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\define\e{\varepsilon}
\topmatter
\title
Limit theorems about the distribution of almost periodic functions
\endtitle
\author P\'eter Major \endauthor
\affil
Mathematical Institute of the Hungarian Academy of Sciences\\
and\\
Bolyai College of the E\"otv\"os Lor\'and University, Budapest
\endaffil
\rightheadtext{Limit theorems about almost periodic functions}
\abstract
We prove a limit theorem about the distribution of an almost periodic
function $F(R)=\sum\limits_{n=1}^\infty a_ne^{2\pi i\lambda_nR}$,
$\sum\limits_{n=1}^\infty |a_n|^2<\infty$, when $R$ is uniformly
distributed in an interval $[0,T]$, and $T\to\infty$. Also a limit
theorem is proved about the distribution of the random vector
$\bigl(F(R), F(R+w(R,T))\bigr)$, $R\in[0,T]$, if the function $w(R,T)$
is appropriately defined. Similar results were proved also in other
papers. (See [2] and [3].) The proofs in this paper are essentially
different from the previous ones, and they may give some new insight to
this problem. Previous proofs were based on ergod theoretical
arguments, while in this paper some standard methods of Fourier
analysis are applied. These investigations were motivated by
the study of the limit behavior of the number of lattice points in
a randomly magnified strip in the plane.
\endabstract
\endtopmatter
\beginsection 1. Introduction
In this paper the following problem is discussed: Let us consider
a function
$$
F(R)=\sum_{n=1}^\infty a_n e^{2\pi i\lambda_nR} \tag1.1
$$
with
$$
\sum_{n=1}^\infty |a_n|^2<\infty\;,\tag1.2
$$
where $\lambda_1$, $\lambda_2$,~\dots are different non-zero real
numbers. We also assume that $\lambda_{2n}=-\lambda_{2n-1}$ and
$a_{2n}=\bar a_{2n-1}$ for $n=1$,~2, \dots. This restriction is not
essential, we only impose it to work with real valued functions.
Define the distribution $\mu_T$ of the function $F(R)$ with respect to
the uniform distribution in the interval $[0,T]$ by the formula
$$
\mu_T(\bold A)=\frac1T\lambda\{R\: 0\le R\le T,\; F(R)\in\bold A\}\tag
1.3
$$
for any measurable set $\bold A\subset \Bbb R^1$, where $\lambda$
denotes
the Lebesgue measure. We want to prove that the measures $\mu_T$ have a
weak limit. We also want to prove a generalization of this result in the
case when the joint distribution of the functions $F(R)$ and
$F(R+w(R,T))$ are investigated with a nice function $w(R,T)$. We shall
study the limit distribution of this vector if $R$ is uniformly
distributed in an interval $[aT,bT]$, $00$
there exists an index $p_0=p_0(\e)$ in such a way that
$$
\limsup_{T\to\infty}\frac1{2T}\int_{-T}^T\left| F(R)-\sum_{n=1}^p a_n
e^{2\pi i\lambda_nR}\right|^2\,dR<\e\tag1.5
$$
for $p>p_0$. The theory of Besicovitch spaces can be found in [1], but in
the present paper we do not need its fine details. Here we only use
relation (1.5). Let us remark that the definition in (1.5) does not
define the function $F(R)$ in a unique way. Indeed, if $F(R)$ and $\bar
F(R)$ are two functions such that
$$
\lim_{T\to\infty}\frac1{2T}\int_{-T}^T|F(R)-\bar F(R)|^2\,dR=0\;,
$$
then the functions $F(R)$ and $\bar F(R)$ simultaneously satisfy or do
not satisfy relation (1.5). Thus the theorems formulated below state in
particular that the limit distribution appearing in them do not depend
on which function $F(R)$ we take from those satisfying formula~(1.5).
Our first result is the following
\proclaim{Theorem 1}\it For all functions $F(R)$ satisfying (1.1) and
(1.2) the probability measures $\mu_T$ defined in (1.3) converge weakly
to a probability measure $\mu$ as $T\to \infty$. Moreover, for all
continuous functions $g(u)$ such that $|g(u)|0$ and $B>0$, the relation
$$
\lim_{T\to\infty}\frac1T\int_0^T g(F(R))\,dR=
\lim_{T\to\infty}\int g(u)\mu_T(\,du)=\int g(u)\mu(\,du)\tag1.6
$$
holds. In particular,
$$
\lim_{T\to\infty}\frac1T \int_0^T F(R)\,dR=0\;,\tag1.7
$$
and
$$
\lim_{T\to\infty}\frac1T \int_0^T
F(R)^2\,dR=\sum_{n=1}^\infty|a_n|^2\;.\tag$1.7'$
$$
\endproclaim
To formulate Theorem 2 first we have to clarify how to define the class
of ``width" functions $w(R,T)$ in it. We consider two different cases.
In case a) this width has constant order, and in a point $R=uT$,
$a\le u\le b$, the value of the ``width" function $w(R,T)$ is close to a
monotone function $K(u)$ for large $T$, and in case b) it tends to
infinity as $T\to\infty$ in a regular way. First we formulate Theorem~2,
and then show that it contains the results of Section~2 in [3] as a
special case.
