\magnification=\magstep1
\input amstex.tex
\documentstyle{amsppt}
\TagsOnRight
\hsize=16truecm
\def\({\left(}
\def\[{\left[}
\def\){\right)}
\def\]{\right]}
\def\e{\varepsilon}
\def\z{{{\Bbb Z}^2}}
\def\r{{\Bbb R}^2}
\def\rpc{\(R+\dfrac c{R}\)}
\def\rmc{\(R-\dfrac c{R}\)}
\def\rpmc{\(R\pm\dfrac cR\)}
\def\const{\text{\rm const.}\,}
% \def\Ar#1{\,\text{\rm Area}\,\(#1\)}
\def\Ar#1{\left|#1\right|}
\def\f{\varphi}
\def\roz{\Bbb {O}_R}
\def\a{\alpha}
\def\ro{\Bbb {O}_R(c,\a)}
\def\s{\sigma}
\def\g{\gamma}
\def\zz{\bold z}
\def\zpl#1{\bold z_{{#1},l}^+}
\def\zpr#1{\bold z_{{#1},r}^+}
\def\zml#1{\bold z_{{#1},l}^-}
\def\zmr#1{\bold z_{{#1},r}^-}
\def\zpml#1{\bold z_{{#1},l}^{\pm}}
\def\zpmr#1{\bold z_{{#1},r}^{\pm}}
\def\Kp#1{K^{+}_{#1}(\zz)}
\def\Km#1{K^{-}_{#1}(\zz)}
\def\Kpm#1{K^{\pm}_{#1}(\zz)}
%\def\01{[0,1]\times[0,1]}
\def\01{[0,1]^2}
\def\A{{\bold A}}
\def\m{\bold m}
\def\at#1{\A^{(2)}(#1)}
\def\xl#1{x_{l,{#1}}(u)}
\def\xr#1{x_{r,{#1}}(u)}
\def\rp{{R^+}}
\def\rmm{{R^-}}
 
 
\leftheadtext{Zheming Cheng, Joel L. Lebowitz and P\'eter Major}
\rightheadtext{On the number of lattice points}
 
\topmatter
\title On the number of lattice points between two\\
enlarged and randomly shifted copies of an oval\endtitle
\author Zheming Cheng$^{(1)}$, Joel L. Lebowitz$^{(2)}$ and P\'eter
Major$^{(3)}$\endauthor
\affil $^{(1)}$Rutgers University, Department of Mathematics,\\
New Brunswick, NJ 08903, USA\\
$^{(2)}$Rutgers University, Department of Mathematics and
Physics,\\ New Brunswick, NJ 08903, USA\\
$^{(3)}$Mathematical Institute of the Hungarian Academy of
Sciences,\\
P.O.B. 127, H--1364, Budapest, Hungary\endaffil
\endtopmatter  \noindent
{\narrower{\narrower
{\it Summary:} Let $\A$ be an oval with a nice boundary in $\r$, $R$ a
large positive number, $c>0$ some fixed number and $\a$  a uniformly
distributed random vector in the unit square $\01$.
We are interested in the number of lattice points in the
shifted annular region consisting of the difference of the sets
$\left\{\rpc \A-\a\right\}$   and   $\left\{\rmc A-\a\right\}$.
We prove that when $R$ tends to infinity,  the expectation and
the variance of this random variable tend to $4c$ times the area of
the set $\A$, i.e.\ to the area of the domain where we are
counting the number of lattice points. This is consistent with
computer studies in the case of a circle or an ellipse which indicate
that the distribution of this random variable tends to the Poisson
law. We also make some comments about possible
generalizations.\par}\par}
 
\beginsection 1. Introduction
 
Using computer simulation Cheng and Lebowitz [4] studied the
distribution of the number of lattice points in
the domain between two concentric circles of radii $R+\dfrac cR$
and $R-\dfrac cR$  whose center is uniformly distributed on the
unit square $\01$. (By lattice points we mean
points from $\z$, i.e.\ from the set of points in $\r$ with
integer coordinates.) This computer study, motivated by works of Sinai
[9], ~[10], Bleher ~[2] and Major ~[8], suggested that for large $R$ this
distribution is asymptotically Poissonian with parameter $4\pi c$, i.e.\ with
the area of the domain where we are counting the number of lattice
points. A first step to check the correctness of this
statement is to investigate whether the variance of this
distribution is asymptotically $4\pi c$, i.e.\ whether the variance
behaves as the Poissonian limit suggests. We answer  this question in
the affirmative. A similar statement
holds for a class of ovals defined as follows.
\proclaim {Definition of an oval} \it
A closed bounded convex set $\A$
is an oval in $\r$ if it contains the origin in its
interior, and its boundary is a smooth four times
differentiable Jordan curve whose curvature is positive at all
points.
\endproclaim
We also introduce the following notations.  Let $\Ar \A$ denote the area
of a measurable set $\A$ in $\r$. Given some set $\A\subset\r$,
$\a\in\r$ and number $R\in {\Bbb R}^+$  define the set $R\A-\a$ as
$$
R\A-\a=\{\bold u\in\r;\quad \bold u=R\bold v-\a,\;\bold v\in\A\}
$$
and for $c>0$, $R^2>c$ introduce the difference  set
$$
\ro=\[\rpc\A-\a\]\setminus\[\rmc\A-\a\]. \tag1.1
$$
Clearly, $|\ro|=4c|\A|$ for all $\a$. The following Theorem is the
main result of this paper.
\proclaim {Theorem} \it
Let $\A$ be an oval, $c>0$ some fixed
positive number and $\a$ a uniformly
distributed random variable on $\01$. For $R>\sqrt c$ let
$\xi_{R}=\xi_{R}(\a)$
denote the number of lattice points in the set $\ro$ defined in
(1.1). Then the relations
$$
\align
E\xi_{R}&=\Ar {\Bbb O_R}=4c|\A|  \tag1.2 \\
\lim_{R\to\infty}\text{\rm Var}\, \xi_{R}&=\Ar{\Bbb O_R}=4c|\A| \tag1.3
\endalign
$$
hold. Here $\Bbb O_R$ denotes $\ro$ with $\a=0$.
\endproclaim
 
The investigation of the number of lattice points in a domain is a
popular subject in number theory. See e.g.\ [6] for a recent
treatment or [7]. This
problem also has  physical motivations, relating to the
investigation of the statistics of eigenvalues in a quantum system
with an integrable classical Hamiltonian. For example, if $\A$ is a circle, the
lattice points $\bold n$ label energy levels $E(\bold n;\a)=|\bold
n-\a|^2$ of the Laplacian $-(\nabla-\a)^2$  on the unit torus. These
energies can be thought of  as points on the real line and their
statistics  can be studied. This problem seems to be very hard. An
easier problem is to consider not a fixed $\a$ but a random one
distributed uniformly on the unit square $\01$, and this is what we
have done. A widely accepted conjecture in the physics
community is that the distribution of levels is, for typical systems
in this class, locally Poissonian [1], i.e.\ the statistics of the
energy levels in the interval $[E,E+L]$, $L$ is fixed and $E$ is uniformly
distributed in an interval $[0,T]$ with $T\to\infty$, behave like
Poisson distributed points with density ~$\pi$.
 In our context the conjecture
is the following:
 
