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%%%%% to appear in the Proceedings of
%%%%% the Workshop ``Emerging Applications of Number Theory", held in
%%%%% July 1996 at the Institute for Mathematics and
%%%%% Its Applications, University of Minnesota.
%%%%%
%%%%% Lev Kapitanski and Igor Rodnianski %%%%%%%%%%%%%%%%%%
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\topmatter
\title Does a Quantum Particle Know the Time?
\endtitle
\author Lev Kapitanski and Igor Rodnianski
\endauthor
\affil Department of Mathematics\\
Kansas State University\\
Manhattan, Kansas 66506
\endaffil
%\thanks{}\endthanks
\endtopmatter
Consider the Hamiltonian
$\,H\,=\,-(1/ 4\pi) (\partial^2/\partial x^2)\,$ on the circle
$\,\Bbb T = \Bbb R/\Bbb Z$. The unitary exponent $\,\exp i t H\,$
is the solution operator for the time-dependent Schr\"odinger equation
$$
{1\over i}\,{\partial\hfill\over\partial t}\,E\,+\,
{1\over 4\pi}{\partial^2\hfill\over\partial x^2}\,E\,=\,0, \tag 0.1a
$$
i.e., $\,\exp i t H\,E^0\,(x)\,$ is the solution of (0.1a)
with the initial condition $\,E^0$. The distributional kernel
of $\,\exp i t H\,$ can be written as a series,
$$
\langle x | \exp i t H | y\rangle\,=\,
\sum_{n\in \Bbb Z}\,\be ( {n^2 t\over 2} + n (x-y))\,.
$$
where we use the notation $\,\be (z)= e^{2 \pi i z}$.
We replace $\,(x-y)\,$ by $\,x\,$ and look at the function
$$
E(t,x)\,=\,
\sum_{n\in \Bbb Z}\,\be ( {n^2 t\over 2} + n x )\,.\tag 0.2
$$
We will be interested in the regularity of $\,E(t,x)\,$ in $\,x\,$
at different times $\,t\in [0,2)$.
%%%%%%%%%%%%
It will be convenient to view the functions on $\,\Bbb T\,$ as periodic
functions on $\,\Bbb R\,$ of period 1. Then (0.2) is the solution of (0.1a)
corresponding to the initial condition -- a comb,
$$
E(0,x)\,=\,E^0(x)\,:=\,\sum_{n\in \Bbb Z}\,\delta(x-n).\tag 0.1b
$$
The usual framework for describing the regularity of the solutions of
the Schr\"odinger equation (0.1a) is the scale
$\,\{H^s, \;s\in\Bbb R\}\,$ of $\,L^2$-Sobolev spaces. Recall that
a distribution $\,f(x)=\sum_m f_m\be(m x)\,$ belongs to the space
$\,H^s\,$ if $\,\sum_m \langle m\rangle^{2s}\,|f_m|^2\,<\infty$,
where $\,\langle m\rangle = (1+m^2)^{1/2}$. The exponent $\,\exp i t H\,$
is a continuous operator in each of the Sobolev spaces $\,H^s$.
The comb-function (0.1b) lies in $\,H^s$, where $\,s\,$ is any number less
than $\,-1/2$. For {\it every\/} $\,t>0$, the solution (0.2),
if viewed through
the telescope of Sobolev spaces, has the same regularity in $\,x\,$ variable,
namely, $\,E(t,\cdot)\in\cup_{s<-1/2}\; H^s$.
\medskip
%%%%%%%%%%%%
The Sobolev spaces are not the only function spaces
that can be applied to the analysis of the solutions of
the Schr\"odinger equation. Of special interest for us here
will be the Besov spaces. We show that the regularity
of $\,E(t,\cdot)$, when measured in the appropriate
Besov spaces, changes with $\,t$. The most drastic difference
in regularity is between the cases when $\,t\,$ is
rational and when $\,t\,$ is irrational.