\proclaim{Theorem 2}\it Let a function $F(R)$ be given, which
satisfies relations (1.1) and (1.2), and let $K(x)$ be a continuously
differentiable, monotone (increasing or decreasing) function with
non-vanishing derivative in an interval $[a,b]$, $00$ for all $x\in [a,b]$. Assume that the function $w(R,T)$,
$T>0$, $aT0$ and $B>0$,
then
$$
\aligned
\lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}g(F(R),F(R+w(R,T))\,dR&=
\lim_{T\to\infty}\int g(u,v)\,\mu_{T,w,(a,b)}(du,dv) \\
&=\int g(u,v)\,\bar\mu(du,dv)
\endaligned
\tag1.9
$$
with $\bar\mu=\mu_{(a,b)}^{K(x)}$ in case a) and
$\bar\mu=\mu^\infty_{(a,b)}=\mu\times\mu$ in case b).
Let us fix some function $K(x)$ which satisfies the conditions imposed
on it in Theorem~2. Then, the probability measures
$\mu^{zK(x)}_{(a,b)}$ depend continuously on $z$ in the weak topology
for $00$ and $B>0$. For a fixed function $g(u,v)$ the
integral at the right-hand side of (1.14) is a continuous and bounded
function of $x$.
The identity
$$
\mu^{K(x)}_{(a,b)}=\frac1{(b-a)}\int_{a}^{b}\nu^{K(x)}\,dx
=\frac1{(b-a)}\int_{K(a)}^{K(b)}\frac{\nu^x}{K'(K^{-1}(x))}\,dx\tag1.15
$$
holds for the function $\mu^{K(x)}_{a,b}$ defined in Theorem~2.
\endproclaim
We shall prove the following corollary of the above results:
\proclaim{Corollary}\it Let $h(x)$ be an integrable function on an
interval $[a,b]$, $00$ and $B>0$.
\endproclaim
This paper consists of four sections. In Section~2 we prove the Theorems
in the special case when the sum (1.1) defining the function $F(R)$
contains finitely many terms. In Section~3 we carry out a limiting
procedure which proves the results of the paper by means of Section~2.
In Section~4 we make some comments and prove some generalizations.
Let us make a short comparison of the method of this paper with
previous ones. The main difference of the proof of Theorem~1 in this
paper and in [2] is that we replace the application of the ergod theorem
by a number theoretical distribution theorem. The formulation
of Theorems~2 and~3 are very close to the results of Section~2 in~[3].
The proofs are essentially different. In paper~[3] these results were
deduced from Theorem~1 by a tricky ergod theoretical argument. Here we
show that a slight modification of the proof of Theorem~1 supplies a
direct proof for them.
\beginsection 2. The proof of the results in a special case
Let us consider the case when the function $F(R)$ is defined by a finite
sum
$$
F(R)=F_p(R)=\sum_{n=1}^p a_n e^{2\pi i\lambda_nR} \tag2.1
$$
with an even number $p$, real non-zero numbers $\lambda_n$ such that
$\lambda_{2n}=-\lambda_{2n-1}$, $a_{2n}=\bar a_{2n-1}$, and the
measures $\mu_T$, $\mu_{T,w,(a,b)}$ and $\nu^x_T$ are defined in
formulas (1.3), (1.4) and (1.13) by means of this function. We prove
in this Section Theorems~1, 2 and~3 in the case when these measures are
determined by a function of the form (2.1). In the next Section we
prove the result for general functions $F(R)$ defined in (1.1) by
approximating them with the functions $F_p$ appearing in (2.1). We
shall indicate the dependence of these measures on the functions $F_p$
by denoting them as $\mu_T(F_p)$, $\mu_{T,w,(a,b)}(F_p)$ and
$\nu^x_T(F_p)$ when necessary. The main results of this Section are the
following
\proclaim{Propositions 1., 2. and 3}\it Let the function $F(R)$ be
defined by the finite trigonometrical sum (2.1), and let the measures
$\mu_T$, $\mu_{T,w,(a,b)}$ and $\nu^x_T$ be defined in formulas (1.3),
(1.4), and (1.13) by means of this function $F(R)$. If the function
$w(R,T)$ and $S(T)$ satisfies the conditions of Theorem~2, then
Theorems~1,~2, and~3 hold with this choice of the corresponding
measures.
\endproclaim
\demo{Proof of Proposition 1}
We have to investigate the asymptotic behavior of the expression
$$
\frac1T\int_0^T g(F(R))\;dR\tag2.2
$$
as $T\to\infty$ in the case when $g(u)$ is a bounded continuous
function. We shall rewrite, following the argument of [2] and [4],
the expression in (2.2) as an integral on a torus with respect to an
appropriate measure. It is useful to work, when handling the function
$F(R)$, with frequencies linearly independent over the rational numbers.
Since the frequencies $\lambda_n$ may not have this property we express
them as a linear combination of some numbers $\tau_1$,\dots,$\tau_s$
linearly independent over the rational numbers
$$
\lambda_n=T_n(\tau_1,\dots,\tau_s)=\sum_{k=1}^s A(n,k)\tau_k\;,\quad
n=1,2,\dots,p, \;k=1,\dots, s\tag2.3
$$
with integer coefficients $A(n,k)$.