Let $P(n;R)$ be the probability that there are exactly $n$ lattice
points in $\ro$. Then
$$
\lim\limits_{R\to\infty} P(n,R)= p(n) \quad\text{with }
p(n)=\dfrac{(4c\Ar {\bold
A} )^n}{n!}e^{-4c\Ar {\bold A} n}\;.\tag1.4
$$
Such a result was proved by Sinai [8] and Major [9] for the
number of lattice points in scaled annuli domains
bounded by {\it very\/} random curves.  Here all the randomness comes
from $\a$, so the proof of (1.4), if indeed it is true, is far from
trivial. Our theorem proves that
the limit of the first and second moment of $P(n;R)$ has the right
behavior when $R\to\infty$. In fact, our result also shows that the
covariance of pairs of distinct
random variables $\eta_R^{(j)}=\xi_{R+jc}-\Ar {{\Bbb O}_{R}}$ vanishes
as $R\to\infty$. This suggests that
$\lim\limits_{c\to\infty}\lim\limits_{R\to\infty} \dfrac{\xi_R-\Ar {{\Bbb
O}_R}}{\sqrt{\Ar {{\Bbb O}_R}}}$ should be a Gaussian random variable.
This is consistent with taking the large parameter limit of the
Poissonian distribution, but may be valid more generally.
 
\beginsection 2. Proof of the Theorem
 
The proof of relation (1.2) is simple. We can write $E\xi_R(\a)$ as
the the sum of the probabilities
$$
E\xi_R(\a)=\sum_{\m\in\z}P(\m\in\ro)=\sum_{\m\in\z}\Ar{\Bbb
O_R\cap (\01-\m)}=\Ar{\Bbb O_R}\;.
$$
Because of (1.2)
formula (1.3) is equivalent to the relation
$$
\lim_{R\to\infty}E\xi_R(\a)(\xi_R(\a)-1)= \Ar{\roz}^2 \;.\tag2.1
$$
We claim that
$$
E\xi_R(\a)(\xi_R(\a)-1)= \sum_{\m\in\z\setminus\{
0\}}\Ar{\roz\cap(\roz-\m)}\;.\tag2.2
$$
Indeed,
$$
\align
E\xi_R(\a)(\xi_R(\a)-1)&= \sum_{\m\in\z}
 \sum_{{\m_1}\in\z\setminus\{\bold m\}}P\(\m\in\ro,\;\m_1\in\ro\)\\
&= \sum_{\m\in\z}
 \sum_{{\m_1}\in\z\setminus\{\bold m\}}\Ar{\01\cap\(\roz-\m\)
\cap\(\roz-\m_1\)}      \\
&= \sum_{\m\in\z}
 \sum_{{\m_1}\in\z\setminus\{0\}}\Ar{(\01+\m)\cap\roz
\cap\(\roz-\m_1\)}      \\
&=\sum_{\m\in\z\setminus\{0\}}\Ar{\roz\cap\(\roz-\m\)}\;.
\endalign
$$
Hence to prove the Theorem it is enough to prove relation (2.1) with the
help of relation (2.2). This requires a good
estimate on the area of $\roz\cap(\roz-\m)$. First we introduce
some notations.
 
Let us denote the boundary of the set $t\A$ for $t>0$, by $\g_t$,
and let $\g=\g_1$. For some $\bold x\in\g_t$ let $\f(\bold x)$ denote
the angle of the vector
$\bold x$ and
$\psi(\bold x)$ the angle of the normal of the curve $\g_t$ at
$\bold x$ (pointing outside of the domain $t\A$) with the vector
$\bold
e_1=(1,0)$.
Given some $\zz\in \r\setminus\{0\}$ let $d(\zz,t)$ denote the
diameter of the set $t\A$ in the direction $\zz$, i.e.
$$
d(\zz,t)=\max\left\{|\zz_1-\zz_2|;\quad\zz_1\in \g_t,\;\zz_2\in \g_t,\;
\zz_1-\zz_2=\lambda\zz,\text{ with some }\lambda>0\right\}\;.
$$
Let $\zz_{d,r}(t)$  and $\zz_{d,l}(t)$ be the end points of this
maximal vector, i.e. $\zz_{d,r}(t) \in\g_t$, $\zz_{d,l}(t)
\in\g_t$, \
$\zz_{d,r}(t)-\zz_{d,l}(t)=\lambda\zz$ with $\lambda>0$ and
$|\zz_{d,r}(t)-\zz_{d,r}(t)| =d(\zz,t)$.
For $\zz\in\r$ let $\zz^{\bot}$ denote the vector $\zz$ rotated by
$+\pi/2$,
and define $\Kp t$ and $\Km t$  as the
half planes whose boundary is the
line going through the points $\zz_{d,r}(t)$ and $\zz_{d,l}(t)$
and which are in
the direction $\zz^{\bot}$ and $-\zz^{\bot}$ of this line respectively.
For $\zz\in\r$ and $0<|\zz|\leq d(\zz,t)$ define the (unique)
points
$\zpr t,\;\zpl t\in\Kp t\cap \g_t$ and $\zmr t,\;\zml t\in\Km t\cap \g_t$  such that $\zpr t-\zpl t=\zz$
and $\zmr t-\zml t=\zz$.
 
For $\zz_1,\;\zz_2\in\g_R$ define the function
$$
F_R(\zz_1,\zz_2)=\frac{4c^2}{R^4}\frac{|\zz_1| |\zz_2|
\cos(\f(\zz_1)-\psi(\zz_1))\cos(\f(\zz_2)-\psi(\zz_2))}
{|\sin(\psi(\zz_1)-\psi(\zz_2))|}
$$
and for some $\zz\in\r$, $0<|\zz|\leq d(\zz,R)$ the functions
$$
f^+_R(\zz)=F_R(\zpl R,\,\zpr R)\;,\qquad f^-_R(\zz)=F_R(\zml
R,\,\zmr R)\;. $$
 
For $\A\subset \z$ and $\bold B\subset \z$ define their sum
$$
\A+\bold B=\{\bold x+\bold y,\quad \bold x\in\A,\;\bold y\in\bold B\}
$$
and
$$
\at R=R\A+(-R\A)\;.
$$
We claim that
$$
\int_{\at R}\[f^+_R(\zz)+f^-_R(\zz)\]\,d\zz=16c^2\Ar{\A}^2\;.\tag2.3
$$
Put
$$
h_R(\zz)=\Ar{\roz\cap\(\roz-\zz\)}\;,\quad \zz\in\r\;.\tag2.4
$$
We then also  claim that
$$
\lim_{R\to\infty}\left\{\sum_{\m\in\z\setminus\{0\}}h_R(\m)-
\int_{\at R}\[f^+_R(\zz)+f^-_R(\zz)\]\,dz\right\}=0\;.\tag2.5
$$
 
Relations (2.2), (2.3) and (2.5) together imply (2.1) hence also the
Theorem. To prove (2.3) we introduce the maps
$$
\align
&G^{\pm}\: \at R\setminus\{0\} \mapsto \g_R\times \g_R\;,\\
&G^+(\zz)=(\zpl R,\zpr R),\quad G^-(\zz)=(\zml R,\zmr R)
\endalign
$$
Observe that the set $\text {Int}\,G^+(\at R\setminus\{0\})
\cap\,\text {Int}\,G^-(\at R\setminus\{0\})$ is empty, the maps $G^\pm$
are diffeomorphisms on $\text {Int}\,\at R\setminus\{0\}$, and
$G^+(\at R\setminus\{0\})
\cup G^-(\at R\setminus\{0\})$ is $\g_R\times\g_R\setminus\{(\zz,\zz),
\;\zz\in \g_r\}$.
 