Within the set of irrational times, although there exists a generic
regularity for generic $\,t$, there are different thin classes
of irrationals which prescribe their particular regularity
to the fundamental solution. These classes are singled out
and characterized by the behavior of the continued fraction
expansions of their members.
\medskip
%%%%%%%%%%%%
Note, that when $\,t\,$ is in the upper
half-plane, then $\,E(t,x)\,$ defined in (0.2) is
(essentially) Jacobi's theta-function. The well-known transformation
properties of theta-functions allow to express
$\,E(t,x)$, when $\,t$ is rational, as a linear
combination of $\,\delta$-functions sitting in
a (depending on $\,t$) finite number of points $\,x\,$
on the circle. This completely answers the question
of regularity at rational times.
%%%%%%%%%%%%
If $\,t\,$ is irrational, the situation is more complicated.
Our choice of Besov spaces to measure the regularity of
$\,E(t,\cdot)$, requires the estimates in $\,L^{\infty}\,$
of the exponential
sums of the form
$$
\sum\limits_{n\in\Bbb Z}\chi(2^{-j}|n|)\,
\be ( {n^2 t\over 2} + n x ),\tag 0.3
$$
for large $\,j$. Here $\,\chi\,$ is a cut-off function, which
is supported on the interval $\,[1/2,\,2]$, and which
is either smooth, or equals $\,1$ on this interval.
The exponential sums (0.3) have been studied extensively,
especially during the last 90 years. Of particular importance for
us are the results of \cite{Hardy \& Littlewood, 1914} with
subsequent developments of \cite{Mordell, 1926}, and
\cite{Fiedler, Jurkat \& K\"orner, 1977}, and \cite{Bombieri, 1990}.
\medskip
%%%%%%%%%%%%
We should mention that K. I. Oskolkov have obtained some nice
results on the regularity of certain solutions of the
Schr\"odinger equation, which exhibit different behavior
for rational and irrational times, see \cite{Oskolkov, 1992}
and further references to his works therein. However, the
questions he discusses and the function spaces he uses
are different from the ones we concern ourselves in the present
paper.
Also related to what we are doing, are the studies of the value distribution
properites of theta-sums in
\cite{Jurkat \& van Horn, 1981, 1982}, \cite{Sarnak, 1981, 1982},
\cite{Marklof, 1996}, and the studies of the geometric patterns
generated by theta-sums in \cite{Dekking \& Mend\`es France, 1981},
\cite{Deshouillers, 1985},
\cite{Berry \& Hannay, 1987}, \cite{Berry \& Goldberg, 1988},
\cite{Coutsias \& Kazarinoff, 1987}.
\medskip
%%%%%%%%%%%%
Our interest in the regularity of $\,E(t,x)\,$ stemmed,
initially, from
our previous work on the regularity of the fundamental
solution for the time-dependent Schr\"odinger equation
in $\,\Bbb R^n\,$ with a growing at infinity potential.
There, the model problem is the follwoing:
$$
{1\over i}\,{\partial\hfill\over\partial t}\,E\,-\,
\Delta_x E\,+\,|x|^{\rho}\,E\, =\,0,\quad t\in\Bbb R^1,\;x\in\Bbb R^d\,;
\qquad E(0,x,y)\, =\,\delta (x-y)\,,
\tag 0.4
$$
$d\ge 1$, and $\,\rho\,$ is a positive constant.
The regularity of $\,E(t,x,y)\,$ depends on the rate of growth of the
potential. If $\,\rho<2$, then $\,E(t,x,y)\,$ is $\,C^{\infty}$-smooth
in $\,(t,x,y)$, $\,t\ne 0$; see \cite{Yajima, 1996},
\cite{Kapitanski \& Rodnianski}. In the case $\,\rho=2\,$ the solution,
$\,E(t,x,y)$, is given by Mehler's formula and shows that the singularities
reappear at resonant times $\,t=m\pi$, $\,m\in\Bbb Z$, while for
all other $\,t\,$ the fundamental solution is smooth. This picture survives
the perturbations of $\,|x|^2\,$ by functions growing slower than
quadratically, \cite{Kapitanski, Rodnianski, Yajima, 1997}.