Let $V$ denote the unit interval with the group action addition
modulo~1. Introduce its $s$-fold and $p$-fold direct products
$$
\Cal V=\undersetbrace{s\;\text{times}} \to{V\times\cdots\times
V} \tag2.4
$$
and
$$
\Cal V'=\undersetbrace{p\;\text{times}} \to{V\times\cdots\times
V}\;.\tag$2.4'$
$$
Define the maps $U\:\Bbb R^1\to \Cal V$
$$
U(R)=\{R\tau_k \text{ (mod}\,1),\quad k=1,\dots, s\}\;.\tag2.5
$$
$V\:\Cal V\to \Cal V'$
$$
V(u_1,\dots,u_s)=\left\{\sum_{n=1}^s A(n,k)u_k\text{
(mod}\,1),\quad n=1,\dots,p\right\} \tag2.6 $$
for $(u_1,\dots,u_s)\in \Cal V$ with the integer coefficients $A(n,k)$
appearing in (2.3) and $G\:\Cal V'\to \Bbb R^1$
$$
G(u_1,\dots,u_p)=\sum_{n=1}^p a_n e^{2\pi i u_n}\;.\tag2.7
$$
Clearly, $F(R)=G\bigl(V(U(R))\bigr)$. Define the probability measure
$\rho_T$ on $\Cal V$ induced by the map $U$ by the formula
$$
\rho_T(\bold A)=\frac1T\lambda\{R\: 0\le R\le T,\; U(R)\in \bold
A\}\tag2.8
$$
for all measurable sets $\bold A\subset \Cal V$.
Then the integral (2.2) can be rewritten as
$$
\frac1T\int_0^T g(F(R))\;dR=\int_{\Cal V}
g(G(V(u)))\,\rho_T(du)\;.\tag2.9
$$
The relation
$$
\rho_T\Rightarrow \rho\quad\text{as }T\to\infty \tag2.10
$$
holds, where $\rho$ denotes the Haar measure on $\Cal V$, and
$\Rightarrow$ means weak convergence of probability measures.
Relation (2.10) is a known result. Nevertheless, we give its proof,
because it is short, and we need its modification in the proof of
Proposition~2. By Weil's lemma (or by the characteristic function method
on commutative compact groups) to prove (2.10) it is enough to check
that, with the notation $(u_1,\dots,u_s)=u\in\Cal V$,
$$
\align
\lim_{T\to\infty}\int\exp\left\{2\pi i\sum_{k=1}^s m_ku_k\right\}
\rho_T(\,du)&=\lim_{T\to\infty}\frac 1T \int_0^T\exp\left\{2\pi
i\sum_{k=1}^s m_k\tau_k R\right\}\,dR\\
&=\lim_{T\to\infty}\frac{\exp\left\{2\pi iT\sum\limits_{k=1}^s
m_k\tau_k\right\}-1} {2\pi iT\sum\limits_{k=1}^s m_k\tau_k}=0
\endalign
$$
if $m_1,m_2,\dots,m_k$ are integers, and not all of them equal zero.
Relations (2.9) and (2.10) imply that
$$
\lim_{T\to\infty}\frac1T\int_0^Tg(F(R))\,dR=\lim_{T\to\infty}\int
g(G(V(u))\rho_T(du)=\int g(G(V(u))\rho(du)\;,
$$
since $g\bigl(G(V(u))\bigr)$ is a bounded, continuous function. The
last relation implies that
$$
\lim_{T\to\infty}\frac1T\int_0^Tg(F(R))\,dR=\int g(u)\,\mu(du)\tag2.11
$$
with the measure $\mu$ defined on $\Bbb R^1$ by the relation
$$
\mu(\bold A)=\rho\{u\: u\in\Cal V,\quad G(V(u))\in\bold A\}\tag2.12
$$
for all measurable sets $\bold A\in\Bbb R^1$. Relations (2.11) and
(2.12)
imply that the measures $\mu_T$ converge weakly to the measure $\mu$
defined in (2.12). To complete the proof of Proposition~1 observe that
the function $|F(R)|$ is bounded by $C_p=\sum_{n=1}^p|a_n|$ for all
$R\in\Bbb R^1$. Hence all measures $\mu_T$ and $\mu$ are concentrated in
the interval $[-C_p,C_p]$, and relation (1.6) follows for all continuous
functions $g(u)$, since they can be replaced by their truncation at $\pm
C_p$, which are bounded continuous functions. Finally, relations (1.7)
and $(1.7')$ follow from the observation that $F(R)$ is a finite sum,
and the relations
$$
\lim_{T\to\infty}\frac1T \int_0^Te^{i\lambda_nR}\,dR=0
$$
$$
\lim_{T\to\infty}\frac1T\int_0^T
e^{i(\lambda_n-\lambda_{n'})R}\,dR=\delta(n,n')
$$
hold.\qed
\enddemo
\demo{Proof of Proposition 2}
The proof of Proposition 2 is very similar to that of
Proposition~1. The main difference is that we have to carry out the
integral transformations by means of different functions $G$, $U$ and
$V$, and the statement formulated in (2.10) has to be generalized.
Define the maps $U\:\Bbb R^1\to \Cal V\times \Cal V$
$$
\aligned
U(R)=U(R,w,T)&=\{R\tau_k\text{ (mod}\,1),\quad k=1,\dots, s\;, \\
&\qquad\qquad (R+w(R,T))\tau_k\text{ (mod}\,1),\quad k=1,\dots, s
\}\;, \endaligned
\tag2.13
$$
$V\:\Cal V\times\Cal V\to \Cal V'\times\Cal V'$
$$
\aligned
V(u_1,\dots,u_{2s})&=\left\{\sum_{k=1}^s A(n,k)u_k\text{
(mod}\,1),\quad n=1,\dots,p\;,\right.\\
&\qquad\qquad \left.
