the
conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<\infty$. Notice that the Voronoi region of $a_{1}$ must contain
$1$. Suppose that the Voronoi region of $a_{1}$ contains the set
$\\{1,2,3,4\\}$. Then,

$V_{2}\geq\sum_{j=1}^{4}f(j)(j-Av[1,4])^{2}=Er[1,4]=\frac{5}{8}>V_{2},$

which yields a contradiction. Hence, we can assume that the Voronoi region of
$a_{1}$ contains only the set $\\{1\\}$ or $\\{1,2\\}$, or the set
$\\{1,2,3\\}$. Suppose that the Voronoi region of $a_{1}$ contains only the
set $\\{1\\}$, and so the Voronoi region of $a_{2}$ contains the set
$\\{n:n\geq 2\\}$. Then, we have

$V_{2}=Er[2,\infty)=\frac{7}{6}>V_{2},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1,2\\}$, or the set $\\{1,2,3\\}$.
Suppose that the Voronoi region of $a_{1}$ contains only the set $\\{1,2\\}$.
Then, the Voronoi region of $a_{2}$ contains $\\{3,4,5,\cdots\\}$ yielding

$V_{2}=Er[1,2]+Er[3,\infty)=\frac{2}{3}>V_{2},$

which gives a contradiction. Hence, we can conclude that the Voronoi region of
$a_{1}$ contains only the set $\\{1,2,3\\}$, and the Voronoi region of $a_{2}$
contains the set $\\{j:j\geq 4\\}$ yielding

$a_{1}=Av[1,3]\text{ and }a_{2}=Av[4,\infty)\text{ with quantization error
}V_{2}=Er[1,3]+Er[4,\infty)=\frac{17}{28}.$

Thus, the proof of the proposition is complete. ∎

######  Proposition 4.1.2.

The set $\\{Av[1,2],Av[3,4],Av[5,\infty)\\}$ forms the optimal set of three-
means with quantization error $V_{3}=\frac{1}{3}$.

###### Proof.

The distortion error due to set $\beta:=\\{Av[1,2],Av[3,4],Av[5,\infty)\\}$ is
given by

$V(P;\beta)=Er[1,2]+Er[3,4]+Er[5,\infty)=\frac{1}{3}.$

Since $V_{3}$ is the quantization error for three-means, we have
$V_{3}\leq\frac{1}{3}$. Let $\alpha:=\\{a_{1},a_{2},a_{3}\\}$ be an optimal
set of three-means. Since the points in an optimal set are the conditional
expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<\infty$. Suppose that the Voronoi region of $a_{1}$ contains
the set $\\{1,2,3\\}$. Then,

$V_{3}\geq\sum_{j=1}^{3}f(j)(j-Av[1,3])^{2}=Er[1,3]=\frac{5}{14}>\frac{1}{3}\geq
V_{3},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1\\}$, or the set $\\{1,2\\}$. For the
sake of contradiction, assume that the Voronoi region of $a_{1}$ contains only
the set $\\{1\\}$. Then, the Voronoi region of $a_{2}$ must contain the
element $2$. Suppose that the Voronoi region of $a_{2}$ contains the set
$\\{2,3,4,5\\}$. Then,

$V_{3}\geq Er[2,5]=\frac{181}{368}=0.491848>V_{3},$

which yields a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{2,3,4\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 5\\}$. Then, the distortion error is

$V_{3}=Er[2,4]+Er[5,\infty)=\frac{9}{22}=0.409091>V_{3},$

which gives a contradiction. Next, assume that the Voronoi region of $a_{2}$
contains only the set $\\{2,3\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 4\\}$. Then, the distortion error is

$V_{3}=Er[2,3]+Er[4,\infty)=\frac{7}{20}>V_{3},$

which leads to a contradiction. Finally, assume that the Voronoi region of
$a_{2}$ contains only the set $\\{2\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 3\\}$. Then, the distortion error is

$V_{3}=Er[3,\infty)=\frac{1}{2}>V_{3},$

which gives a contradiction. Thus, we can conclude that the Voronoi region of
$a_{1}$ contains only the set $\\{1,2\\}$. Then, the Voronoi region of $a_{2}$
must contain the element $3$. Suppose that the Voronoi region of $a_{2}$
contains the set $\\{3,4,5,6\\}$. Then,

$V_{3}\geq\sum_{j=1}^{2}f(j)(j-Av[1,2])^{2}+\sum_{j=3}^{6}f(j)(j-Av[3,6])^{2}=Er[1,2]+Er[3,6]=\frac{59}{160}=0.36875>V_{3},$

which yields a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{3,4,5\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 6\\}$. Then, the distortion error is

$V_{3}=Er[1,2]+Er[3,5]+Er[6,\infty)=\frac{29}{84}=0.345238>V_{3},$

which gives a contradiction. Next, assume that the Voronoi region of $a_{2}$
contains only the element $3$, and so the Voronoi region of $a_{3}$ contains
the set $\\{n:n\geq 4\\}$. Then, the distortion error is

$V_{3}=Er[1,2]+Er[4,\infty)=\frac{5}{12}=0.416667>V_{3},$

which yields a contradiction. Hence, we can conclude that the Voronoi region
of $a_{2}$ contains only the set $\\{3,4\\}$ yielding $a_{1}=Av[1,2]$,
$a_{2}=Av[3,4]$, and $a_{3}=Av[5,\infty)$ with quantization error
$V_{3}=\frac{1}{3}$. Thus, the proof of the proposition is complete. ∎

######  Proposition 4.1.3.

The sets $\\{1,2,Av[3,4],Av[5,\infty)\\}$ forms the optimal sets of four-means
with quantization error $V_{4}=\frac{1}{6}$.

###### Proof.

The distortion error due to set $\beta:=\\{1,2,Av[3,4],Av[5,\infty)\\}$ is
given by

$V(P;\beta)=Er[3,4]+Er[5,\infty)=\frac{1}{6}.$

Since $V_{4}$ is the quantization error for four-means, we have
$V_{4}\leq\frac{1}{6}$. Let $\alpha:=\\{a_{1},a_{2},a_{3},a_{4}\\}$ be an
optimal set of four-means. Since the points in an optimal set are the
conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<a_{4}<\infty$. Clearly, the Voronoi region of $a_{1}$
contains the point $1$. Suppose that the Voronoi region of $a_{1}$ contains
the set $\\{1,2,3\\}$. Then,

$V_{3}\geq\sum_{j=1}^{3}f(j)(j-Av[1,3])^{2}=Er[1,3]=\frac{5}{14}>\frac{1}{6}\geq
V_{4},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1\\}$, or the set $\\{1,2\\}$. Suppose
that the Voronoi region of $a_{1}$ contains only the set $\\{1,2\\}$. Then,
the remaining elements of the set of natural numbers are contained in the
union of the Voronoi regions of $a_{2},a_{3}$ and $a_{4}$. Notice that the
total distortion error contributed by the points $a_{2},a_{3}$ and $a_{4}$ are
positive. Hence,

$V_{4}>\text{distortion error contributed by the point
}a_{1}=Er[1,2]=\frac{1}{6}=V_{4},$

which leads to a contradiction. Hence, the Voronoi region of $a_{1}$ cannot
contain $\\{1,2\\}$, i.e., the Voronoi region of $a_{1}$ contains only set
$\\{1\\}$, i.e., $a_{1}=1$. Then, the Voronoi region of $a_{2}$ must contain
$2$. Suppose that the Voronoi region of $a_{2}$ contains the set
$\\{2,3,4\\}$. Then,