The inverses of the maps $G^\pm(\zz)$ have Jacobians $\left|\sin
\(\psi( \zpmr R)- \psi(\zpml R)\)\right|$. To see this we make the
following
observation: Let $[\zz_1, \zz_1+d\zz_1]$ and $[\zz_2, \zz_2+d\zz_2]$
be two small curves on $\g_R$ starting from some points $\zz_1$ and
$\zz_2$ respectively. Then the inverse of the map $G^\pm$ maps the set
$[\zz_1, \zz_1+d\zz_1]\times[\zz_2, \zz_2+d\zz_2]$ approximately to
$\zz_2-\zz_1+\Delta(d\zz_1,d\zz_2)$, where $\Delta(d\zz_1,d\zz_2)$ is a
parallelogram with one vertex at the origin, whose sides are the vectors
$d\zz_1$ and $d\zz_2$. The area of this parallelogram is $|d\zz_1|
|d\zz_2||\sin(\psi(\zz_1)-\psi(\zz_2))|$. We can  approximate the
area of the image of the above domain by the inverse map
$\(G^\pm\)^{-1}$ with the area of this parallelogram. Since this
approximation gives only an error of order $o(|d\zz_1||d\zz_2|)$ the
Jacobian has the form we have stated.
 
The above relations imply that
$$
\align
&\int_{\at R}\[f^+_R(\zz)+f^-_R(\zz)\]\,d\zz\\
&\qquad=\int_{\g_R\times\g_R}\frac{4c^2}{R^4}|\zz_1|
|\zz_2|\cos(\psi(\zz_1)-\f(\zz_1))  \cos(\psi(\zz_2)-\f( \zz_2))
\,d\zz_1\,d\zz_2\\
&\qquad=\[\frac{2c}{R^2}\int_{\g_R}|\zz| \cos(\psi(\zz)
-\f(\zz))\,d\zz\]^2\\
&\qquad=\[2c\int_{\g_1}|\zz|
\cos(\psi(\zz)-\f(\zz))\,d\zz\]^2 =16c^2\Ar{\A}^2\;,
\endalign
$$
hence relation (2.3) holds. To prove relation (2.5) we need some
geometrical facts formulated in relations (2.9) and (2.10) and a lemma
about the value of $h_R(\zz)$. They will be proved in
the next Section.
\proclaim {Lemma 1} \it
There is some $\e>0$ such that the function $h_R(\zz)$ defined in (2.4)
satisfies the following estimates:
\medskip
\item{a)} For $1\le|\zz|<\e R$, \quad $h_R(\zz)<\dfrac{\const}{R\,|\zz|}$.
\item{b)} For all $0<\eta\le\e$ and $\eta R\le |\zz|\le (1-\eta)d(\zz,R)$
$$
R^2\left\{h_R(\zz)-\[f^+_R(\zz)+f^-_R(\zz)\]\right\}\to 0\quad\text{as }
R\to \infty\;,
$$
and the convergence is uniform in $\zz$.
\item{c)} Let us fix some positive constant $B>0$. Then the following
inequalities hold:
\itemitem{c1)} $h_R(\zz)<\dfrac{\const}{R^{3/2}\sqrt{d(\zz,R)-|\zz|}}$
\quad if $(1-\e)d(\zz,R)\leq|\zz|\leq d(\zz,R)-\dfrac BR$.
\itemitem{c2)} $h_R(\zz)<\dfrac{\const}R$ \quad if
$d(\zz,R)-\dfrac BR\leq|\zz|\leq d(\zz,R)+\dfrac BR$.
\item{d)} $h_R(\zz)=0$ if $|\zz|>d(\zz,R)+\dfrac BR$, and $B$ is larger
than $c$ times the diameter of the set $\A$.
\endproclaim
To prove relation (2.5) let us introduce the sets
$$
\align
D^\e_1(R)&=\left\{\zz\in\r,\; 0<|\zz|<\e R\right\}\\
D^\e_2(R)&=(1-\e)\at R\setminus D^\e_1(R)\\
D^\e_3(R)&=\(1-\dfrac{B}{R^2}\)\at R\setminus(1-\e)\at R\\
D^\e_4(R)&=\(1+\dfrac{B}{R^2}\)\at R\setminus\(1-\dfrac{B}{R^2}\)\at
R\;. \endalign
$$
Define the discrete measure $\mu$ on the positive half-line
$[0,\infty]$,
$$
\mu([0,x])=\left\{\text{the number of lattice points in the set
}x\A^{(2)} \right\}\;,
$$
where $\A^{(2)}$ denotes $\at R$ with $R=1$. Define also the signed
measure
$$
\nu([0,x])=\mu([0,x])-x^2\text{Area}\,(\A^{(2)})\;.
$$
If $\A$ is an oval, then $\A^{(2)}$ is again an oval. (See
[3]). (This means that the boundary of $\A^{(2)}$ is again strictly
convex, smooth, and has positive curvature at all points.) Hence the
results known for ovals can be applied to $\A^{(2)}$. In particular,
we can
state because of a result of Colin de Verdi\`ere [5] that
$$
|\nu([0,x])|<\const x^{2/3}\quad\text{for all }x>1\; .\tag2.6
$$
 
Let us also remark that the normals of $\g_R$ in the points
$\zz_{d,r}(R)$ and $\zz_{d,l}(R)$ satisfy the relation
$$
\psi(\zz_{d,r}(R))=\psi(\zz_{d,l}(R))+\pi \tag2.7
$$
for all $\zz\in\r\setminus\{0\}$, i.e.\ the normals in the points
$\zz_{d,r}(R)$ and $\zz_{d,l}(R)$ point in opposite
directions. The half-line $\lambda\zz$, $\lambda>0$, intersects the
boundary of $\A^{(2)}(R)$
at distance $d(\zz,R)$ from the origin. Hence Part c1) of  Lemma ~1
bounds the value of $h_R(\zz)$ for $\zz\in D^\e_3(R)$ and Part
c2) bounds the value of $h_R(\zz)$ for $\zz\in D^\e_4(R)$.
 