\sum_{k=1}^s A(n,k)u_{k+s}\text{ (mod}\,1),\quad n=1,\dots,p\right\}
\endaligned
\tag2.14
$$
for $(u_1,\dots,u_{2s})\in \Cal V\times\Cal V$ with the integer
coefficients $A(n,k)$ appearing in (2.3) and $G\:\Cal V'\times\Cal V'\to
\Bbb R^2$
$$
G(u_1,\dots,u_{2p})=\left( \sum_{n=1}^p a_n e^{2\pi i u_n},\;\;
\sum_{n=1}^p a_n e^{2\pi i u_{n+p}}\right)\;.\tag2.15
$$
Then $\bigl(F(R),F(R+w(R,T))\bigr)=G\bigl(V(U(R))\bigr)$.
Define the probability measure $\bar\rho_{T}=\bar\rho_{T,w,(a,b)}$
on $\Cal V\times\Cal V$ induced by the map $U$ by the formula
$$
\bar\rho_{T,w,(a,b)}(\bold A)=\frac1{(b-a)T}\lambda\{R\: aT\le R\le
bT,\quad U(R,w,T))\in \bold A\}\tag2.16
$$
for any measurable set $\bold A\subset \Cal V\times\Cal V$.
We claim that if $w(R,T)$ satisfies the conditions of Theorem~2, then
the limit relation
$$
\bar\rho_{T,w,(a,b)}\Rightarrow\bar \rho\quad \text{as }
T\to\infty\tag2.17
$$
holds with a probability measure $\bar\rho$ on $\Cal V\times\Cal V$.
Moreover, we claim that
$$
\bar\rho=\bar\rho_{(a,b)}^{K(x)} \tag$2.17'$
$$
if $w(R,T)$ satisfies the conditions of case a) of Theorem~2 with
$K(x)$, i.e.\ the limit depends only on the function $K(x)$ in this
case, and
$$
\bar\rho=\bar\rho^\infty_{a,b}=\,\text{the Haar measure
}\rho\times\rho\text{ on }\Cal V\times\Cal V\, \tag $2.17''$
$$
if $w(R,T)$ satisfies the conditions of case b) of Theorem~2.
To prove (2.17) we show that the Fourier coefficients
$$
L_{w,T}(m_1,\dots,m_{2s})=\int_{\Cal V\times\Cal V}\exp\left\{2\pi
i\sum_{k=1}^{2s}m_ku_k\right\}\,\bar\rho_{T,w,(a,b)}(\,du) $$
with $u=(u_1,\dots,u_{2s})\in\Cal V\times \Cal V$ have a limit
$$
\lim_{T\to
\infty}L_{w,T}(m_1,\dots,m_{2s})=L(m_1,\dots,m_{2s})\tag2.18
$$
for all integers $m_1,\dots, m_{2s}$. These Fourier coefficients can
be rewritten by an integral transformation as
$$
L_{w,T}(m_1,\dots,m_{2s})=\frac1{(b-a)T}\int_{aT}^{bT}
e^{i(A(m_1,\dots,m_{2s})R+B(m_1,\dots,m_{2s})w(R,T))}\,dR \tag2.19
$$
with
$$
\aligned
A(m_1,\dots,m_{2s})&=2\pi\sum_{k=1}^{s}(m_k+m_{k+s})\tau_k\;,\\
B(m_1,\dots,m_{2s})&=2\pi\sum_{k=1}^{s}m_{k+s}\tau_k\;.
\endaligned \tag$2.19'$
$$
Because of the linear independence of the numbers $\tau_k$ both
expressions $A(m_1,\dots,m_{2s})$ $B(m_1,\dots,m_{2s})$ can disappear
simultaneously only if all coefficients $m_k$ are zero, which is a
trivial case. Otherwise we claim that
$$
\lim_{T\to\infty}L_{w,T}(m_1,\dots,m_{2s})=\cases
0&\text{if }A(m_1,\dots,m_{2s})\neq0\\
\frac1{(b-a)}\int_{a}^{b}e^{iBK(u)}\,du
&\text{if }A(m_1,\dots,m_{2s})= 0 \text{ and}\\
&\quad w(R,T)\text{ satisfies case a)}\\
0&\text{if }A(m_1,\dots,m_{2s})= 0 \text{ and} \\
&\quad w(R,T)\text{ satisfies case b)}
\endcases\;. \tag2.20
$$
The first line in relation (2.20) can be proved by means of relation
(2.19) with the change of variables $AR+Bw(R,T)=u$. Let us observe
that because of the conditions of Theorem~2 $\frac{du}{dR}=A+o(1)$
uniformly for $aT\le R\le bT$, and the boundaries of the domain of
integration after the change of variables are $aAT(1+o(1))$ and
$bAT(1+o(1))$. Hence we get that
$$
\lim_{T\to\infty}L_{w,T}(m_1,\dots,m_{2s})
=\lim_{T\to\infty}\frac{(1+o(1))}{(b-a)A(m_1,\dots,m_{2s})T}
\int_{aA(m_1,\dots,m_{2s})T}^{bA(m_1,\dots,m_{2s})T}
e^{iu}\,du=0
$$
in this case. If the conditions of the second line of (2.20) hold, i.e.\
when the conditions of case~a) of Theorem~2 hold, and $A=0$, then we can
calculate the expression (2.19) with the change of variables $u=\dfrac
RT$. Simple calculation shows that
$$
\frac{1}{(b-a)T}\int_{aT}^{bT}
e^{iBw(R,T)}\,dR\to\frac1{(b-a)}\int_{a}^{b}e^{iBK(u)}\,du\;,
$$
and the second relation of (2.20) holds. The third line of (2.20) (this
case holds when $A=0$ and condition b) is satisfied.) can be proved
similarly with the change of variables $u=\dfrac RT$ and
$v=\dfrac1{L(T)}w(uT,T)$. Some calculation shows that $v=K(u)(1+o(1))$,
$\dfrac{\partial v}{\partial u}=K'(u)(1+o(1))$, and
$$
\align
\frac1{(b-a)T}\int_{aT}^{bT}
e^{iBw(R,T)}\,dR&=\frac{1}{(b-a)}\int_{a}^{b}
e^{iBw(uT,T)} \,du\\ &=
\frac{1}{(b-a)}\int_{K(a)}^{K(b)}
e^{iBL(T)v}\frac1{K'(K^{-1}(v))}(1+o(1))\,dv\to0
\endalign
$$
by the Riemann lemma. The convergence of the Fourier coefficients
formulated in relation (2.20) implies formulas (2.17), $(2.17')$ and
$(2.17'')$. In particular, relation $(2.17'')$ holds, since in the case
when $w(R,T)$ satisfies case~b) of Theorem~2, then all non-trivial
Fourier coefficients of $\bar\rho$ equal zero.