$V_{4}\geq Er[2,4]=\frac{25}{88}>V_{4},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{2}$ contains only the set $\\{2\\}$, or the set $\\{2,3\\}$. Suppose
that the Voronoi region of $a_{2}$ contains only the set $\\{2,3\\}$. Assume
that the Voronoi region of $a_{3}$ contains the set $\\{4,5,6,7\\}$. Then,

$V_{4}\geq Er[2,3]+Er[4,7]=\frac{193}{960}>V_{4},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{3}$ contains only the set $\\{4\\}$, $\\{4,5\\}$, or $\\{4,5,6\\}$.
Suppose that the Voronoi region of $a_{3}$ contains only the set
$\\{4,5,6\\}$. Then, the Voronoi region of $a_{4}$ contains the set
$\\{n:n\geq 7\\}$. Then,

$V_{4}=Er[2,3]+Er[4,6]+Er[7,\infty)=\frac{53}{280}>V_{4},$

which is a contradiction. Suppose that the Voronoi region of $a_{3}$ contains
only the set $\\{4,5\\}$. Then, the Voronoi region of $a_{4}$ contains the set
$\\{n:n\geq 6\\}$. Then,

$V_{4}=Er[2,3]+Er[4,5]+Er[6,\infty)=\frac{11}{60}>V_{4},$

which leads to a contradiction. Suppose that the Voronoi region of $a_{3}$
contains only the set $\\{4\\}$. Then, the Voronoi region of $a_{4}$ contains
the set $\\{n:n\geq 5\\}$. Then,

$V_{4}=Er[2,3]+Er[5,\infty)=\frac{9}{40}>V_{4},$

which leads to a contradiction. Thus, we see that if the Voronoi region of
$a_{2}$ contains only the set $\\{2,3\\}$, then a contradiction arises. Hence,
we can conclude that the Voronoi region of $a_{2}$ contains only the set
$\\{2\\}$, in other words, we have $a_{2}=2$. Then, the Voronoi region of
$a_{3}$ contains the set $\\{3\\}$. Suppose that the Voronoi region of $a_{3}$
contains the set $\\{3,4,5,6\\}$, then as before we see a contradiction
arises. Hence, the Voronoi region of $a_{3}$ contains only the set $\\{3\\}$,
$\\{3,4\\}$, or the set $\\{3,4,5\\}$. Notice that if the Voronoi region of
$a_{3}$ contains only the set $\\{3,4,5\\}$, then the Voronoi region of
$a_{4}$ contains the set $\\{n:n\geq 6\\}$, and if the Voronoi region of
$a_{3}$ contains only the set $\\{3\\}$, then the Voronoi region of $a_{4}$
contains the set $\\{n:n\geq 4\\}$. In either of the cases, proceeding as
before, we see that a contradiction arises. Hence, we can conclude that the
Voronoi region of $a_{3}$ contains only the set $\\{3,4\\}$. Hence, the
Voronoi region of $a_{4}$ contains $\\{n:n\geq 5\\}$. Thus, we have

$a_{1}=1,a_{2}=2,a_{3}=[3,4],\text{ and }a_{4}=[5,\infty)\text{ with
}V_{4}=\frac{1}{6}.$

Thus, the proof of the proposition is complete. ∎

######  Proposition 4.1.4.

The sets $\\{1,2,\cdots,n-3,Av[n-2,n-1],Av[n,n+1],Av[n+2,\infty)\\}$ and
$\\{1,2,\cdots,n-3,n-2,Av[n-1,n],Av[n+1,\infty)\\}$ form the optimal sets of
$n$-means for all $n\geq 5$ with the quantization error
$V_{n}=\frac{2^{3-n}}{3}.$

###### Proof.

The proof follows by Theorem 1.3 and Conjecture 1.5 under the assumption that
Conjecture 1.5 is true. ∎

###  4.2. Optimal quantization for $P$ when
$(p_{1},p_{2},p_{3},p_{4},\cdots)=(\frac{1}{2^{3}},\frac{1}{2^{2}},\frac{1}{2},\frac{1}{2^{4}},\frac{1}{2^{5}},\cdots)$

Let us give the results in the following propositions.

######  Proposition 4.2.1.

The optimal set of two-means is given by $\\{Av[1,3],Av[4,\infty)\\}$ with
quantization error $V_{2}=\frac{5}{7}$.

###### Proof.

We see that $Av[1,3]=\frac{17}{7}$, and $Av[4,\infty)=5$. Since
$3<\frac{1}{2}(\frac{17}{7}+5)=\frac{26}{7}<4$, the distortion error due to
the set $\beta:=\\{\frac{17}{7},5\\}$ is given by

$V(P;\beta)=Er[1,3]+Er[4,\infty)=\frac{5}{7}.$

Since $V_{2}$ is the quantization error for two-means, we have
$V_{2}\leq\frac{5}{7}$. Let $\alpha:=\\{a_{1},a_{2}\\}$ be an optimal set of
two-means such that $a_{1}<a_{2}$. Since the points in an optimal set are the
conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<\infty$. Notice that the Voronoi region of $a_{1}$ must contain
$1$. Suppose that the Voronoi region of $a_{1}$ contains the set
$\\{1,2,3,4,5\\}$. Then,

$V_{2}\geq Er[1,5]=\frac{393}{496}>V_{2},$

which yields a contradiction. Thus, we can conclude that the Voronoi region of
$a_{1}$ does not contain the point $5$. Suppose that the Voronoi region of
$a_{1}$ contains only the $\\{1,2,3,4\\}$, and so the Voronoi region of
$a_{2}$ contains the set $\\{n:n\geq 5\\}$. Then, we have

$V_{2}=Er[1,4]+Er[5,\infty)=\frac{11}{15}>V_{2},$

which leads to a contradiction. Similarly, we can show that if the Voronoi
region of $a_{1}$ contains only the set $\\{1\\}$, or the set $\\{1,2\\}$,
then we get a contradiction. Hence, we can assume that the Voronoi region of
$a_{1}$ contains only the set $\\{1,2,3\\}$, and so the Voronoi region of
$a_{2}$ contains only the set $\\{n:n\geq 4\\}$. Thus, we have

$a_{1}=Av[1,3],\ a_{2}=Av[4,\infty)\text{ with }V_{2}=\frac{5}{7}.$

Thus, the proof of the proposition is complete. ∎

######  Proposition 4.2.2.

The sets $\\{Av[1,2],Av[3,4],Av[5,\infty)\\}$ forms the optimal sets of three-
means with quantization error $V_{3}=\frac{19}{72}$.