%%%%%%%%%%%%
When $\,\rho>2\,$ and $\,d>1$, nothing is known about
the regularity of $\,E(t,x,y)\,$ (except, of course, for what the standard
energy estimates give). However, it is likely that
the fundamental solution is nowhere smooth. This conjecture is
supported by a remarkable recent result of K.~Yajima, who showed
that in the one-dimensional case ($d=1$), if $\,\rho>2$, then
$\,E(t,x,y)\,$ is not even in the local Bessel-Sobolev space
$\,\Cal L^{1/\rho}_{1,\,loc}(\Bbb R^3)$, see \cite{Yajima, 1996}.
%%%%%%%%%%%%
The initial boundary value problem (IBVP) for the Schr\"odinger equation
may be viewed as the extreme limit case of (0.4) as $\,\rho\to\infty$.
Yajima's technique, when applied to the one-dimensional IBVPs,
gives the corresponding nonsmoothness results. In particular,
it turns out that
the distributional kernel $\,E(t,x,y)\,$
of the operator $\,\exp\{-i t \Cal H\}$, where
$\,\Cal H=-(1/4\pi)\,d^2/dx^2\,$ on the interval $\,[0,1/2]\,$
with Dirichlet boundary conditions, i.e., the function
$\,E(t,x,y)=4\sum_{n=1}^{\infty}
e^{-\pi itn^2}\sin\,2\pi nx\,\sin\,2\pi ny\,$
is nowhere locally integrable (in $\,\Bbb R^3$),
see \cite{Yajima}, Remark 4.
Our results will add to this by revealing the fine changes
in regularity of $\,E(t,\cdot,\cdot)\,$ at different times $\,t$.
\medskip
%%%%%%%%%%%%
{\bf ACKNOWLEDGMENTS.\/} This work was partially supported by NSF grant DMS-9623520.
The authors thank Peter Sarnak and Jens Marklof
for valuable discussions.
The second author also thanks the Institute for Mathematics and
Its Applications at the University of Minnesota, and the organizers of
the workshop ``Emerging Applications of Number Theory" at IMA, for their
hospitality.
\medskip
%%%%%%%%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%
\head {\bf 1\/}\ \ Statements of Results \endhead
We start with the definitions of the Besov spaces
we need to state the results. For reference on Besov spaces
see \cite{Bergh \& L\"ofstr\"om}, \cite{Triebel}.
Let $\, \chi\,$ be a $\,C_0^{\infty}\,$
function on $\,\Bbb R_+\,$ with the following properties:
$$
\text{supp}\,\chi\,=\,[2^{-1}, 2], \tag 1.1a
$$
and
$$
\sum\limits_{j=-\infty}^{\infty}\,\chi (2^{-j}\xi)\,=\,1,
\qquad\forall \xi >0.\tag 1.1b
$$
Define the functions
$$
\chi_j(\xi) =\chi (2^{-j}\xi), \qquad j=1,\,2, \dots;\qquad
\chi_0(\xi) =
1-\sum\limits_{j=0}^{\infty}\,\chi_j (\xi)\,. \tag 1.2
$$
With each of these functions we associate an operator $\,K_j$, which
maps a distribution $\,f(x)=\sum_m f_m\, \be ( m x)\,$
to a (finite) exponential sum,
$$
K_j(f)(x)=
\sum_{m=-\infty}^{\infty}\, \chi_j(|m|)\, f_m\, \be ( m x),
\quad j=0,\,1,\dots.\tag 1.3a
$$
We define the Besov spaces $\,B^s_{u,v}\,$ on $\,\Bbb T$
for the following values of parameters $\,s$, $\,u$, and $\,v$:\
$\;-\infty~~0.