Let us also show that the Fourier coefficients in the second line of
formula (2.20) corresponding to the function $zK(x)$ tend to zero as
$z\to\infty$. This relation holds, because $B\neq0$ in this case, and
the Riemann lemma yields that
$$
\frac1{(b-a)}\int_a^b \!e^{izB(m_1,\dots,m_{2s})K(u)}\,du=
\frac1{(b-a)}\int_{K_1(a)}^{K_1(b)}\frac{e^{izB(m_1,\dots,m_{2s})u}}
{K'(K^{-1}(u))}\,du\to 0\quad\text{as }z\to\infty\tag2.21.
$$
Let $g(u,v)$ be a bounded continuous function. We get similarly to the
argument of Proposition~1 from relations (2.17) $(2.17')$ and
$(2.17'')$ that
$$
\aligned
&\lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}\limits
g(F(R),F(R+w(R,T))\,dR=
\lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}\limits
g\bigl(G\!\bigl(V(U(R))\bigr)\bigr)\,dR\\ &\qquad=\lim_{T\to\infty}\int
g(G(V(u))\bar\rho_{T,w,(a,b)}\,(du)=
\int g(G(V(u))\bar\rho\,(du) \endaligned\tag2.22
$$
with $\bar\rho=\bar\rho_{(a,b)}^{K(x)}$ if case~a) and
$\bar\rho=\bar\rho^\infty_{(a,b)}$ if case~b) of Theorem~2 holds.
Hence
$$ \lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}g(F(R),F(R+w(R,T))\,dR=
\int g(u,v)\bar\mu(du,dv) \tag2.23
$$
with
$$
\bar\mu(\bold A)=\bar\rho\{u\: u\in\Cal V\times\Cal
V,\quad G(V(u))\in\bold A\} \tag2.24
$$
for all measurable sets $\bold A\in \Bbb R^2$.
Relation (2.23) implies the weak convergence of the measures
$\mu_{T,w,(a,b)}$ to $\bar\mu$. Formula (2.24) together with the form
of the measures $\bar\rho$ imply that the measure $\mu$ has the
prescribed form if cases~a) of Theorem~2 holds, i.e.\ it depends only
on the function $K(x)$. If case~b) of Theorem~2 holds, then a
comparison of formulas (2.12) and (2.24) together with relation
$(2.17'')$ and the product form of the functions $G$ and $V$ in
formulas (2.14) and (2.15) imply formula~(1.8). Relation (1.9) can be
deduced from the weak convergence in the same way as the analogous
result in Proposition~1. To prove formula (1.10) and continuity of the
measures $\mu^{zK(x)}_{(a,b)}$ (in the variable $z$) it is enough to
prove the continuity of the expression $\int g(G(V(u))\bar
\rho_{(a,b)}^{zK(x)}\,(du)$. Because of the Weierstrass approximation
theorem it is enough to check the continuity of the Fourier
coefficients. This follows from relations (2.20) and (2.21).
Proposition~2 is proved.\qed \enddemo
\demo {Proof of Proposition 3}
The proof is based on a representation similar to Proposition~2.
Define the maps $U\:\Bbb R^1\to \Cal V\times \Cal V$
$$
\aligned
U(R)=U(R,x)&=\{R\tau_k\text{ (mod}\,1),\quad k=1,\dots, s\;, \\
&\qquad\qquad (R+x)\tau_k\text{ (mod}\,1),\quad k=1,\dots, s \}\;.
\endaligned
\tag2.25
$$
and the maps $V(u_1,\dots,u_{2s})$ and $G(u_1,\dots,u_{2p})$ by formulas
(2.14) and (2.15) as in the proof of Proposition~2. Introduce the
measures
$$
\hat\rho_{T,x}(\bold A)=\frac1{T}\lambda\{R\: 0\le R\le
T,\quad U(R,x)\in \bold A\}\tag2.26
$$
for any measurable set $\bold A\subset \Cal V\times\Cal V$. Then
$$
\hat\rho_{T,x}\Rightarrow\hat\rho^x\quad \text{as }T\to\infty\tag2.27
$$
with Fourier coefficients
$$
L^x(m_1,\dots,m_{2s})=\cases
0&\text{if }A(m_1,\dots,m_{2s})\neq0\\
\exp\{i B(m_1,\dots,m_{2s})x\} &\text{if }
A(m_1,\dots,m_{2s})= 0
\endcases\;. \tag2.28
$$
with the functions $A(m_1,\dots,m_{2s})$ and $B(m_1,\dots,m_{2s})$
defined in $(2.19')$. Then we get the proof of the identity (1.14) as in
the proof of Proposition~2 with the limit measure
$$
\nu^x(\bold A)=\hat\rho^x\{u\: u\in\Cal V\times\Cal
V,\quad G(V(u))\in\bold A\} \tag2.29
$$
for all measurable $\bold A\in \Bbb R^2$. The expression at the
right-hand side of (1.14) is clearly a bounded function, and it is a
continuous function of $x$, because the Fourier coefficients of
$\hat\rho^x$ are continuous functions of $x$. A
comparison of the Fourier coefficients in (2.20) and (2.28) yields that
$$
\bar\rho^{K(x)}_{(a,b)}=\frac1{(b-a)}\int_{a} ^{b} \hat\rho^{K(x)}\,dx=
\frac z{(b-a)}\int_{K(a)} ^{K(b)} \frac1{K'K^{-1}(x)}\hat\rho^x\,dx\;.