###### Proof.

The distortion error due to set $\beta:=\\{Av[1,2],Av[3,4],Av[5,\infty)\\}$ is
given by

$V(P;\beta)=Er[1,2]+Er[3,4]+Er[5,\infty)=\frac{19}{72}.$

Since $V_{3}$ is the quantization error for three-means, we have
$V_{3}\leq\frac{19}{72}=0.263889$. Let $\alpha:=\\{a_{1},a_{2},a_{3}\\}$ be an
optimal set of three-means. Since the points in an optimal set are the
conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<\infty$. Suppose that the Voronoi region of $a_{1}$ contains
the set $\\{1,2,3\\}$. Then,

$V_{3}\geq Er[1,3]=\frac{13}{28}>\frac{19}{72}\geq V_{3},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1\\}$, or the set $\\{1,2\\}$. Suppose
that the Voronoi region of $a_{1}$ contains only the set $\\{1\\}$. In this
case, the Voronoi region of $a_{2}$ must contain the element $2$. Suppose that
the Voronoi region of $a_{2}$ contains the set $\\{2,3,4\\}$. Then,

$V_{3}\geq Er[2,4]=\frac{7}{26}=0.269231>V_{3},$

which yields a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{2,3\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 4\\}$. Then, the distortion error is

$V_{3}=Er[2,3]+Er[4,\infty)=\frac{5}{12}>V_{3},$

which leads to a contradiction. Finally, assume that the Voronoi region of
$a_{2}$ contains only the set $\\{2\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 3\\}$. Then, the distortion error is

$V_{3}=Er[3,\infty)=\frac{13}{20}>V_{3},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1,2\\}$. Then, the Voronoi region of
$a_{2}$ must contain the set $\\{3\\}$. Suppose that the Voronoi region of
$a_{2}$ contains the set $\\{3,4,5,6\\}$. Then,

$V_{3}\geq Er[1,2]+Er[3,6]=\frac{151}{416}>V_{3},$

which yields a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{3,4,5\\}$, and so the Voronoi region of $a_{3}$
contains the set $\\{n:n\geq 6\\}$. Then, the distortion error is

$V_{3}=Er[1,2]+Er[3,5]+Er[6,\infty)=\frac{35}{114}>V_{3},$

which gives a contradiction. Next, assume that the Voronoi region of $a_{2}$
contains only the element $3$, and so the Voronoi region of $a_{3}$ contains
the set $\\{n:n\geq 4\\}$. Then, the distortion error is

$V_{3}=Er[1,2]+Er[4,\infty)=\frac{1}{3}>V_{3},$

which yields a contradiction. Hence, we can conclude that the Voronoi region
of $a_{2}$ contains only the set $\\{3,4\\}$ yielding $a_{1}=Av[1,2]$,
$a_{2}=Av[3,4]$, and $a_{3}=Av[5,\infty)$ with quantization error
$V_{3}=\frac{19}{72}$. ∎

######  Proposition 4.2.3.

The sets $\\{Av[1,2],3,Av[4,5],Av[6,\infty)\\}$ forms the optimal sets of
four-means with quantization error $V_{4}=\frac{1}{6}$.

###### Proof.

The distortion error due to set $\beta:=\\{Av[1,2],3,Av[4,5],Av[6,\infty)\\}$
is given by

$V(P;\beta)=Er[1,2]+Er[4,5]+Er[6,\infty)=\frac{1}{6}.$

Since $V_{4}$ is the quantization error for four-means, we have
$V_{4}\leq\frac{1}{6}=0.166667$. Let $\alpha:=\\{a_{1},a_{2},a_{3},a_{4}\\}$
be an optimal set of four-means. Since the points in an optimal set are the
conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<a_{4}<\infty$. Suppose that the Voronoi region of $a_{1}$
contains the set $\\{1,2,3\\}$. Then,

$V_{4}\geq Er[1,3]=\frac{13}{28}>V_{4},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the set $\\{1\\}$, or the set $\\{1,2\\}$. Suppose
that the Voronoi region of $a_{1}$ contains only the set $\\{1\\}$. In this
case, the Voronoi region of $a_{2}$ must contain the element $2$. Suppose that
the Voronoi region of $a_{2}$ contains the set $\\{2,3,4\\}$. Then,

$V_{4}\geq Er[2,4]=\frac{7}{26}=0.269231>V_{4},$

which yields a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{2,3\\}$. Then, notice that the Voronoi regions of
$a_{3}$ and $a_{4}$ contain all the elements $\\{n:n\geq 4\\}$. Thus, the
total distortion error contributed by $a_{3}$ and $a_{4}$ must be positive.
This leads to the fact that

$V_{4}>Er[2,3]=\frac{1}{6}\geq V_{4},$

which gives a contradiction. Assume that the Voronoi region of $a_{2}$
contains only the set $\\{2\\}$. Then, as before, we see that a contradiction
arises. Hence, we can assume that the Voronoi region of $a_{1}$ contains only
the set $\\{1,2\\}$. Then, the Voronoi region of $a_{2}$ must contain $3$. If
the Voronoi region of $a_{2}$ contains more points using the similar arguments
as before, we can show that a contradiction arises. Hence, we can conclude
that $a_{2}=3$. Again, using the similar arguments, we can show that the
Voronoi region of $a_{3}$ contains only the set $\\{4,5\\}$, and the Voronoi
region of $a_{4}$ contains only the set $\\{n:n\geq 6\\}$. Thus, we have

$a_{1}=Av[1,2],a_{2}=3,a_{3}=Av[4,5],\text{ and }a_{4}=Av[6,\infty)\text{ with
quantization error }V_{4}=\frac{1}{6}.$

Thus, the proof of the proposition is complete. ∎

######  Proposition 4.2.4.

The sets $\\{1,2,3,Av[4,5],Av[6,\infty)\\}$ forms the optimal sets of five-
means with quantization error $V_{5}=\frac{1}{12}$.

###### Proof.

The distortion error due to set $\beta:=\\{1,2,3,Av[4,5],Av[6,\infty)\\}$ is
given by

$V(P;\beta)=Er[4,5]+Er[6,\infty)=\frac{1}{12}.$

Since $V_{5}$ is the quantization error for five-means, we have
$V_{5}\leq\frac{1}{12}=0.0833333$. Let
$\alpha:=\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}$ be an optimal set of five-means
such that $a_{1}<a_{2}<a_{3}<a_{4}<a_{5}$. Since the points in an optimal set
are the conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<a_{4}<a_{5}<\infty$. Clearly, the Voronoi region of $a_{1}$
contains the point $1$. For the sake of contradiction, assume that the Voronoi
region of $a_{1}$ contains the set $\\{1,2,3\\}$. Then,

$V_{5}\geq Er[1,3]=\frac{13}{28}>V_{5},$

which is a contradiction. Next, assume that the Voronoi region of $a_{1}$
contains only the set $\\{1,2\\}$. Then, notice that the union Voronoi regions
of $a_{2}$, $a_{3}$, $a_{4}$, and $a_{5}$ contain all the elements $\\{n:n\geq
3\\}$. Hence, we must have

$V_{5}>Er[1,2]=\frac{1}{12}\geq V_{5},$

which is a contradiction. Hence, we can conclude that the Voronoi region of
$a_{1}$ contains only the element $1$, i.e., $a_{1}=1$. Clearly, the Voronoi
region of $a_{2}$ contains the element $2$. Suppose the Voronoi region of
$a_{2}$ contains the set $\\{2,3\\}$. Then, we have

$V_{5}\geq Er[2,3]=\frac{1}{6}>V_{5},$

which give a contradiction. Hence, the Voronoi region of $a_{2}$ contains only
the element $2$, i.e., $a_{2}=2$. Similarly, we can show that $a_{3}=3$. The
rest of the proof follows in the similar lines as given in Proposition 4.1.3.
Thus, we see that $a_{4}=Av[4,5]$ and $a_{5}=Av[6,\infty)$ with quantization
error $V_{5}=\frac{1}{12}$. Thus, the proof of the proposition is complete. ∎