\endsplit
\tag 1.3b
$$
The norm in $\,\lceil B\rceil^s_{u,v}$ is defined as in
(1.4), with $\,K_j\,$ replaced by $\,\lceil K\rceil_j$.
It is known that if $\,1~~__0}\,B^{s+\epsilon}_{\infty}$.
A similar convention will be in force for the spaces
$\,\lceil B\rceil^s_{\infty}$ and $\,H^s$.
%%%%%%%%%%%%%%%%
Closely related to the persistence of regularity
of solutions of (0.1a) in $\,H^s\,$ is the following fact.
\proclaim{Remark 1.1} For all $\,t$,
the distribution $\,E(t,\cdot)\,$ belongs to
$\,B^{-1/2}_{2,\infty}\,$ sharp.
\endproclaim
\proclaim{Remark 1.2}
That $\,E(t,\cdot)\,$ belongs to the Besov space
$\,\lceil B\rceil^{-\alpha}_{\infty}\,$ is equivalent to the existence of a
constant $\,\lceil C\rceil\,$ such that
$$
\sup_x\,
\big | \sum\limits_{n=2^{j-1}}^{2^{j+1}} \be ({n^2t\over 2}+nx)\big |
\le \lceil C\rceil\,2^{\alpha j},
\quad\text{\rm for all sufficiently large}\quad j\,.
\tag 1.5a
$$
That $\,E(t,\cdot)\,$ belongs to the Besov space
$\, B^{-\alpha}_{\infty}\,$ is equivalent to the existence of a
constant $\,C\,$ such that
$$\multline
\sup_x\,
\big | \sum\limits_{n=2^{j-1}}^{2^{j+1}}
\chi(2^{-j}n) \be ({n^2t\over 2}+nx)+
\chi(2^{-j}n) \be ({n^2t\over 2}-nx)
\big |
\le C\,2^{\alpha j},\\
\hfill\text{\rm for all sufficiently large}\quad j\,.\hfill
\endmultline
\tag 1.5b
$$
\endproclaim
\qed
%%%%%%%%%%%%%%%%
The regularity properties of $\,E(t,\cdot)\,$ in Besov spaces
$\,B^s_{\infty}\,$ and $\,\lceil B\rceil^s_{\infty}$
depend on the
continued fraction representation of $\,t$.
We refer the reader to \cite{Khinchin, 1964} and \cite{Schmidt, 1980}
for the basic
theory of continued fractions.
Consider first the case of a rational $\,t=\frac pq \in [0,2)$.
In this case $\,t\,$ has a finite continued fraction expansion:
$$
t=\frac pq=[a_0,a_1,\dots , a_n]\,=\,a_0+\,
{1\over\displaystyle a_1+
{\strut 1\over\displaystyle \dots
{\strut \dots\over
{\strut 1\over\displaystyle a_n}}}}\;.
$$
Note, that the expansion is not unique: we also have
$\,\frac pq = [a_0,a_1,\dots, a_{n-1}, a_n-1,1]$, if $\,a_n\ne 1$,
and $\,\frac pq = [a_0,a_1,\dots, a_{n-1}+1]$, if $\,a_n=1$.
%%%%%%%%%%%%%%
If $\,t=[a_0,a_1,\dots , a_n]$, then the numbers
$\,{p_k/ q_k}\,=\,[a_0,a_1,\dots , a_k]$,
$\,k=1,\dots, n-1$, are the corresponding convergents, and
$\,p_n=p$, $\,q_n=q$.
\proclaim{Theorem {\rom I\/}} Let $\,t\in [0,2)$ be a rational number,
and $\,t=\frac pq$ -- its simple fraction representation.