$$
This relation together with (2.24) and (2.29) imply relation (1.15).
Proposition~3 is proved. \qed
\enddemo
\beginsection 3. Proof of the Theorems
First we formulate an estimate which enables us to generalize the
results of Section~2 to functions satisfying (1.1) and (1.2).
Let $g(u,v)$ be a continuous function such that $|g(u,v)|0$ and $B>0$, $F(R)$ a function satisfying (1.1) and (1.2),
$F_p(R)$ the trigonometrical series containing the first $p$ terms of
$F(R)$, and let some numbers $0< a**0$ there are some thresholds $p_0=p_0(\e)$ and $T_0=T_0(p,\e)$
for $p>p_0$ such that
$$
\aligned
&\left|\,\frac1{(b-a)T}\int_{aT}^{bT}
g\bigl(F(R),F(R+w(R,T))\bigr)\,dR \right. \\
&\qquad\qquad\left.-\frac1{(b-a)T}\int_{aT}^{bT}
g\bigl(F_p(R),F_p(R+w(R,T))\bigr)\,dR\,\right|<\e
\endaligned \tag3.1
$$
for any $p>p_0$ and $T>T_0(p,\e)$. Moreover, the threshold $p_0$ can be
chosen depending only on $\e$, $a$, $b$ and $g(u,v)$, but not depending on
the choice of the function $w(R,T)$ of which we only require that it
satisfied the conditions of Theorem~2.
To prove relation (3.1) first we make the following observation: For
any $(u,v)\in \Bbb R^2$ and $(u_0,v_0)\in \Bbb R^2$ and $1>\eta>0$ there
exist some constants $K=K(\eta)$ and $K_0$ depending only on the
function $g(u,v)$ such that
$$
\aligned
|g(u,v)-g(u_0,v_0)|&<\eta+K((u-u_0)^2+(v-v_0)^2)
+K_0(u_0^2+v_0^2)I(\{u_0^2+v_0^2>\eta^{-1}\})\\
&\qquad+K_0(u^2+v^2)I(\{u^2+v^2>\eta^{-1}\})\;,
\endaligned \tag3.2
$$
where $I(A)$ denotes the indicator function of the set $A$.
Indeed, relation (3.2) holds with
$$
K_0=2\sup\limits_{u^2+v^2\ge 1}\frac{|A(u^2+v^2)+B|}{u^2+v^2}\le2(A+B)
$$
if $u_0^2+v_0^2>\eta^{-1}$ or $u^2+v^2>\eta^{-1}$. On the complementary
set this inequality holds if $(u_0-u)^2+(v_0-v)^2<\delta$ with some
$\delta=\delta(\eta)$ because of the uniform continuity of the function
$g(u,v)$ on this set. Finally, relation (3.2) holds on the remaining
set if $K=K(\eta)$ is chosen sufficiently large. Relation (3.2) can be
rewritten in a simpler form. We can apply the inequality
$$
(u^2+v^2)I(\{u^2+v^2>\eta^{-1}\})\le 2u^2I(\{u^2>(2\eta)^{-1}\}+
2v^2I(\{v^2>(2\eta)^{-1}\}\,,
$$
and write with the help of this relation that
$$
\aligned
|g(u,v)-g(u_0,v_0)|&<\eta+K\bigl((u-u_0)^2+(v-v_0)^2\bigr)\\
&\qquad+K_0\bigr[(u_0^2I(\{u_0^2>\eta^{-1}\}+v_0^2I\{v_0^2>\eta^{-1}\}
\bigr] \\
&\qquad+K_0\bigr[(u^2I(\{u^2>\eta^{-1}\}+v^2I\{v^2>\eta^{-1}\}\bigr]\;.
\endaligned
\tag$3.2'$
$$
with a new constant $K$ which corresponds to the bound $\eta/2$ and
with a new constant $K_0$ which is the double of the original one.
We shall prove (3.1) from $(3.2')$ with the
choice $(u_0,v_0)=(F_p(R),F_p(R+w(R,T)))$, $(u,v)=(F(R),F(R+w(R,T)))$
with an appropriate $\eta>0$ and $p=p(\eta)$
and then by integration with respect to $R$.
By relation $(3.2')$
$$
\align
&\bigl|g\bigl(F(R),F(R+w(R,T))\bigr) -g\bigl(F_p(R),F_p(R+w(R,T))\bigr)
\bigr| \\
&\qquad<\eta+K\bigl[(F(R)-F_p(R))^2+(F(R+w(R,T))-F_p(R+w(R,T)))^2\bigr]\\
&\qquad
+K_0\bigl[F_p(R)^2I\{F_p(R)^2>\eta^{-1}\}+F_p(R+w(R,T))^2
I\{F_p(R+w(R,T))^2>\eta^{-1}\}\bigr]\\
&\qquad
+K_0\bigl[F(R)^2I\{F(R)^2>\eta^{-1}\}+F(R+w(R,T))^2I\{F(R+w(R,T))^2
>\eta^{-1}\}\bigr].