######  Proposition 4.2.5.

The sets $\\{1,2,\cdots,n-3,Av[n-2,n-1],Av[n,n+1],Av[n+2,\infty)\\}$ and
$\\{1,2,\cdots,n-3,n-2,Av[n-1,n],Av[n+1,\infty)\\}$ form the optimal sets of
$n$-means for all $n\geq 6$ with the quantization error
$V_{n}=\frac{2^{3-n}}{3}.$

###### Proof.

The proof follows by Theorem 1.3 and Conjecture 1.5 under the assumption that
Conjecture 1.5 is true. ∎

##  5\. Observation and Remarks

In Conjecture 1.5 the probability measure $P$ is defined as
$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j}$,
where $(p_{1},p_{2},\cdots,p_{k-1})$ is a permutation of the set
$\\{\frac{1}{2},\frac{1}{2^{2}},\cdots,\frac{1}{2^{k-1}}\\}$, where
$k\in\mathbb{N}$ with $k\geq 2$. If
$P:=\sum_{j=1}^{k-1}p_{j}\delta_{j}+\sum_{j=k}^{\infty}\frac{1}{2^{j}}\delta_{j}$,
and $(p_{1},p_{2},\cdots,p_{k-1})$ is not a permutation of the set
$\\{\frac{1}{2},\frac{1}{2^{2}},\cdots,\frac{1}{2^{k-1}}\\}$, then Conjecture
1.5 is not true. In this regard, we give the following proposition.

######  Proposition 5.1.

For the probability measure $P$ given by
$P:=\frac{149}{200}\delta_{1}+\frac{1}{200}\delta_{2}+\sum_{j=3}^{\infty}\frac{1}{2^{j}}\delta_{j}$
the optimal set of five-means is given by

$\\{1,Av[2,3],4,Av[5,6],Av[7,\infty)\\},\text{ or
}\\{1,Av[2,3],Av[4,5],Av[6,7],Av[8,\infty)\\}$

with quantization error $V_{5}=\frac{29}{624}$.

###### Proof.

The distortion error due to set
$\beta:=\\{1,Av[2,3],4,Av[5,6],Av[7,\infty)\\}$ is given by

$V(P;\beta)=Er[2,3]+Er[5,6]+Er[7,\infty)=\frac{29}{624}.$

Since $V_{5}$ is the quantization error for five-means, we have
$V_{5}\leq\frac{29}{624}=0.0464744$. Let us assume that
$\alpha:=\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}$ is an optimal set of five-means
such that $a_{1}<a_{2}<a_{3}<a_{4}<a_{5}$. Since the points in an optimal set
are the conditional expectations in their own Voronoi regions, we have $1\leq
a_{1}<a_{2}<a_{3}<a_{4}<a_{5}<\infty$. Clearly, the Voronoi region of $a_{1}$
contains the point $1$. For the sake of contradiction, assume that the Voronoi
region of $a_{1}$ contains the set $\\{1,2,3\\}$. Then,

$V_{5}\geq Er[1,3]=\frac{7537}{17500}=0.430686>V_{5},$

which is a contradiction. Hence, we can assume that the Voronoi region of
$a_{1}$ contains only the set $\\{1\\}$ or the set $\\{1,2\\}$. Suppose that
the Voronoi region of $a_{1}$ contains only the set $\\{1,2\\}$. Then, the
Voronoi region of $a_{2}$ must contain the element $3$. Suppose that the
Voronoi region of $a_{2}$ contains the set $\\{3,4\\}$. Then,

$V_{5}\geq Er[1,2]+Er[3,4]=\frac{1399}{30000}=0.0466333>V_{5},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{2}$ contain only the element $3$, i.e., $a_{2}=3$. Then, the union of
the Voronoi regions of $a_{3},a_{4},a_{5}$ contains the set
$\\{4,5,6,\cdots\\}$ with associated probability $\frac{1}{2^{j}}$ for each
$j\in\\{4,5,6,\cdots\\}$. Hence, using the similar lines as described in the
proof of Theorem 1.3, we can show that

(2) $\displaystyle\\{a_{3},a_{4},a_{5}\\}\text{ equals the set
}\\{4,Av[5,6],Av[7,\infty)\\},\text{ or }\\{Av[4,5],Av[6,7],Av[8,\infty)\\}$

with the quantization error

$V_{5}=Er[1,2]+Er[5,6]+Er[7,\infty)=\frac{1399}{30000}=0.0466333>V_{5},$

which leads to a contradiction. Hence, we can assume that the Voronoi region
of $a_{1}$ contains only the element $1$, i.e., $a_{1}=1$. Then, the Voronoi
region of $a_{2}$ must contain $2$. Suppose that the Voronoi region of $a_{2}$
contains the set $\\{2,3,4\\}$. Then,

$V_{5}\geq Er[2,4]=\frac{31}{616}>V_{5},$

which leads to a contradiction. Hence, the Voronoi region of $a_{2}$ contains
only the set $\\{2\\}$, or $\\{2,3\\}$. Suppose that the Voronoi region of
$a_{2}$ contains only the set $\\{2\\}$, i.e., $a_{2}=2$. Then, as $a_{1}=1$,
$a_{2}=2$, using the similar lines as described in the proof of Theorem 1.3,
we can show that $a_{3}=3,a_{4}=Av[4,5],a_{5}=Av[6,\infty)$; or
$a_{3}=Av[3,4],a_{4}=Av[5,6],a_{5}=Av[7,\infty)$ with quantization error
$V_{5}=\frac{1}{12}>V_{5}$, which is a contradiction. Hence, we can assume
that the Voronoi region of $a_{2}$ contains only the set $\\{2,3\\}$. Again,
using the similar lines as described in the proof of Theorem 1.3, we can show
that $\\{a_{3},a_{4},a_{5}\\}$ equals the set $\\{4,Av[5,6],Av[7,\infty)\\}$;
or $\\{Av[4,5],Av[6,7],Av[8,\infty)\\}$. Thus, we conclude that the optimal
set of five-means is either $\\{1,Av[2,3],4,Av[5,6],Av[7,\infty)\\}$ or
$\\{1,Av[2,3],Av[4,5],Av[6,7],Av[8,\infty)\\}$ with quantization error
$V_{5}=\frac{29}{624}$. This completes the proof of the proposition. ∎

######  Remark 5.2.

Proposition 5.1 implies that Conjecture 1.5 is not true for an arbitrary
probability distribution $(p_{1},p_{2},p_{3},\cdots)$ associated with the set
of positive integers $\mathbb{N}$.

######  Remark 5.3.

Conjecture 1.5 is verified by two examples given in Subsection 3.2 and
Subsection 3.3. We still could not give a general proof of the conjecture. It
will be worthwhile to investigate the general proof of the conjecture.