By Part a) of Lemma 1 $$
\sum_{\m\in\z\cap D^\e_1(R)\setminus\{0\}} h_R(\m) <\const\frac1R
\sum\Sb\m\in\z\\ 0<|\m|<\e R \endSb \frac1{|\m|}<\const \e.
$$
Since $R^2\(f^+_R(R\zz)+f^-_R(R\zz)\)$ is uniformly continuous in the
set $\dfrac
1R D^\e_2(R)$, hence
$$
\sum_{\m\in\z\cap D^\e_2(R)} h_R(\m) -\int_{D^\e_2(R)}\[f^+_R(\zz)
+f^-_R(\zz)\]\,d\zz\to 0\quad\text{as }R\to\infty
$$
by Part b) of Lemma 1. Put
$$
H_R(t)=\sup_{\zz\in t\partial\A^{(2)}}h_R(\zz)\;,
$$
where $\partial\A^{(2)}$ denotes the boundary of $\A^{(2)}$. By
using Part c1) of Lemma~1, integrating by parts and applying (2.6)  we
can write
$$
\align
\sum_{\m\in\z\cap D^\e_3(R)} h_R(\m) &\leq\int_{(1-\e)R}^{\(R-\frac
BR \)} H_R(x)\,\mu(dx)\\
&<\frac{\const}{R^{3/2}}\int_{(1-\e)R}^{\(R-\frac BR\)}\frac x
{\sqrt{R-x}}\,dx
+\frac{\const}{R^{3/2}}\int_{(1-\e)R}^{\(R-\frac BR\)}\frac {\nu(\,dx)}
{\sqrt{R-x}}\\
&<\const\sqrt\e+\frac{\const}{R^{3/2}}\[\frac{\nu([0,x])}{\sqrt{R-x}}
\]^{\(R-\frac BR\)}_{(1-\e)R}\\
&\qquad+\const\int_{(1-\e)R}^{\(R-\frac BR\)}\frac
{\nu([0,x])}{R^{3/2}\(R-x\)^{3/2}}\,dx \\
&<\const\(\sqrt\e+R^{-1/3}+R^{-5/6}
\int_{(1-\e)R}^{\(R-\frac BR\)}\frac
{1}{\(R-x\)^{3/2}}\,dx \)\\
&<\const \[R^{-1/3}+\sqrt\e\].
\endalign
$$
Similarly, by Part c2) of Lemma 1
$$
\align
\sum_{\m\in\z\cap D^\e_4(R)} h_R(\m) &\leq\const\int_{\(R-\frac
BR\)}^{\(R+\frac BR\)}\frac 1R\, \mu(\,dx)\\
 &\leq\const\int_{\(R-\frac BR\)}^{\(R+\frac BR\)}\frac xR \,dx
 +\frac{\const}R\nu\( \[R-\frac BR,R+\frac BR\]\)\\
 &<\const \[\frac 1R+R^{-1/3}\]<\const R^{-1/3} \;.
 \endalign
$$
The above relations together with Part d) of Lemma~1 imply that
$$
\sum_{\m\in\z\setminus\{0\}}h_R(\m)-\int_{D^\e_2(R)}\[f^+_R(\zz)
+f^-_R(\zz)\]\,d\zz=O\(\sqrt\e+R^{-1/3}\)\;.  \tag2.8
$$
We claim that
$$
I_1=\int_{D^\e_1(R)}\[f^+_R(\zz)
+f^-_R(\zz)\]\,d\zz=O(\e)  \tag$2.8'$
$$
and
$$
I_2=\int_{\A^{(2)}(R)\setminus(1-\e)\A^{(2)}(R)}
\[f^+_R(\zz)+f^-_R(\zz)\]
\,d\zz=O\(\sqrt\e\)\;.  \tag$2.8''$
$$
Since $\e>0$ can be chosen arbitrary small the above relations imply
(2.5).
 
In Section 3 we shall prove the following statements. There is
some
$\e>0$ such that if $|\zz|=uR$ with some $0<u<\e$, then the normals of
$\g_R$ satisfy the inequality
$$
\left|\psi(\zpmr R)-\psi(\zpml R)\right|>\const u\;,\tag2.9
$$
and if $|\zz|=(1-u)d(\zz,R)$ with some $0<u<\e$, then
$$
\left|\psi(\zpmr R)-\psi(\zpml R)-\pi\right|>\const\sqrt u\;.\tag2.10
$$
In the first case we get that
$$
|f^+_R(\zz)| +|f^-_R(\zz)|<\const \frac1{R^2 u} \tag2.11
$$
and in the second case
$$
|f^+_R(\zz)| +|f^-_R(\zz)|<\const \frac1{R^2\sqrt u}\;. \tag2.12
$$
Thus we get, by integrating in a polar coordinate system and applying
the estimate (2.11), that
$$
I_1<\const\frac 1{R^2}\int_0^\e R^2u\frac 1u\,du<\const \e\;.
$$
Relation (2.12)  implies that for $(1-\e)R\le t\le R$
$$
\align
\oint_{t\partial \bold A^{(2)}}
\[f^+_R(\zz)+f^-_R(\zz)\]\,d\zz&\le\const
\oint_{t\partial \bold A^{(2)}} (1-t)^{-1/2}R^{-3/2}\,d\zz \\
&\le\const \frac1{\sqrt{(1-t)R}}\;.
\endalign
$$
Integrating first on the curves
$\g_t$, $(1-\e)R<t<R$, we get that
$$
I_2<\const\int^{\e R}_0 R^{-1/2}u^{-1/2}\,du<\const\sqrt \e\;.
$$
Hence we proved the Theorem with the help of Lemma~1 and formulas
(2.9) and (2.10).
 
\beginsection 3. Proof of  Lemma 1
 
We shall need a result about convex sets in the proof. To formulate it
we
introduce some notations. Let us fix some vector $\zz\in\r$, $\zz\neq0$.
Introduce the coordinate system whose coordinate axis $x$ is in the
direction $\dfrac {\zz}{|\zz|}$ and  the coordinate axis $y$ is its
rotation with $+\frac\pi 2$, in the direction $\dfrac
{\zz^\bot}{|\zz^\bot|}$. In this new coordinate system let
$(x_0(t), y_0(t))$ and $(x_1(t),
y_1(t))$ be the points of tangency of the curve $\g_t$ with the
line parallel to the vector $\zz$ in the half-spaces $\Kp t$ and
$\Km t$ respectively. For $y_1(t)<u<y_0(t)$  the line $y=u$ intersects
the curve
$\g_t$ in the points $\xl t$ and $\xr t$, \ $\xl t<\xr t$. We shall
prove the following
\proclaim {Lemma 2}  \it
There are some positive constants $A>0$, \
$0<B_1<B_2$ depending only on the curve $\g$ such that
$$
\align
B_1\sqrt{t(y_0(t)-u)}&<\xr t- x_0(t)  <B_2
\sqrt{t(y_0(t)-u)}\;,\\
B_1\sqrt{t(y_0(t)-u)}&< x_0(t)-\xl t<B_2 \sqrt{t(y_0(t)-u)} \\
\frac{B_1 \sqrt t}{\sqrt{y_0(t)-u}}&<-\frac d{du}\xr t<
\frac{B_2 \sqrt t}{\sqrt{y_0(t)-u}}\;,      \\
\frac{B_1 \sqrt t}{\sqrt{y_0(t)-u}}&<\frac d{du}\xl t<
\frac{B_2 \sqrt t}{\sqrt{y_0(t)-u}}\;,
\endalign
$$
if $y_0(t)-tA<u<y_0(t)$, and
$$
\align
B_1\sqrt{t(u-y_1(t))}&< \xr t-x_1(t)<B_2
\sqrt{t(u-y_1(t))}\;,\\
B_1\sqrt{t(u-y_1(t))}&<x_1(t)-\xl t <B_2 \sqrt{t(u-y_1(t))} \\
\frac{B_1 \sqrt t}{\sqrt{u-y_1(t)}}&<\frac d{du}\xr t<
\frac{B_2 \sqrt t}{\sqrt{u-y_1(t)}}\;,\\
\frac{B_1 \sqrt t}{\sqrt{u-y_1(t)}}&<-\frac d{du}\xl t<
\frac{B_2 \sqrt t}{\sqrt{u-y_1(t)}}\;,
\endalign
$$
if $y_1(t)<u<y_1(t)+tA$.
\endproclaim
\demo {Proof of Lemma 2}
Let us first restrict our attention to the
case $t=1$.  It is more convenient to work with the inverse of the
function $\xr 1$. Let $(x, g(x))$ be a small part of the curve $\g$
in the neighborhood of the point $(x_0(1),y_0(1))$, and let
$\rho(x)$ be the curvature of the curve $\g$ in the point
$(x,g(x))$. We can write in an interval $[x_0(1),x_0(1)+\eta]$, \
$\eta>0$,
$$
\rho(x)=-\frac{\[1+g'(x)^2\]^{3/2}}{g''(x)}\;.
$$
 
