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
$$
-\frac{C_2}2(x-x_0(1))^2__0$. The derivative $\dfrac d{du}
x_{r,\rp}(u)$ is a monotone decreasing function of $u$, hence it
follows from the third relation in Lemma~2 that
$$
\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 0$.
By Lemma 2
$$
\left|y^\pm_\rp(u)\right|<\const \sqrt{Ru}\quad\text{if } 00$. We shall show, using again Lemma~2, arguing similarly as in the
proof of relation (3.4) in the proof of Part~a) that
$$
\left|y^\pm_\rp(u)- y^\pm_\rmm(u)\right|<\frac{\const} {\sqrt{Ru}}\,,
\quad\text{if }\frac KR0$ 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
\item{1.} Berry, M. V., Tabor, M.: Level clustering in the regular
spectrum. Proc.\ R. Soc.\ London A {\bf356} (1977), 375--394
\item{2.} Bleher, P. M.: Quasi-classical expansion and the
theory of quantum chaos. Lecture Notes in Mathematics. {\bf
1469}, (1991) 60--89
\item{3.} Bonnesen, T., Fenchel, W.: Theorie der konvexen
K\"orper. Ergebnisse der Mathematik 3. Bd. Heft 1, New York,
Chelsea (1948)
\item{4.} Cheng, Zh., Lebowitz, L. J.: Statistics of energy levels
in integrable quantum systems. Phys.\ Rev.\ {\bf A}, {\bf 44} Nr.~6
(1991) 3399--3402
\item{5.} Colin de Verdi\`ere, Y: Nombre de points entiers
dans une famille homoth\'etique de domaines de {\bf R}$^n$.
Ann.\ Scient.\ \'Ec.\ Norm.\ Sup., $4^{\text{e}}$ s\'erie,
{\bf 10} (1977) 559--576
\item{6.}Huxley, M. N.: Exponential sums and lattice points.
Proc.\ London Math.\ Soc., {\bf 60}~(3) (1990) 470--502
\item{7.} Kendall, D. G.: On the number of lattice points inside a
random oval. Quart. J. Math.\ Oxford, {\bf 19} (1948) 1--26
\item{8.} Major, P: Poisson law for the number of lattice
points in a random strip with finite area. Probab.\ Theory
Relat.\ Fields, {\bf92} (1992) 423--464
\item{9.} Sinai, Ya.\ G.: Poisson distribution in a geometric
problem, In: Advances in Soviet Mathematics, v. 3,
ed. Ya. G. Sinai, AMS, Providence, (1991) 199--214.
\item{10.} Sinai, Ya.\ G.: Mathematical problems in the theory of
quantum chaos. Lecture Notes in Mathematics. {\bf 1469} (1991) 41--52
\bye
__