## References

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  * [GG] A. Gersho and R.M. Gray, _Vector quantization and signal compression_ , Kluwer Academy publishers: Boston, 1992.
  * [GKL] R.M. Gray, J.C. Kieffer and Y. Linde, _Locally optimal block quantizer design_ , Information and Control, 45 (1980), pp. 178-198.
  * [GL1] A. György and T. Linder, _On the structure of optimal entropy-constrained scalar quantizers_ , IEEE transactions on information theory, vol. 48, no. 2, February 2002.
  * [GL2] S. Graf and H. Luschgy, _Foundations of quantization for probability distributions_ , Lecture Notes in Mathematics 1730, Springer, Berlin, 2000.
  * [GN] R.M. Gray and D.L. Neuhoff, _Quantization_ , IEEE Transactions on Information Theory, Vol. 44, No. 6, October 1998, 2325-2383.
  * [P] D. Pollard, _Quantization and the Method of $k$-Means_, IEEE Transactions on Information Theory, 28 (1982), 199-205.
  * [R1] M.K. Roychowdhury, _Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets_ , Journal of Fractal Geometry, 4 (2017), 127-146.
  * [Z1] P.L. Zador, _Asymptotic Quantization Error of Continuous Signals and the Quantization Dimension_ , IEEE Transactions on Information Theory, 28 (1982), 139-149.
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00footnotetext: Research partially supported by NSF Grants DMS-0111298 and
DMS-0805594

# Hyper-Kähler Fourfolds Fibered by Elliptic Products

Ljudmila Kamenova<EMAIL_ADDRESS>Department of Mathematics, SUNY
Stony Brook

###### Abstract.

Every fibration of a projective hyper-Kähler fourfold has fibers which are
Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two
curve, these have been classified by Markushevich. We classify those cases
where the Abelian surface is a product of two elliptic curves, under some mild
genericity hypotheses. Unlike the genus two case, we prove there are
infinitely many deformation families for hyper-Kähler fourfolds fibered by
elliptic products.

##  1\. Introduction

Among projective complex manifolds with zero first Chern class, hyper-Kähler
manifolds play a distinguished role, generalizing the class of K3 surfaces. In
fact the hyper-Kähler manifolds of (complex) dimension 2 are precisely the K3
surfaces. In higher dimensions, dimension 4 and higher, few hyper-Kähler
manifolds are known: two infinite classes introduced by Beauville ([2]) and
two exceptional cases discovered by O’Grady ([12] and [13]). Nonetheless, even
in the first case of hyper-Kähler fourfolds, we are still far from a
classification or even a proof that hyper-Kähler fourfolds form a bounded
family.

Just as elliptically fibered K3 surfaces are more amenable to study, so also
fibered hyper-Kähler manifolds are better understood. By works of Matsushita
and Hwang, ([11] and [9]), every holomorphic fibration whose total space is a
$2n$-dimensional hyper-Kähler has base manifold $\mathbb{P}^{n}$ and has
general fiber an $n$-dimensional Abelian variety, up to a finite (unramified)
cover. A hyper-Kähler manifold fibered over $\mathbb{P}^{n}$ with general
fiber an Abelian variety is called an _Abelian fibration_. Every Abelian
fibered manifold is a “Tate-Shafarevich twist” of an Abelian fibration with a
section, and the set of these twists coming from a given Abelian fibration is
classified by a well-studied Tate-Shafarevich group. Thus from now on we
assume that the Abelian fibration admits a section.

The principal case is when the fibers are _principally polarized_ Abelian
varieties. For hyper-Kähler fourfolds, the fiber will be a principally
polarized Abelian surface. There are two types of such surfaces, depending on
whether or not the polarizing divisor is irreducible or reducible, i.e.,
whether the Abelian surface is the Jacobian of a genus $2$ curve or a product
of elliptic curves. Markushevich classified the first case, when the Abelian
surface is the Jacobian of a genus $2$ curve.

######  Theorem 1.1.

[10, Markushevich] Every irreducible holomorphic symplectic fourfold $X$ which
is fibered by Jacobians of genus two curves and such that the fibration
$X\rightarrow\mathbb{P}^{2}$ admits a section is birational to ${\it
Hilb}^{2}(S)$, for a K3 surface $S$ which is a double cover of
$\mathbb{P}^{2}$ branched over a plane sextic.

Our goal is to classify those Abelian fibrations where the generic fiber is a
product of two elliptic curves, an _elliptic product_. But unlike
Markushevich’s theorem, there are infinitely many deformation classes of
projective hyper-Kähler fourfolds together with a fibration by elliptic
products. Moreover, these hyper-Kähler fourfolds can become “wild” when the
fibration is allowed to become pathologically degenerate. In fact we
conjecture that these pathological degenerations never occur, at least for a
generic member of a deformation family. But, at the moment, we add our
genericity conditions as hypotheses: the conditions are quite natural and are
satisfied in every case we know of.

In order to explain the genericity conditions, we first introduce some
notation.

As we shall see, every hyper-Kähler fibration by elliptic products
$p:X\rightarrow\mathbb{P}^{2}$ arises from a degree-2 branched cover
$f:\bar{V}\rightarrow\mathbb{P}^{2}$, where $\bar{V}$ is a normal variety, and
a family of elliptic curves $\pi:{\mathcal{E}}\rightarrow\bar{V}$. Let $D$ be
the branch locus of $f$, $\tilde{D}=f^{-1}(D)$ and let $G$ be the discriminant
locus of $\pi$. These are our _genericity conditions_ :

  1. (1)

Over every generic point of $G$, the fiber of $\pi$ is irreducible and
semistable.

  2. (2)

The image $f(G)$ contains no singular point of $D$. In fact it suffices to
assume that $f(G)$ contains no intersection point of two irreducible
components of $D$ which are lines.

Here is the main theorem.

######  Theorem 1.2.

Let $p:X\rightarrow\mathbb{P}^{2}$ be a projective hyper-Kähler fibration
whose general fiber is a product of two elliptic curves and which satisfies
the genericity conditions above. Assume that the fibration admits a section
$\tau$. Then $X$ is birational to $Hilb^{2}(S)$ for a projective, elliptic K3
surface $S$ with a section.

Notice that for every projective, elliptic K3 surface $S$ with a section,
$\text{Hilb}^{2}(S)$ is a projective hyper-Kähler fourfold fibered by elliptic
products and which admits a section. There are countably many families of
projective, elliptic K3 surfaces $S$, hence countably many families of
elliptic product hyper-Kähler fourfolds. And for the general member of each of
these families, the genericity conditions are satisfied. As mentioned, we
conjecture that every sufficiently general elliptic product hyper-Kähler
fourfold satisfies the genericity conditions. All such are deformation
equivalent to one of the countably many families arising from projective,
elliptic K3 surfaces.

##  2\. Preliminaries

First we define our main objects of study, irreducible holomorphic symplectic
manifolds or irreducible hyper-Kähler manifolds.

######  Definition 2.1.

A compact complex Kähler manifold $X$ is called irreducible holomorphic
symplectic if it is simply connected and if $H^{0}(X,\Omega_{X}^{2})$ is
spanned by an everywhere non-degenerate 2-form $\omega$.

Any holomorphic two-form $\sigma$ induces a homomorphism
${\mathcal{T}}_{X}\to\Omega_{X}$. The two-form is everywhere non-degenerate if
${\mathcal{T}}_{X}\to\Omega_{X}$ is bijective. The last condition in the
definition implies that $h^{2,0}(X)=h^{0,2}(X)=1$ and
$K_{X}\cong{\mathcal{O}}_{X}$, i.e., $c_{1}(X)=0$.

######  Definition 2.2.

A compact connected $4n$-dimensional Riemannian manifold $(M,g)$ is called
irreducible hyper-Kähler if its holonomy is ${\rm Sp}(n)$.