Put $z(x)=g'(x)$. Since $z(x_0(1))=0$, the last relation implies that
$$
\int_0^{z(x)}\frac{dt}{(1+t^2)^{3/2}}=-\int_{x_0(1)}^x\frac1{\rho(t)}\,dt
$$
or
$$
\frac{z(x)}{\sqrt{1+z^2(x)}}=-P(x)
$$
with $\dsize P(x)=\int_{x_0(1)}^x\frac1{\rho(t)}\,dt$. Since the
curvature
of $\g$ is separated both from zero and infinity, there are some
constants $K_2>K_1>0$ such that
$$
K_1(x-x_0(1))<P(x)<K_2(x-x_0(1))
$$
 
Since $z(x)=\dfrac{-P(x)}{\sqrt{1-P^2(x)}}$ the last relation
implies that
$$
C_1(x-x_0(1))<-g'(x)<C_2(x-x_0(1))
$$
with some $C_2>C_1>0$ in an interval $x\in[x_0(1),x_0(1)+\eta)$.
Since $g(x_0(1))=y_0(1)$ we get by integrating that
$$
-\frac{C_2}2(x-x_0(1))^2<g(x)-y_0(1)<-\frac{C_1}2(x-x_0(1))^2.
$$
These formulas imply that $\xr 1$, the inverse of $g(x)$, satisfies
Lemma 1 for $t=1$ in an interval $[y_0(1)-A,y_0(1)]$. The remaining
statements of Lemma 1 for $t=1$ can be proved in the same way. The case
of general $t>0$ follows from the case $t=1$ because of the
homogeneity properties of $\g_t$. \qed
\enddemo
Now we turn to the proof of relations (2.9) and (2.10). Here again we
can restrict our attention to the case $R=1$. Let us recall that the
curvature $\rho(\bold x)$ of $\g$ in a point $\bold x\in\g$ and the
angle of the normal $\psi(\bold x)$ in this point satisfy the
relation $$
\frac{d\bold \psi(\bold x )}{ds(\bold x)}=\rho(\bold x)\;,\tag3.1
$$
where $s(\bold x)$ is the length of the arc $(\bold x_0, \bold x)$ of
$\gamma$ with some fixed $\bold x_0\in\g$.
 
Under the conditions imposed for formula (2.9) the
length of the arc $(\zpml 1,\zpmr
1)$ is greater than $ u$. Since $\rho (\bold x)$ is
bounded from below by a positive constant,  we get formula (2.9) by
integrating (3.1).
 
The proof
of (2.10) is similar. Here we can apply formula (2.7). Because of this
formula it is enough to show that
$$
\aligned
|\psi(\zpml 1)-\psi(\zz_{d,l}(1))|&>\const\sqrt u \\
|\psi(\zpmr 1)-\psi(\zz_{d,r}(1))|&>\const\sqrt u
\endaligned  \tag3.2
$$
under the conditions imposed for (2.10).
 
Given two vectors $\zz_1$ and $\zz_2$ let $\angle(\zz_1,\zz_2)$
denote the angle between them. For $\zz\in\g_1$ let $n(\zz)$
denote the
normal vector to the curve $\g_1$ at  $\zz$. We make the following
observation: For any $\zz\in\r\setminus\{0\}$ consider the
end points
$\zz_{d,l}=\zz_{d,l}(1)$ and $\zz_{d,r}=\zz_{d,r}(1)$ of the interval
with maximal length
in $\A$ in the direction of $\zz$. The normal of $\g_1$ in these points
cannot be almost orthogonal to $\zz$. More explicitly, there is some
$\eta>0$ such that
$$
-\frac\pi 2+\eta< \angle\(-\zz,n(\zz_{d,l})\)< \frac\pi 2-\eta \;.
\tag3.3
$$
(This statement is equivalent to the following one: If $\zz$ is a
boundary point of the oval $\A^{(2)}$, then the vector $\zz$ cannot
be almost orthogonal to the normal of the boundary of $\A^{(2)}$ in this
point. The equivalence of these two statements follows from the
following argument. The vector $\zz_{d,l}-\zz_{d,r}$ is on the
boundary of
$\A^{(2)}$, and it is parallel to $\zz$. The normal of $\A^{(2)}$ in this
point is parallel to $n(\zz_{d,l})$. The proof of the second
statement is simpler.)
 
We claim that under the conditions imposed for formula (2.10) the
distance of the parallel lines going through
the points $\zpml 1$ and $\zpmr 1$ and through the points $\zz_{d,l}(1)$
and $\zz_{d,r}(1)$ is bigger than $\const \sqrt u$. Put
$m_l=m_l(\zz)=\dfrac1{\cos\angle(\zz,n(\zz_{d,l}))}$  and
$m_r=m_r(\zz)=\dfrac1{\cos\angle(\zz,n(\zz_{d,r}))}$. Then
$m_r=-m_l$ by (2.7), and
by (3.3) there is some $\infty>K>0$ such that $-K<m_r(\zz)<K$ for all
$\zz\in\r\setminus\{0\}$.
 
Let us fix a new coordinate system with the origin in a point of the
line going through the points $\zz_{l,r}$ and $\zz_{d,r}$, with the
$x$ axis in the direction of the
vector $\zz$ and $y$ axis in the direction $\zz^\bot$, and let  us
work in it. For $\e>v>0$ let
$\zz_l^+(v)=(z_l^+(v),v)$ and $\zz_r^+(v)=(z_r^+(v),v)$,
$z_l^+(v)<z_r^+(v)$, be the two intersections of $\g$ and the line
$y=v$, and put $\zz(v)=\zz_l^+(v)-\zz_r^+(v)$. It follows from Lemma~2
(with its application in the coordinate system with coordinate axes
parallel to the normal and to the tangent vector of the curve $\g_1$ in
the points $\zz_{d,l}(1)$ and $\zz_{d,r}(1)$ respectively)
 that there is some $C>0$ such that
$z_l^+(v)<z_{d,l}(1)+vm_l+Cv^2$
 and $z_r^+(v)>z_{d,r}(1)+vm_l-Cv^2$. The above relations imply
that  $|\zz(v)|=z_r^+(v)-z_l^+(v)>d(\zz,1)-2Cv^2$.
Since we imposed the condition  $|\zz|=d(\zz,1)-ud(\zz,1)$, this
relation implies that the distance between the parallel lines going
through the points
$\zpl 1$ and $\zpr 1$ and through the points $\zz_{d,l}(1)$
and $\zz_{d,r}(1)$ is greater than $\const\sqrt u$. The same
statement holds if  $\zpl 1$ and $\zpr 1$ are replaced by $\zml 1$ and
$\zmr 1$. Hence the arcs $\zz^\pm_{1,l},\zz_{d,l}$ and the arcs
$\zz^\pm_{1,r},\zz_{d,r}$ are longer than $\const \sqrt u$. Hence
relation (3.1) and the strict positivity of $\rho(\bold x)$
imply formula (3.2) and hence formula (2.10) too.
 