Let $\,{p_k\over q_k}$, $\,k=1,\dots, n-1$, be partial convergents
to $\,t$ determined by a finite continued fraction
$\,[a_0,a_1,\dots , a_n]$ with an odd number of quotients
(i.e., $\,n\,$ is even). Then,
1) (formula)
$$
E(\frac pq,x)={\varkappa_0(t)\over\sqrt{q}}\,
\be (-\frac 12 q_{n-1}q x^2+
\frac 12 q\eta x -
\frac 18 \xi\eta)\,
\sum\limits_{n} \delta ({n+\frac 12\xi\over q} - x)\,, \tag 1.6
$$
where
$$
\xi = p\cdot q\;(\text{\rm mod\/}\;2),
\qquad \eta = p_{n-1}\cdot q_{n-1}\;(\text{\rm mod}\;2)\,,
$$
and $\,\varkappa_0(t)\,$ is an eighth root of $\,1$;
2) (regularity)
$$
E(t,\cdot)\,\in\,B^{-1}_{\infty}\cap \lceil B\rceil^{-1}_{\infty}
\quad
\text{sharp}\,.\tag 1.7
$$
\endproclaim
\medskip
When $\,t\in(0,2)\,$ is irrational, its continued fraction expansion
$\,t=[a_0,a_1,\dots , a_n,\dots]\,$
is infinite and unique.
The regularity of $\,E(t,\cdot)\,$ for generic irrational $\,t\,$
is given by the following
\proclaim{Theorem {\rom {II}}}
(i) For almost all irrational $\,t$,
$\;
E(t,\cdot)\,\in\,\cap_{\ve>0} ( B^{-1/2-\ve}_{\infty}\cap \lceil B\rceil^{-1/2-\ve}_{\infty} )\,$.
(ii) If $\,t\,$ is an irrational number with bounded quotients,
i.e.,
there is a constant $\,C>0\,$ such that $\,a_n\le C$, for all $\,n$,
then $\;
E(t,\cdot)\,\in\,B^{-1/2}_{\infty}\cap \lceil B\rceil^{-1/2}_{\infty}
\quad
\text{sharp}$.
(iii) There is no $\,t\,$ for which
$\,E(t,\cdot)\,$ belongs to
$\, B^{(-1/2) +\ve}_{\infty}\cup \lceil B\rceil^{(-1/2)+\ve}_{\infty}$ with
any positive $\,\ve$.
\endproclaim
\medskip
We now define a few (narrower) classes of irrational numbers
using restrictions on the growth of the denominators $\,q_n$ of
their convergents $\,p_n/q_n$.
For $\,\sigma\ge 0$, denote by $\,\Cal I(\le\sigma)\,$ the set of all
irrational $\,t\,$ such that for each of them there exists a constant
$\,C_t\,$ such that
$$
q_{n+1}\,\le\,C_t\,q_n^{1+\sigma}\,,\qquad
\text{for all sufficiently large $\,n$}.\tag 1.8
$$
Denote by $\,\Cal I(\ge\sigma)\,$ the set of all
irrational $\,t\,$ such that for each of them there exists a constant
$\,c_t>0\,$ such that
$$
q_{n+1}\,\ge\,c_t\,q_n^{1+\sigma}\,,\qquad
\text{for an infinite number of $\,n$}.\tag 1.9
$$
Finally, denote
$\,\Cal I(\sigma)\,=\,\Cal I(\le\sigma)\,\cap\,\Cal I(\ge\sigma)$.