As Huybrechts notes [8], irreducible holomorphic symplectic manifolds with a
fixed Kähler class are equivalent to irreducible hyper-Kähler manifolds. In
the rest of the paper, we refer to irreducible hyper-Kähler manifolds just as
hyper-Kähler manifolds for simplicity.

######  Definition 2.3.

An Abelian fibration on a $2n$-dimensional hyper-Kähler manifold $X$ is the
structure of a fibration over ${\mathbb{P}}^{n}$ whose generic fibre is a
smooth abelian variety of dimension $n$.

This is a higher dimensional analogue of elliptic fibrations on K3 surfaces.
Any fibration structure of a projective hyper-Kahler manifold is an abelian
fibration due to the following theorems by Matsushita [11] and Hwang [9]:

######  Theorem 2.4.

(Matsushita, [11]) For a projective holomorphic symplectic manifold $X$, let
$f:X\rightarrow B$ be a proper surjective morphism such that the generic fibre
$F$ is connected. Assume that $B$ is smooth and
$0<{\mathrm{dim}}B<{\mathrm{dim}}X$. Then

(1) $F$ is an abelian variety up to a finite unramified cover,

(2) $B$ is $n$-dimensional and has the same Hodge numbers as
${\mathbb{P}}^{n}$,

(3) the fibration is Lagrangian with respect to the holomorphic symplectic
form.

######  Theorem 2.5.

(Hwang, [9]) In the setting of Matsushita’s theorem, $B$ is biholomorphic to
${\mathbb{P}}^{n}$.

Fogarty gives the following description of the Hilbert schemes of complex
surfaces.

######  Theorem 2.6.

(Fogarty, [4]) For a nonsingular surface $X$ and $n\in\mathbb{N}$
(1) ${\it Hilb}^{n}(X)$ is non-singular of dimension $2n$ and
(2) $\pi:{\it Hilb}^{n}(X)\rightarrow S^{n}X$ is a resolution of
singularities, where $S^{n}X$ is the n-th symmetric product of $X$.

##  3\. Proof of the Main Theorem

In this section we prove our main theorem.

###### Proof.

Let us be more precise about the hypothesis that the general singular fiber is
irreducible and semistable. Precisely, there exists a closed codimension two
subscheme $N$ of $\mathbb{P}^{2}$ such that the fiber of $p$ over every point
outside of $N$ is either smooth or irreducible and semi-stable.

Take the open subset $U\subset\mathbb{P}^{2}$ over which the fibers of $p$ are
smooth. $U$ is algebraic. The fibers over $U$ are of the form $E^{1}_{t}\times
E^{2}_{t}$. We can form the fibration $Y\rightarrow U$ with fibers of the form
$E^{1}_{t}\cup E^{2}_{t}$, where the two elliptic curves are glued along the
section $\tau$. Take the normalization ${\tilde{Y}}\rightarrow Y\rightarrow
U$. Since U is algebraic, we can apply Stein factorization. The morphism
${\tilde{Y}}\rightarrow U$ factors through a smooth proper morphism with
connected fibers and an étale morphism of degree 2: ${\tilde{Y}}\rightarrow
V\rightarrow U$.

According to Grothendieck’s lemma ( [5], Section 6.3.), there is a unique
normal variety $\bar{V}$, a finite degree-2 morphism
$f:\bar{V}\rightarrow\mathbb{P}^{2}$ and a fiber diagram:

$\begin{array}[]{ccc}V&\stackrel{{\scriptstyle 2:1}}{{\longrightarrow}}&U\\\
\curvearrowright&&\curvearrowright\\\ \bar{V}&\stackrel{{\scriptstyle
2:1}}{{\longrightarrow}}&\mathbb{P}^{2}\\\ \end{array}$

Since the general fiber of $X$ is semi-stable, the fibration
${\tilde{Y}}\rightarrow V$ extends to a minimal family of elliptic curves
$\pi:{\mathcal{E}}\rightarrow\bar{V}$ with a general fiber being semi-stable.
The singular fibers are from Kodaira’s list of degenerations of elliptic
curves. In codimension one there are no multiple fibers. In codimension two
there might be multiple fibers, the fiber dimension can jump or $\mathcal{E}$
might not be even defined. However, we are interested in codimension one.
Notice that $\mathcal{E}$ is a Néron model [3] since the fibers are abelian
varieties. There is an induced section $\sigma$ of the fibration.

Since the map $f:\bar{V}\rightarrow\mathbb{P}^{2}$ is 2:1, there is an
involution $i$ acting on $\bar{V}$ which interchanges the sheets of the
fibers. The involution is well defined on $V$ and from the fiber diagram above
it is well defined on $\bar{V}$ as well, because with $i^{-1}V$ we can
construct a similar fiber diagram since the maps to $\mathbb{P}^{2}$ are the
same. Therefore there will be an involution on $\bar{V}$ compatible with the
involution on $V$. We recall the data from the Introduction: denote the
branched locus of $f$ by $D$ and $f^{-1}(D)=\tilde{D}$. Let $G$ be the
discriminant locus of $\pi:{\mathcal{E}}\rightarrow\bar{V}$. Note that the
intersection $G\cap\tilde{D}$ consists of finitely many points. Indeed, if it
wasn’t true, then $f(G)$ and $D$ would have a whole component in common. The
fibers of $p$ above this component would be very degenerate (they will be
products of two degenerate elliptic curves). According to the genericity
conditions, in codimension one the fibers of the original fibration have at
worst simple normal crossing singularities, so this cannot happen.

The section $\sigma$ induces a section ($\sigma$, $\sigma\circ i$) of the map:

$pr_{\bar{V}}:{\mathcal{E}}\times_{\bar{V}}i^{\ast}{\mathcal{E}}\rightarrow\bar{V}$

and the involution $i:\bar{V}\circlearrowleft$ induces an involution
$i_{\mathcal{E}}$ on ${\mathcal{E}}\times_{\bar{V}}i^{\ast}{\mathcal{E}}$.

Consider the non-branched locus $\mathbb{P}^{2}-D$ and its pre-image $X_{0}$
in $X$. Denote the induced fibration by
$p_{0}:X_{0}\rightarrow\mathbb{P}^{2}-D$. By construction,

(1)
$X_{0}\cong{\mathcal{E}}\times_{\bar{V}}i^{\ast}{\mathcal{E}}|_{V}/i_{\mathcal{E}}$

Let $L\subset\mathbb{P}^{2}$ be a line intersecting $D$ transversally at
general points of $D$. Denote the pre-image of $L$ in $\bar{V}$ by $L_{V}$.
The restricted map $f_{V}:L_{V}\rightarrow L$ is 2:1. We choose $L$ general so
that $L_{V}$ doesn’t intersect $\tilde{D}\cap G$, so every fiber of
${\mathcal{E}}_{L_{V}}\times_{L_{V}}i^{\ast}{\mathcal{E}}_{L_{V}}\rightarrow
L_{V}$ over a point of $L_{V}\cap\tilde{D}$ is smooth. After pulling back our
construction to $L_{V}$, there a rational morphism:

$L_{V}\times_{\mathbb{P}^{2}}X\dashrightarrow{\mathcal{E}}_{L_{V}}\times_{L_{V}}i^{\ast}{\mathcal{E}}_{L_{V}}$

which by Weil’s extension theorem [3] is regular on the smooth locus of
$L_{V}\times_{\mathbb{P}^{2}}X\rightarrow L_{V}$ since the fibers are abelian
varieties.