Now we turn to the proof of Lemma~1. We introduce the abbreviation
$\rp$ instead of $\rpc$ and $\rmm$ instead of $\rmc$.
\demo
{\it Proof of Part a)} Let us introduce the
coordinate system
with $x$ axis parallel to $\zz$ and $y$ axis parallel to $\zz^\bot$. We
shall estimate, by means of Lemma ~2, the length of the intersection of
the set $\roz\cap(\roz-\zz)$ with the lines $y=u$ for different $u$-s.
 
Define $u_\rp(\zz)$ as the (unique) solution of the equation
$$
x_{r,\rp}(u)-x_{l,\rp}(u)=|\zz|,\quad u>u_{d,\rp}\;,
$$
where $u_{d,R^{\pm}}$ is defined by the formula
$$
x_{r,R^{\pm}}\(u_{d,R^{\pm}}\)-x_{l,R^\pm}\(u_{l,R^\pm}\)=
\max_u\[ x_{r,R^\pm}(u)-x_{l,R^\pm}(u)\]\;,
$$
that is, it is the level $u$ at which the horizontal line $y=u$ has the
longest intersection with the set $R^\pm \A$.
 
We claim that there is some $K>0$ such that
$$
\align
x_{r,\rp}(u)-x_{l,\rp}(u)&<|\zz|+\frac{\const}{|\zz|}\quad\text
{if }u_\rp(\zz)-\frac KR\leq u\leq u_\rp(\zz)\;,    \tag3.4    \\
x_{r,\rmm}(u)-x_{l,\rmm}(u)&>|\zz|\quad\text {if }
u_{d,\rmm} \leq u\leq u_\rp(\zz)-\frac KR\;, \tag$3.4'$
\endalign
$$
and
$$
\left.
\aligned
x_{r,\rp}(u)-x_{r,\rmm}(u)&<\frac12 \\
x_{l,\rp}(u)-x_{l,\rmm}(u)&<\frac12
\endaligned
\right\}\quad\text{if }u_{d,\rmm} \leq u\leq u_\rp(\zz)-\frac KR\;.
\tag$3.4''$
$$
 
First we show that relations (3.4)---$(3.4'')$ imply that
$$
\Ar{\roz\cap(\roz-\zz)\cap\Kp \rmm}<\frac{\const}{R\,|\zz|}\;.\tag3.5
$$
To see this we show that the intersection of $\roz\cap(\roz-\zz)$ with
the line $y=u$ has a length smaller than $\dfrac {\const}{|\zz|}$ if
$u_\rp(\zz)-\dfrac KR\leq u\leq u_\rp(\zz)$,
and it is empty if $u>u_\rp(\zz)$ or
$u_{d,\rmm} \leq u\leq u_\rp(\zz)-\dfrac KR$.
The above relations imply (3.5).
 
The above intersections are contained in the interval $[x_{l,\rp}(u),
x_{r,\rp}(u)-|\zz|]$ whose length is less than $\dfrac{\const}{|\zz|}$
if $u_\rp(\zz)-\dfrac KR\leq u\leq u_\rp(\zz)$ by (3.4). The distance
$x_{r,\rp}(u)- x_{l,\rp}(u)$ is less than $|\zz|$ for $u>u_\rp(\zz)$,
because it is a (convex) monotone decreasing function of $u$ in the
interval $\[u_{d,\rp},y_0(\rp)\]$ and  it equals $|\zz|$ for
$u_\rp(\zz)$. This implies that the intersection
$\roz\cap(\roz-\zz)\cap\{(x,u),\;x\in{\Bbb R}^1\}$ is empty for $u>
u_\rp(\zz)$. To see that it is empty for
$u_{d,\rmm} \leq u\leq u_\rp(\zz)-\dfrac KR$ too, observe first that the
intersection of $\roz$ with the line $y=u$ consists of two intervals,
$\[x_{l,\rp}(u),x_{l,\rmm}(u)\]$ and $\[x_{r,\rmm}(u),x_{r,\rp}(u)\]$.
The intersections
$$
\[x_{l,\rp}(u),x_{l,\rmm}(u)\]\cap
\(\[x_{l,\rp}(u),x_{l,\rmm}(u)\]-|\zz|\)
$$
and
$$
\[x_{r,\rmm}(u),x_{l,\rp}(u)\]\cap
\(\[x_{r,\rmm}(u),x_{l,\rp}(u)\]-|\zz|\)
$$
are empty because of ($3.4''$) and the condition $|\zz|>1$. The
intersection
$$
\[x_{l,\rp}(u),x_{l,\rmm}(u)\]\cap
\(\[x_{r,\rmm}(u),x_{l,\rp}(u)\]-|\zz|\)
$$
is empty because of ($3.4'$),
and the intersection
$$
\[x_{r,\rmm}(u),x_{l,\rp}(u)\]\cap
\(\[x_{l,\rp}(u),x_{l,\rmm}(u)\]-|\zz|\)
$$
is always empty. In such a way we deduced (3.5) from (3.4)---($3.4''$).
 
Let us recall the following notation.  The point
$(x_0(R^+),y_R^+)$ is the point of tangency of the curve
$\gamma_{R^+}$ with the line parallel to $\zz$ in the half-space
$K^+_{R^+}(\zz)$.
To prove relations (3.4)---($3.4''$) let us first observe that
because of the first two relations in Lemma~2
$$
B_1 R(y_0(R^+0)-u_\rp(\zz)<\zz^2<B_2 R(y_0(R^+0)-u_\rp(\zz)\;.
$$
In particular, for $|\zz|\ge1$ \
$y_0(R^+)-u_\rp(\zz)>\dfrac{\const}R$. Hence, by the third
and fourth relations in Lemma~2
$$
C_1\frac R{|\zz|}<-\frac d{du}\(x_{r,\rp}(u)-x_{l,\rp}(u)\)<C_2
\frac R{|\zz|}\quad \text{for }u_\rp(\zz) -\frac KR<u<u_\rp(\zz) \tag3.6
$$
with some $C_2>C_1>0$. Since
$x_{r,\rp}\(u_\rp(\zz)\)-x_{l,\rp}\(u_\rp(\zz)\)=|\zz|$ we get relation
(3.4) by integrating the right-hand side of (3.6) in the interval
$[u,u_\rp(\zz)]$.
 
Because of the left-hand side of (3.6) we can choose for any $D>0$ a
number $C=C(D)>0$ such that for $u_0=u_\rp(\zz)-\dfrac CR$
$
x_{r,\rp}(u_0)-x_{l,\rp}(u_0)>|\zz|+\frac D{|\zz|} %\;.
$.
We rewrite this relation by turning from $\g_\rp$ to $\g_\rmm$. In this
calculation we exploit that $|\zz|<\const R$ if $\zz\in\g_\rp$.
Putting $u_1=(1-\eta)u_0=\(1-\dfrac c{R^2}\)\(1+\dfrac c{R^2}\)^{-1}
u_0$ we have $u_1>u_\rp(\zz)-\dfrac {K}R$ with an appropriate
$K>0$, \ $\eta=O(R^{-2})$ and
$$
\align
x_{r,\rmm}(u_1)-x_{l,\rmm}(u_1)&= (1-\eta)
\(x_{r,\rp}(u_0)-x_{l,\rp}(u_0)\)\\
&>(1-\eta)\(|\zz|+\frac D{|\zz|}\)>|\zz|
\endalign
$$
if $D>0$ is sufficiently large. This means that relation $(3.4')$ holds
for $u_1>u_\rp(\zz)-\dfrac {
K}R$. Because of the monotonicity  of $x_{r,\rmm}(u_0)-x_{l,\rmm}(u_0)$
for $u_{d,\rmm}<u<y_0(\rmm)$ relation ($3.4'$) holds.
 