\proclaim{Theorem {\rom {III}}}
(i) If $\,t\in\Cal I(\le\sigma)\,$, then
$$
E(t,\cdot)\in B^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\,\cap\,
\lceil B\rceil^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\,. \tag 1.10a
$$
(ii) If $\,t\in\Cal I(\ge\sigma)\,$, then
$$
E(t,\cdot)\notin
\left(
\cup_{\epsilon>0}B^{{}^{-{1+\sigma\over 2+\sigma}+\epsilon}}_{\infty}
\right)\,
\cup\,
\left(\cup_{\epsilon>0}\lceil
B\rceil^{{}^{-{1+\sigma\over 2+\sigma}+\epsilon}}_{\infty}
\right)\,. \tag 1.10b
$$
(iii) If $\,t\in\Cal I(\sigma)\,$, then
$$
E(t,\cdot)\in B^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\,\cap\,
\lceil B\rceil^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\quad
\text{sharp}. \tag 1.10c
$$
\endproclaim
\medskip
%%%%%%%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%%%%%%%%%
\head {\bf 2\/}\ \ Estimates for Exponential Sums \endhead
\medskip
In this section we study two basic exponential sums,
$$
\lceil S\rceil_M^N\,(t,x)\,=\,
\sum_{M\le |n|\le N}\,\be\,({n^2t\over 2}\,+\,n x)\,,\tag 2.1
$$
and
$$
S_M^N\,(t,x)\,=\,
\sum_{M\le |n|\le N}\,\omega_n\,\be\,({n^2t\over 2}\,+\,n x)\,.\tag 2.2
$$
The first sum is needed for the estimates of the norm of $\,E(t,\cdot)\,$
in the space $\,\lceil B\rceil^s_{\infty}$. The second
is for the space $\, B^s_{\infty}$. There, the r\^ole of coefficients $\,\omega_n\,$
will be assigned to $\,\chi(2^{-j}|n|)$, see (0.3).
However, in this section we do not restrict ourselves
to this particular choice of $\,\omega_n$.
%%%%%%%%%%%%%%
The estimates from above on
$\,|\sum_{n=1}^N\,\be\,({n^2t\over 2}\,+\,n x)|\,$
go back to \cite{Hardy \& Littlewood}. They introduced the method
of an approximate functional equation for incomplete theta-sums.
This method was developed further by \cite{Fiedler, Jurkat \& K\"orner},
who established, in particular, the result we need, Theorem 2.1 below.
This result was later proved by \cite{Bombieri}
using a different technique
(of maximal operators and Hunt-Carleson theorem).
\proclaim{Theorem 2.1} Let $\,t\,$ be real, and
$\,|t-\frac pq|\le {1\over q^2}$, for some co-prime integers
$\,p\,$ and $\,q$. Then there exists a constant $\,C>0\,$ such that,
for all real $\,x$,
$$
|\sum_{n=1}^N\,\be\,({n^2t\over 2}\,+\, n x)|\,\le\,
C\,\left( {N\over \sqrt{q}}\,+\,\sqrt{q}\right)\,,\tag 2.3
$$
for any integer $\,N>0$.
\endproclaim
This theorem implies immediately the following estimate.
\proclaim{Corollary 2.2} In the assumptions of Theorem 2.1,
$$
\| \lceil S\rceil_M^N\,(t,\cdot)\|_{L^{\infty}}\,\le\,
\lceil C\rceil\,\left( {N-M\over\sqrt q}\,+\,\sqrt{q}\right)\,,\tag 2.4
$$
for all integers $\,M\,$ and $\,N>M$.
\endproclaim
\medskip
%%%%%%%%%%%%%%%%%%
The estimate from above on the sum (2.2) is given by the following theorem
(compare \cite{Bourgain, Lemma 3.18}).
\proclaim{Theorem 2.3} Let $\,t\,$ be real, and
$\,|t-\frac pq|\le {1\over q^2}$, for some co-prime integers
$\,p\,$ and $\,q$.
Assume that the coefficients $\,\omega_n\,$
satisfy the following conditions:
$$
\omega_n = 0,\quad\text{for}\; n\,N,\tag 2.5a
$$
and
$$
\sum_M^N\,|\omega_{n+1}-\omega_n|\,\le\,\varkappa\,. \tag 2.5b
$$
Then
$$
\| \sum_M^N\,\omega_n\,\be\,({n^2t\over 2}+n\cdot)\|_{L^{\infty}}\,\le\,
\varkappa\,
\lceil C\rceil\,\left( {N-M\over\sqrt q}\,+\,\sqrt{q}\right)\,,\tag 2.6
$$
where $\,\lceil C\rceil\,$ is the same as in (2.4).