Every section of $p:X\rightarrow\mathbb{P}^{2}$ is contained in the smooth
locus of $X\rightarrow\mathbb{P}^{2}$, so it pulls-back to a curve in the
smooth locus of $L_{V}\times_{\mathbb{P}^{2}}X\rightarrow L_{V}$ via the
section $\tau f_{V}$.

Since ${\mathcal{K}}_{X}={\mathcal{O}}_{X}$, the relative sheaf of
$p:X\rightarrow\mathbb{P}^{2}$ is
$\omega_{X/\mathbb{P}^{2}}=p^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(3)$ and
therefore
$\tau^{\ast}\omega_{X/\mathbb{P}^{2}}={\mathcal{O}}_{\mathbb{P}^{2}}(3)$.

Consider the relative tangent bundle with the natural isomorphism:

$(\tau f_{V})^{\ast}N_{\tau
f_{V}(L_{V})/L_{V}\times_{\mathbb{P}^{2}}X}=f_{V}^{\ast}(\tau^{\ast}T_{X/\mathbb{P}^{2}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\sigma,\sigma\circ
i)^{\ast}T_{{\mathcal{E}}\times i^{\ast}{\mathcal{E}}/\bar{V}}|_{L_{V}}$

For every point $p\in L_{V}\cap\tilde{D}$, the map is $L_{1}\oplus
L_{2}\rightarrow L_{1}\oplus L_{2}(p)$, where $L_{1}\in
T_{\Delta_{\mathcal{E}}}|_{p}$ and $L_{2}\in
N_{\Delta_{\mathcal{E}}/{\mathcal{E}}\times{\mathcal{E}}}\cong
T_{\mathcal{E}}$, and $\Delta_{\mathcal{E}}$ is the diagonal. Outside the
branched locus, the map is the identity.

From the above we get:

$f_{V}^{\ast}(\tau^{\ast}\Lambda^{2}T_{X/\mathbb{P}^{2}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\sigma,\sigma\circ
i)^{\ast}\Lambda^{2}T_{{\mathcal{E}}\times
i^{\ast}{\mathcal{E}}/\bar{V}}|_{L_{V}},$

or equivalently,

$f_{V}^{\ast}{{\omega}_{X/\mathbb{P}^{2}}}\check{}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\sigma,\sigma\circ
i)^{\ast}\omega_{{\mathcal{E}}\times
i^{\ast}{\mathcal{E}}/{\bar{V}}}\check{}~{}|_{L_{V}}$

Also, by the description of the map above, after twisting with the branched
locus, we have:

(2) $\omega_{{\mathcal{E}}\times i^{\ast}{\mathcal{E}}/\bar{V}}|_{L_{V}}\cong
f_{V}^{\ast}\omega_{X/\mathbb{P}^{2}}(-L_{V}\cap\tilde{D})\cong
f^{\ast}[{\mathcal{O}}_{\mathbb{P}^{2}}(3)(-\frac{1}{2}D)]|_{L_{V}},$

and the last isomorphism holds because $\tilde{D}\rightarrow D$ is 2:1.

However, $\omega_{{\mathcal{E}}\times
i^{\ast}{\mathcal{E}}/\bar{V}}=\omega_{{\mathcal{E}}/\bar{V}}\otimes\omega_{i^{\ast}{\mathcal{E}}/\bar{V}}$.
Denote by $\bar{\mathcal{M}}_{1,1}$ the coarse moduli space of marked elliptic
curves. The singular locus of a family of elliptic curves maps to the boundary
of $\bar{\mathcal{M}}_{1,1}$. If we denote the pull-back of the boundary of
$\bar{\mathcal{M}}_{1,1}$ by $\delta$, then it is well known that
$\omega_{{\mathcal{E}}/\bar{V}}=\frac{\delta}{12}$ (see for example [6]).

Therefore,

(3) $(\sigma,\sigma\circ i)^{\ast}\omega_{{\mathcal{E}}\times
i^{\ast}{\mathcal{E}}/\bar{V}}\cong{\mathcal{O}}_{\bar{V}}\Big{(}\frac{G+i^{-1}G}{12}\Big{)}={\mathcal{O}}_{\bar{V}}\Big{(}\frac{f^{-1}(f(G))}{12}\Big{)}=f^{\ast}{\mathcal{O}}_{\mathbb{P}^{2}}\Big{(}\frac{f(G)}{12}\Big{)}$

When we compare the isomorphisms (2) and (3), we get:

$f^{\ast}{\mathcal{O}}_{\mathbb{P}^{2}}\Big{(}\frac{f(G)}{12}\Big{)}|_{L_{V}}\cong
f^{\ast}[{\mathcal{O}}_{\mathbb{P}^{2}}(3)(-\frac{1}{2}D)]|_{L_{V}},$

or equivalently,

$f^{\ast}{\mathcal{O}}_{\mathbb{P}^{2}}\Big{(}\frac{D}{2}+\frac{f(G)}{12}\Big{)}|_{L_{V}}\cong
f^{\ast}{\mathcal{O}}_{\mathbb{P}^{2}}(3)|_{L_{V}}$

Comparing the degrees, we obtain the relation:

$\frac{1}{2}deg(D)+\frac{1}{12}degf(G)=3$

The degrees of $D$ and $f(G)$ are positive even integers (otherwise we would
have trivial fibrations), hence there are two possibilities:
$(deg(D),deg(G))=(2,24)$ or $(4,12)$.

Case 1: $(deg(D),deg(f(G)))=(4,12)$

First we consider the case when $D$ is smooth. Since $deg(D)=4$, $\bar{V}$ is
a del Pezzo surface ($K_{\bar{V}}<0$). We want to show that ${\mathcal{E}}$ is
rationally connected. Take two general points $p,q\in{\mathcal{E}}$. Then
$f(\pi(p)),f(\pi(q))$ are two general points in $\mathbb{P}^{2}$. Then
$f(\pi(p))\in L_{p}$, where $L_{p}$ is a tangent line to $D$ at $f(\pi(p))$
and $f(\pi(q))\in L_{q}$, where $L_{q}$ is also a tangent line. Let $L_{p}\cap
L_{q}=\\{r\\}$.

Take a tangent line $L$ to $D$ and pull it back to $\bar{V}$:
$L_{\bar{V}}\doteq L\times_{\mathbb{P}^{2}}\bar{V}$. Its normalization is
$\tilde{L}_{\bar{V}}$ and ${\mathcal{E}}|_{\tilde{L}_{\bar{V}}}$ is an
elliptic fibration over $\mathbb{P}^{1}$ with 12 nodal fibers. The surface is
rational, because it is deformation equivalent to $\mathbb{P}^{2}$ blown-up at
9 points in the base locus of a pencil of plane cubics ([1], section 5.12).

We can lift the lines $L_{p}$ and $L_{q}$ to ${\mathcal{E}}$ and get
${\mathcal{E}}|_{\tilde{L}_{p,\bar{V}}}$ and
${\mathcal{E}}|_{\tilde{L}_{q,\bar{V}}}$ which are rational surfaces. We can
connect any two points on a rational surface with a rational curve. Connect
$p$ to $\tilde{r}$ and $q$ to $\tilde{r}$, where $\tilde{r}\in(f\pi)^{-1}(r)$,
so $p$ is rationally chain connected to $q$.