\endalign
$$
The inequality
$$
\left|\frac{K_0}{(b-a)T}\int_{aT}^{bT}
F(R)^2I\{F(R)^2>\eta^{-1}\}\,dR\right|<\frac\e8 \tag3.3
$$
holds, if $\eta<\eta(\e)$ and $T>T(\e)$. Indeed, by relation (1.5)
there is some $\bar p=\bar p(\e)$, and $T(\e)$ in such a way that
$$
\frac{K_0}{(b-a)T}\int_{aT}^{bT}|
F(R)-F_{\bar p}(R)|^2\,dR<\frac\e{32}\;.
$$
for $T>T(\e)$. The function $|F_{\bar p}(R)|$ is bounded. Put $\eta=
\inf\limits_R 4|F_{\bar p}(R)|^{-2}$. Then the last inequality implies
(3.3), since $F(R)^2<4|F(R)-F_{\bar p}(R)|^2$ on the set
$\{F(R)^2>\eta^{-1}\}$. We also claim that
$$
\left|\frac{K_0}{(b-a)T}\int_{aT}^{bT}
F(R+w(R,T))^2I\{F(R+w(R,T))^2>\eta^{-1}\}\,dR\right|<\frac\e8 \tag$3.3'$
$$
if $\eta<\eta(\e)$ and $T>T(\e)$. This can be proved similarly to (3.3)
with some modification. Apply the change of variables $u=R+w(R,T)$ in
the integral in $(3.3')$. Since $w(R,T)$ satisfies the conditions
Theorem~2, $\frac{du}{dR}\to1$ uniformly for $aT\le R\le bT$, as
$T\to\infty$. The domain of integration after this change of variable
is the interval $[aT(1+o(1)), bT(1+o(1))]$. Hence after this change of
variables the integral in $(3.3')$ can be estimated in the same way as
in (3.3). Relations (3.3) and $(3.3')$ remain valid if the function
$F(R)$ is replaced by $F_p(R)$ with $p>\bar p$, and $T>T(\e,p)$.
Choose $\eta$ so that relations (3.3) and $(3.3')$ and their variants
for the function $F_p(R)$ hold and $\eta<\e/4$. Then, because of
relation (1.5) and the argument in the proof of $(3.3')$ some
thresholds $p_0=p(\eta)$ and $T_0=T_0(\eta,p)$ can be chosen in
such a way that for $p>p_0$ and $T>T_0$
$$
\frac{K}{(b-a)T}\int_{aT}^{bT} (F(R)-F_p(R))^2\,dR<\frac\e8\tag3.4
$$
and
$$
\frac{K}{(b-a)T}\int_{aT}^{bT} (F(R+w(R,T))-F_p(R+w(R,T)))^2\,dR<\frac\e8
\tag$3.4'$
$$
with the constant $K=K(\eta)$ appearing in formula $(3.2')$. Formulas
(3.3), $(3.3')$, their variants for the function $F_p(R)$, (3.4) and
$(3.4')$ together with the relation $\eta<\e/4$ imply (3.1).
It follows from (3.1) and relation (1.9) already proved for the
function $F_p$ that
$$
\aligned
\limsup_{T\to\infty}&\left|\frac1{(b-a)T}\int_{aT}^{bT}
g\bigl(F(R),F(R+w(R,T))\bigr)\,dR \right. \\
&\qquad\left.-\int g(u,v)\bar\mu(F_p)(\,du,\,dv)\,\right|<\e
\endaligned
$$
for all $p>p(\e)$ with $\bar\mu(F_p)=\mu_{(a,b)}^{K(x)}(F_p)$ if
$w(R,T)$ satisfies the conditions of case~a) and with
$\bar\mu(F_p)=\mu_{(a,b)}^\infty(F_p)$ if it satisfies the conditions
of case~b) of Theorem~2. This relation implies that
$$
\aligned
\lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}
g\bigl(F(R),F(R+w(R,T))\bigr)\,dR&=\lim_{T\to \infty}\int
g(u,v)\mu_{T,w,(a,b)}(\,du,\,dv)\\
&=\lim_{p\to\infty}\int g(u,v)\,\bar\mu(F_p)(du,\,dv)\;.
\endaligned
\tag3.5
$$
with the same choice of the measure $\bar\mu(F_p)$ as in the previous
formula. The last relation also means that all limits in this formula
exist. Since relation (3.5) also holds for $g(u,v)=u^2+v^2$, hence the
measures $\mu_{T,w,(a,b)}$ are uniformly tight, and we get by applying
relation (3.5) that there exists the limit
$$
\bar\mu=\lim_{T\to\infty}\mu_{T,w,(a,b)}=\lim_{p\to\infty}
\bar\mu(F_p)\;,\tag3.6
$$
with the same measures $\bar\mu(F_p)$ as in (3.5) and in the previous
formula, and also relation (1.9) holds with the function $F(R)$.