To prove relation  $(3.4'')$  first we observe that
$x_{r,\rp}(\bar u)=(1+\beta)x_{r,\rmm}(u)$ with $1+\beta=\(1+\dfrac
c{R^2}\)\(1-\dfrac c{R^2}\)^{-1}$ and $\bar u=(1+\beta)u$. Hence
$x_{r,\rp}(\bar u)=x_{r,\rmm}(u)+O\(\dfrac 1R\)$ and $|u-\bar
u|<\dfrac LR$ with some $L>0$. The derivative $\dfrac d{du}
x_{r,\rp}(u)$ is a monotone decreasing function of $u$, hence it
follows from the third relation in Lemma~2 that
$$
 \left|\dfrac d{du} x_{r,\rp}(u)\right|<\frac R{3L}
$$
if $y_1(\rp)-\dfrac KR \leq u\leq y_0(\rp)-\dfrac KR$, and $K>0$ is
chosen sufficiently large. Hence
$$
 \left| x_{r,\rp}(u )- x_{r,\rp}(\bar u)\right|<\frac R{3L}  |u-\bar
u|+O\(\frac 1R\)<\frac12\;.
$$
The first relation of ($3.4''$) is proved, and the second one can be
proved in the same way.
 
We have proved relation (3.5). It can be proved in the same way if $\Kp
\rp$ is replaced by $\Km \rp$. These two relations together imply Part
~a).
\enddemo
\demo {Proof of Part b)} We define two parallelograms $P^+(\zz)$ and
$P^-(\zz)$. The
parallelogram $P^+(\zz)$ is bounded by two pairs of parallel lines, the
lines of the first
pair are going through the points $\(1-\dfrac c{R^2}\)\zpl R$ and
$\(1+\dfrac c{R^2}\)\zpl R$ and they have normal $\psi(\zpl R)$,
the lines of the second pair are going through the points
$$
\(1-\dfrac c{R^2}\)\zpr R-\zz\quad\text{and}\quad
\(1+\dfrac c{R^2}\)\zpr R-\zz\;,
$$
and they have normal $\psi(\zpr R)$. The parallel pairs of lines
bounding $P^-(\zz)$
are going through the points $\(1-\dfrac c{R^2}\)\zml R$ and
$\(1+\dfrac c{R^2}\)\zml R$ with normal $\zml R(\psi)$, and
through the points $\(1-\dfrac c{R^2}\)\zmr R-\zz$ and
$\(1+\dfrac c{R^2}\)\zmr R-\zz$ with normal $\zmr R(\psi)$. The
parallelograms $P^+(\zz)$ and $P^-(\zz)$ have area $f^+_R(\zz)$ and
$f^-_R(\zz)$ respectively, and they are disjoint if $|\zz|>\eta R$.
Since the difference of these parallelograms and the domains
$\roz\cap(\roz-\zz)\cap \Kp R$ and $\roz\cap(\roz-\zz)\cap \Km R$ have
an area of order $o\(R^{-2}\)$, the above relations imply Part ~b) of
Lemma~1.
\enddemo
\demo {Proof of Part c)} Let us work in the coordinate system with
origin $\zz_{d,l}(\rp)$, with $x$-coordinate axis in the direction
$-n(\zz_{d,l}(\rp))$, the normal of $\g_\rp$ in the point
$\zz_{d,l}(\rp)$ showing inside the domain $\A_\rp$, and with $y$-axis
in the direction $\zz_e=-n(\zz_{d,l}(\rp))^\bot$, the
tangent of $\g_\rp$ in this point which is obtained when the $x$ axis
is rotated with angle $+\pi/2$. Let $y^\pm_\rp(u)$ be the $y$ coordinate
of the intersection of the set $\g_{R^+}\cap \Kpm \rp$ with the
line $x=u$ and  $y^\pm_\rmm(u)$ the $y$ coordinate of
the intersection of the set $\g_{R^-} \cap \Kpm \rmm$ with this line.
We shall estimate the length of the intersection of $\roz$ with the line
$x=u$. We shall prove that the Lebesgue measure of this intersection
satisfies the inequality
$$
\lambda\(\roz\cap\{(u,y),\;y\in{\Bbb R}^{(1)}\}\)<\const\max\(\frac
1{\sqrt {Ru}},1\)\quad\text{if } 0<u<\eta R \tag3.7
$$
with some $\eta >0$.
 
By Lemma 2
$$
\left|y^\pm_\rp(u)\right|<\const \sqrt{Ru}\quad\text{if } 0<u<AR\;,
$$
and this inequality implies (3.7) in the case $u<\dfrac KR$ with some
$K>0$. We shall show, using again Lemma~2, arguing similarly as in the
proof of relation (3.4) in the proof of Part~a) that
$$
\left|y^\pm_\rp(u)- y^\pm_\rmm(u)\right|<\frac{\const} {\sqrt{Ru}}\,,
\quad\text{if }\frac KR<u<\eta R\;,\tag3.8
$$
which relation completes the proof of (3.7). To prove (3.8) we have to
express $y^\pm_\rmm(\cdot)$ by $y^\pm_\rp(\cdot)$ and to exploit that by
Lemma~2
$\left|\dfrac d{dv}y^\pm_\rmm(v)\right|<\const \dfrac{\sqrt R}{\sqrt
u}$
if $\dfrac KR<u<v<u+\dfrac KR$.
 
Let $\bold v^+_\rp=(v^+_{\rp,1}, v^+_{\rp,2})$ and
$\bold v^-_\rp=(v^-_{\rp,1}, v^-_{\rp,2})$ be the points of intersection
of $\g_\rp$ and $\g_\rp-\zz$ in the half planes $\Kp \rp$ and $\Km \rp$
respectively. We shall prove that there is some constant $K>0$ such that
$$
\roz\cap(\roz-\zz)\cap \Kp \rp\cap\{(v,y),\;y\in {\Bbb R}^1\}=\emptyset
\quad\text {if }v\notin \[v^+_{\rp,1}-\frac KR, v^+_{\rp,1}+\frac KR\]
\tag3.9
$$
and
$$
\roz\cap(\roz-\zz)\cap \Km \rp\cap\{(v,y),\;y\in {\Bbb R}^1\}=\emptyset
\quad\text {if }v\notin \[v^-_{\rp,1}-\frac KR, v^-_{\rp,1}+\frac KR\]\;.
\tag$3.9'$
$$
We also claim that under the conditions of Part ~c1) or
Part~c2) of Lemma~1
$$
v^{\pm}_{R^+,1}>\const (d(\zz,R)-|\zz|)\;. \tag3.10
$$
Relations (3.7), (3.9), ($3.9'$) and (3.10) together imply Part ~c) of
Lemma~1. To prove relation (3.10) let us consider the projection of the
vectors $\bold v^\pm_{R^+}$ and $(d(\zz,R)-|\zz|)\dfrac {\zz}{|\zz
|}-\bold v^\pm_{R^+}$ to the direction of the vector
$-n(\zz_{d,l}(R^+))$. The sum of these two vectors, which is the
projection of $d(\zz,R)-|\zz|$ to $-n(\zz_{d,l}(R^+))$ is longer than
$\const (d(\zz,R)-|\zz|)$ because of relation (3.3). On the other hand,
the proportion of the length of these two vectors is separated both from
zero and infinity because of relation (3.3) which implies this relation
if the projection is done in the orthogonal direction
$(d(\zz,R)-|\zz|)^\bot$ and Lemma~2.
 