The point $\tilde{r}$ is a smooth point of ${\mathcal{E}}$. In characteristic
0, we can smooth the nodal rational curve if we have a general pair $p,q$ and
if $\tilde{r}$ is smooth. Therefore, ${\mathcal{E}}$ is rationally connected.

Fix a section $s$ of $i^{\ast}{\mathcal{E}}\rightarrow\bar{V}$. Then we get a
section $\tilde{s}$ of
${\mathcal{E}}\times_{\bar{V}}i^{\ast}{\mathcal{E}}\rightarrow{\mathcal{E}}$.
Since we have a finite morphism
${\mathcal{E}}\times_{\bar{V}}i^{\ast}{\mathcal{E}}\rightarrow X$, we get a
finite morphism ${\mathcal{E}}\rightarrow X$.

The image of a finite morphism from a rationally connected variety to a hyper-
Kähler manifold is of dimension at most $\frac{1}{2}dim(X)$, because we have a
$(2,0)$ form on $X$ and there are no holomorphic $(2,0)$ or $(1,0)$ forms on a
rational variety. However, $dim({\mathcal{E}})=3$ and it is bigger than
$\frac{1}{2}dim(X)=2$ \- a contradiction.

Now assume $D$ is singular. Consider the linear system of lines $l$ containing
a fixed singular point $r$ of $D$. Consider the associated linear system
$f^{-1}(l)$. These divisors will typically be all singular because $\bar{V}$
is singular at $f^{-1}(r)$. However, the linear system of strict transforms
$\widetilde{f^{-1}(l)}$ on $Bl_{f^{-1}(r)}{\bar{V}}$ is a basepoint free
pencil of divisors on the normal surface $Bl_{f^{-1}(r)}{\bar{V}}$. Thus, by
Bertini’s theorem, a general member of this pencil is smooth and intersects
$G$ transversally. In particular, the surface
${\mathcal{E}}|_{\widetilde{f^{-1}(l)}}$ is smooth for such a member.

Let $L$ be any line passing through a singular point $r\in D$. Every component
of $f^{-1}(L)$ is a rational curve. Without loss of generality we shall
consider the irreducible case. The normalization $\widetilde{f^{-1}(L)}$ is
isomorphic to $\mathbb{P}^{1}$. Therefore,
${\mathcal{E}}|_{\widetilde{f^{-1}(L)}}$ is an elliptic surface over
$\mathbb{P}^{1}$. Since there is the following relation of intersection
numbers: $f^{-1}(L)\cdot G=L\cdot f(G)=12$, it follows that
${\mathcal{E}}|_{\widetilde{f^{-1}(L)}}$ is a rational surface.

Consider the 1-parameter family $\mathcal{F}$ of such rational surfaces
parametrized by $\mathbb{P}^{1}$ (since the line $L$ varies in
$\mathbb{P}^{1}$). The 3-fold $\mathcal{F}$ is rationally connected inside a
4-dimensional hyper-Kähler manifold which is impossible.

We ruled out the first case completely and the only remaining case is:

Case 2: $(deg(D),deg(f(G)))=(2,24)$

First we consider the case when $D$ is smooth. Then
$\bar{V}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Take a tangent line $L$ to the conic and pull it back to $\bar{V}$:
$L_{\bar{V}}\doteq L\times_{\mathbb{P}^{2}}\bar{V}$. Its normalization
$\tilde{L}_{\bar{V}}$ is reducible and consists of two copies of
$\mathbb{P}^{1}$, say $\tilde{L}_{\bar{V},1}$ and $\tilde{L}_{\bar{V},2}$.

Case 2.1:
${\mathcal{E}}|_{\tilde{L}_{\bar{V},k}}$ is an elliptic fibration over
$\mathbb{P}^{1}$ with 12 nodal fibers, $k=1,2$. Then we repeat the same
argument as in Case 1 in order to exclude this case.

Case 2.2:
${\mathcal{E}}|_{\tilde{L}_{\bar{V},1}}$ is an elliptic fibration over
$\mathbb{P}^{1}$ with 24 nodal fibers and
${\mathcal{E}}|_{\tilde{L}_{\bar{V},2}}$ is an elliptic fibration over
$\mathbb{P}^{1}$ with no singular fibers. Then
${\mathcal{E}}|_{\tilde{L}_{\bar{V},2}}$ is the trivial fibration and
therefore, ${\mathcal{E}}$ is the pull back of an elliptic fibration on
$\mathbb{P}^{1}$ through the projection on this factor. And, since the
elliptic fibration on $\mathbb{P}^{1}$ has 24 nodal fibers, it is an elliptic
K3 surface $S\rightarrow\mathbb{P}^{1}$.

After considering all the cases, we see that:

${\mathcal{E}}\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}i^{\ast}{\mathcal{E}}=(S\times\mathbb{P}^{1})\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}(\mathbb{P}^{1}\times
S).$

We want to prove that
$(S\times\mathbb{P}^{1})\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}(\mathbb{P}^{1}\times
S)\cong S\times S$. Indeed, we have the following commutative fiber diagram:

$\begin{array}[]{ccc}S\times S&\longrightarrow&\mathbb{P}^{1}\times S\\\
\downarrow&\searrow&\downarrow\\\
S\times\mathbb{P}^{1}&\longrightarrow&\mathbb{P}^{1}\times\mathbb{P}^{1}\\\
\end{array}$

Therefore,

${\mathcal{E}}\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}i^{\ast}{\mathcal{E}}=(S\times\mathbb{P}^{1})\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}(\mathbb{P}^{1}\times
S)\cong S\times S.$

But $X$ is birational to the desingularization of
${\mathcal{E}}\times_{(\mathbb{P}^{1}\times\mathbb{P}^{1})}i^{\ast}{\mathcal{E}}/{\tilde{i}}$
(by (1)) which is birational to $S\times S/{\mathbb{Z}_{2}}$. Therefore, $X$
is birational to the desingularization of $S\times S/{\mathbb{Z}_{2}}$ which
is $\text{Hilb}^{2}(S)$ by Fogarty’s theorem (Theorem 2.6.)

Now consider the case when $D$ is singular. Since $\bar{V}$ is normal, $D$
cannot be a double line. Therefore, $D$ is the union of two lines and
$\bar{V}$ is a singular quadric cone.

Let $L$ be a line in ${\mathbb{P}}^{2}$ passing through the node $r\in D$.
Then $f^{-1}(L)$ is the union of two lines $L_{1}$ and $L_{2}$ each one of
which is a line in the cone $\bar{V}$ passing through its vertex. On a quadric
cone all lines are algebracially equivalent, and in particular $L_{1}$ and
$L_{2}$ are algebraically equivalent. Since $deg(G)=24$, $L_{1}\cdot
G=L_{2}\cdot G=12$. Therefore, the surface ${\mathcal{E}}|_{L_{i}}$ is
rational (for $i=1,2$). We constructed a rationally parametrized 1-parameter
family of rational surfaces which is impossible to exist in a 4-dimensional
hyper-Kähler manifold. With this we finish the proof of our main theorem.

∎

Acknowledgments. The author is thankful to Prof. G. Tian for suggesting the
problem and for the helpful discussions. She also thanks the Institute of
Advanced Study for the opportunity to be there during the 2006-07 academic
year.

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