Moreover, for a fixed bounded continuous function $g(u,v)$ the limit
$$
\lim_{p\to\infty}\int g(u,v)\,\mu^{zK(x)}_{(a,b)}(F_p)(du,\,dv)=
\int g(u,v)\,\mu^{zK(x)}_{(a,b)}(du,\,dv)
$$
is uniform in $z$, and this fact together with the continuity
properties of the measures $\mu^{zK(x)}_{(a,b)}(F_p)$ imply the
continuity of the measures $\mu^{zK(x)}_{a,b}$ for $00$. Some
of the results follow automatically also for $a=0$ from our results,
but to generalize all statements of Theorem~2 to the case when the
parameter $a$ can take also the value zero some additional conditions
must be imposed. To carry out all required limiting procedures we must
know that
$$
\lim_{\e\to0}\limsup_{T\to\infty}\frac1T\int_0^{\e T}F(R+w(R,T))\,dR=0
$$
To guarantee the last relation some additional properties should be
imposed on the function $w(R,T)$. Since the restriction $a>0$ is not
essential in applications we proved Theorem~2 only under the condition
$a>0$.
The results of this paper were proved originally for finite
trigonometrical sums in Section~2, and then in Section~3 these results
were generalized to functions which can be well approximated by finite
trigonometrical sums. The content of formulas (1.1) and (1.2) was the
possibility of such a good approximation. In applications this condition
can be checked. On the other hand, the weak convergence of the
random variables $F(R)$, $F\bigl((R),F(R+w(R,T))\bigr)$ or
$\bigl(F(R),F(R+x)\bigr)$ in Theorems 1,~2 and~3 also hold if formulas
(1.1) and~(1.2) are replaced by the following weaker condition: There
exists a sequence of finite trigonometrical sums $F_p(R)$,
$p=1,2,\dots$, such that
$$
\lim_{p\to\infty}\limsup_{T\to\infty}\frac1{2T}\int_{-T}^T
\min\{1,|F_p(R)-F(R)|\}\,dR=0\;.\tag4.1
$$
Similar conditions were formulated in paper~[2] or~[4].
We only briefly explain why formula (4.1) implies the weak convergence
in Theorems~1, 2~and~3. If we consider functions of the form
$g(u,v)=g_{s,t}(u,v)=e^{i(su+tv)}$, then one can show by means
of condition (4.1) and the relation $\frac{\partial}{\partial
R}(R+w(R,T))=1+o(1)$ that the under the conditions of Theorem~2
the limits
$$
\aligned
&\lim_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}
g\bigl(F(R),F(R+w(R,T))\bigr)\,dR \\
&\qquad=\lim_{T\to\infty}\lim_{p\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}
g\bigr(F_p(R),F_p(R+w(R,T))\bigr)\,dR
\endaligned \tag4.2
$$
exist. Hence to prove the weak convergence of the distribution of the
random vectors $\bigl(g(F(R),g(F(R+w(R,T))\bigr)$, $aR\le T\le bT$, it
is enough to check the compactness of these distributions in the weak
topology. To do this it is enough to show that for any $\e>0$ there
exits a constant $K=K(\e)$ such that the following relation holds.
The function $h(u,v)=h_K(u,v)=H_K(u^2+v^2)$, where $H_K(u)$ is defined
by the relations $H_K(u)=0$ for $|u|\le K$, $H_K(u)=1$
for $|u|\ge2K$, and $H_K(u)$ is given by linear interpolation for
$K<|u|<2K$, satisfies the inequality
$$
\limsup_{T\to\infty}\frac1{(b-a)T}\int_{aT}^{bT}
h\bigr(F_p(R),F_p(R+w(R,T))\bigr)\,du\,dv<\e\;.
$$
To prove this relation, choose a number $\bar p$ such that for
$T>T(\bar p)$
$$
\frac1{2T}\int_{-T}^T
\min\{1,|F_{\bar p}(R)-F(R)|\}\,dR<\frac{(b-a)}2\e\;.
$$
Then, since $F_{\bar p}(R)$ is a bounded function we can choose
$K=1+2\sup\limits_R F_{\bar p}(R)$. It is not difficult to see that
$\int h\bigl(F_{\bar p}(R),F_{\bar p}(R+w(R,T))\bigr)\,dR=0$, and
relation (4.2) holds with this choice of the function $h_K(u,v)$. The
analogue of Theorem~2 under condition (4.1) can be proved by working out
the details. The modified version of Theorems~1 and~3 can be proved
similarly.
In this paper we did not discuss such functions $w(R,T)$ which satisfy
the relation
$$
\lim\limits_{T\to0}\sup\limits_{0\le R\le T}w(R,T)=0\;.
$$
The reason for this omission is not our disinterest for this case.
Actually, the description of this case is a very exciting problem. This
is related to the investigation of the limit behavior of the number of
lattice points in randomly chosen thin strips. This is a very
interesting problem with many unsolved conjectures and few rigorous
results. The methods of the present paper are not sufficient to study
such problems. Here some essentially new ideas are needed.
\bigskip
\noindent {\bf References}
\item{1.} Besicovitch, A. S.: Almost periodic functions. Dover
Publications, New York 1958
\item{2.} Bleher, P. M.: On the distribution of the number of lattice
points inside a family of convex ovals. {\it Duke Math. Journ.} {\bf
67}, 3, 461--481 (1992)
\item{3.} Bleher, P. M. and Lebowitz, J. L.: Energy--level statistics of
model quantum systems: Universality and scaling in a lattice-point
problem {\it Journal of Statistical Physics}\/ (to appear)
\item{4.} Heath--Brown, D. R.: The distribution and moments of the error
term in the Dirichlet divisor problem. {\it Acta Arithmetica}\/ {\bf
LX}, 389--415 (1992)
\bye
**