To prove relations (3.9) and ($3.9'$) we introduce the following
notation. Let $s^\pm_{R^\pm}(u)$ be the $y$ coordinate
of the intersection of the set $(\g_{R^\pm}-\zz)\cap \Kpm {R^\pm}$ with
the line $x=u$. Since $s^+_\rp(v^+_{\rp,2})=y^+_\rp(v^+_{\rp,2})$ we get
by expressing $s^+_\rmm(\cdot)$ through $s^+_\rp(\cdot)$, exploiting
the lower bound on the derivative of the function $s^+_\rp(\cdot)$ given
by Lemma~2 and arguing similarly to the proof of relation (3.7) that
$s^+_\rmm(v)>x^+_\rp(v)$ or $v<0$ if $v< v^\pm_{\rp,1}-\dfrac KR$ with
some sufficiently large $K>0$. If $v<0$, then the set
$\roz\cap\{(v,y),\;y\in {\Bbb R}^1\}$ is empty. Hence
$$
\roz\cap\{(v,y),\;y\in {\Bbb R}^1\}=\emptyset\quad\text{if }v<
v^+_{\rp,1}-\frac KR \;.
$$
By changing the role of $\g_R$ and $\g_R-\zz$ we get that
$$
\roz\cap\{(v,y),\;y\in {\Bbb R}^1\}=\emptyset\quad\text{if }v>
v^+_{\rp,1}+\frac KR\;.
$$
The last two relations together imply (3.9). The proof of ($3.9'$) is
similar.
 
In such a way we have proved Part c) of Lemma~1. The proof of Part d)
is trivial, since in this case even the set $\rp\A\cap(\rp\A-\zz)$ is
empty. \qed
\enddemo
 
\beginsection 4. Some concluding remarks
 
In this Section we discuss the conjecture about the
Poissonian distribution of a randomly placed circle suggested by
the computer study of Cheng and Lebowitz ~[4]  and briefly explain
what kind of approach is suggested by the present paper.
 
It is relatively easy to show that the Poissonian limit for
the number of lattice points in $\ro$ would follow from the
following generalization of formula (2.1):
$$
\lim_{R\to\infty}E\xi_R(\a)(\xi_R(\a)-1)\cdots (\xi_R(\a)-k+1)=
\Ar{\roz}^k =\[4c\Ar{\A}\]^k  \quad\text{for all
}k\ge1\;.\tag4.1
$$
Actually relation (4.1) is equivalent to the statement that all
moments of the random variable  $\xi_R(a)$ converge to the
corresponding moments of a Poissonian random variable with
parameter $\Ar{\roz}=4c\Ar{\A}$. Some modification of the argument
leading to the proof of formula (2.2) gives that
$$
\align
&E\xi_R(\a)(\xi_R(\a)-1)\cdots(\xi_R(\a)-k+1)\\&\qquad=
\!\!\!\!\sum\Sb\m_1\in\z\setminus\{0\},\dots,\m_{k-1}\in\z
\setminus\{0\}\\
\text{the points }\m_1,\dots, \m_{k-1}\text{ are different.}
\endSb\!\!\!\!
\Ar{\roz\cap(\roz-\m_1)\cap\dots\cap(\roz-\m_{k-1})}\;.\tag4.2
\endalign
$$
 
It is relatively simple to prove the following identity:
$$
\int
\Ar{\roz\cap(\roz-\zz_1)\cap\dots\cap(\roz-\zz_{k-1})}
\,d\zz_1\dots\,d\zz_{k-1}=\[4c\Ar{\A}\]^k\;.\tag4.3
$$
 
Hence to prove the Poissonian limit it would be enough to show
that for large $R$ the replacement of the sum in (4.2) by the
integral in (4.3) causes a negligible error for all $k=1$,
2,~\dots.
Actually, this fact was proved for $k=1$ and~2 in this paper. But
the proof for larger $k$ is much harder. In our proof we exploited
the independence caused by the random shift $\alpha\in\01$. But
this independence is not sufficient for the proof of (4.3) if $k\ge3$.
To prove formula (4.3) in this case some deep number theoretical
statement would be needed which states that certain functions of the
$k$-tuples of lattice points $(\bold m_1,\dots,\bold m_k)$ are almost
uniformly distributed. We could give an explicit formulation of this
statement, but since this would require complicated notations and would
lead to a problem that we cannot handle we omit it.
 
We also discuss briefly the higher dimensional version of the
problem in this paper.
 
Let $\A$ be a convex set with a nice boundary, $\a\in{\Bbb R}^d$ a
vector, uniformly distributed in the $d$-dimensional unit cube, $c>0$
a fixed number and
$$
\ro=\[\(R+\frac c{R^{d-1}}\)\A-\a\]\setminus
\[\(R-\frac c{R^{d-1}}\)\A-\a\]\;.
$$
The volume of the set $\ro$ is $2cd\,\text{Volume}\,(\A)+O\(\dfrac 1R\)$.
We are interested in the number of lattice points $\xi_{R}(\a)$ in the
randomly shifted
set $\ro$. It can be proved that the first two moments of $\xi_{R}$
tend to the first two moments of a Poissonian random variable with
parameter $2d\,\text{Volume}\,(\A)$. This can be proved by methods
similar to those  of the present paper. Moreover, in this
case a stronger result can be proved. It can be shown that in the
$d$-dimensional case
$$
\lim_{R\to\infty}E\xi_R(\a)(\xi_R(\a)-1)\cdots (\xi_R(\a)-k+1)=
\[2cd\,\text{Volume}\,(\A)\]^k  \quad\text{for } 1\le k\le d\;.
\tag4.4
$$
 
This means that the first $d$ moments of the random variable $\xi_{R}$
tend to that of a Poissonian random variable with parameter
$2cd\,\text{Volume}\,(\A)$ as $R\to\infty$. To explain why relation
(4.5)
holds let us remark that relations (4.2) and (4.3) remain valid for all
$d\ge2$ if the area is replaced by Volume, $\z\setminus\{0\}$ by
${\Bbb Z}^d\setminus\{0\}$ and $4c\Ar \A$ by $2cd\,\text{Volume}\,(\A)$.
We have to show that by replacing the sum in (4.2) by the integral
in (4.3) we commit a negligible error. Let us also observe that for
$k\le d$ the expression
$$
h(\zz_1,\dots,\zz_{k-1})
=\text{Volume}\,\(\roz\cap(\roz-\zz_1)\cap\dots\cap(\roz-\zz_{k-1})\)
$$
changes very little if the arguments in this expression are changing
with an order of constant. Hence the same technique works for $k\le d$,
$d>2$, as for the case $d=2$ in the second Section of
the present paper. Actually some technical difficulties have to be
overcome if we
want to carry out this program. We do not go into the details.
\bigskip\noindent
{\it Acknowledgement:}\/ The authors would like
to thank E. Makai Jr.\ for some useful explanations on convex geometry.
J. L. ~Lebowitz and P.~Major would like to thank the IHES in
Bures--Sur--Yvette where this work was completed. The work of
Zh. ~Cheng and J.~L.~Lebowitz was supported in part by the NSF
Grant DMR~89~18902.
\bigskip  \noindent
{\bf References:} \parindent=18pt
 
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\bye
 