\endproclaim
%%%%%%%%%%%%%%%%%%%%%%%%%
\demo{Proof} Using summation by parts, we obtain
$$\multline
|\,\sum_M^N\,\omega_n\,\be\,({n^2t\over 2}+n x)\,|\,=\,
|\,\sum_M^N\,(\omega_{n+1}-\omega_n)\,
\sum_{k=M}^n\,\be\,({k^2t\over 2}+k x)\,|\,\\
\le\,
\varkappa\,\sup_{M\le n\le N}\,
|\,\sum_{k=M}^n\,\be\,({n^2t\over 2}+k x)\,|,\hfill
\endmultline
$$
and (2.6) follows from (2.4).
\qed
\enddemo
\proclaim{Corollary 2.4} Let $\,t\,$ be real, and
$\,|t-\frac pq|\le {1\over q^2}$, for some co-prime integers
$\,p\,$ and $\,q$.
Assume that the coefficients $\,\omega_n\,$
satisfy the following conditions:
$$
\omega_n = 0,\quad\text{for}\; |n|N,\tag 2.5a
$$
and
$$
\sum_{M\le |n|\le N}\,|\omega_{n+1}-\omega_n|\,\le\,\varkappa\,. \tag 2.5b
$$
Then
$$
\| S_M^N(t,\cdot)\|_{L^{\infty}}\le \,
2\,\varkappa\,
\lceil C\rceil\,\left( {N-M\over\sqrt q}\,+\,\sqrt{q}\right)\,.\tag 2.7
$$
\endproclaim
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the sharpness part in Theorems {\rom I - III\/} we need to
estimate the exponential sums from below. We do this in two steps.
First, we estimate $\lceil S\rceil_M^N\,(t,x)\,$ and
$\, S_M^N\,(t,x)\,$ for {\it rational\/} $\,t$, and then show
that the (suprema of the) sums for an irrational $\,t\,$ are
close to those with a sufficiently good rational approximation to
$\,t$. This approach is quite standard, see \cite{Montgomery,
Chapter 3}.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%
\proclaim{Theorem 2.5} Let $\lceil S\rceil_M^N\,(t,x)\,$
and $\, S_M^N\,(t,x)\,$ be as in (2.1), (2.2). Assume that
$\,\omega_n$ satisfies (2.5a).
Let $\,p\,$ and $\,q\,$ be co-prime positive integers.
For each of four cases below, there exists $\,x=\frac hq\in [0,1]$,
with integer $\,h$,
such that
$$
\align
| S_M^N\,(\frac pq, x) |\,& \ge\,
{ \sum\limits_{n=M}^N \omega_n+\omega_{-n} \over \sqrt 2\sqrt{N-M} }\,,
\qquad
\text{if}\quad
q\ge N-M+1\,; \tag 2.8a\\
| S_M^N\,(\frac pq, x) |\,& \ge\,
{ \sum\limits_{n=M}^N \omega_n+\omega_{-n} \over \sqrt{2q} }\,,\qquad
\text{if}\quad
q< N-M+1\,;\tag 2.8b \\
| \lceil S\rceil_M^N\,(\frac pq, x) |\,& \ge\,
\sqrt{2}\sqrt{N-M}\,,\;\qquad\qquad
\text{if}\quad
q\ge N-M+1\,;\tag 2.9a \\
| \lceil S\rceil_M^N\,(\frac pq, x) |\,& \ge\,
{\sqrt{2}(N-M)\over \sqrt{q}}\,,\qquad\qquad
\text{if}\quad
q < N-M+1\,.\tag 2.9b
\endalign
$$
\endproclaim
\demo{Proof} Because of the orthogonality relations among the additive
characters $(\text{\rm mod}\,2q)$, we have the following equalities:
$$
\sum\limits_{h=1-(M-1)p}^{2q-(M-1)p}\,
\left|S_M^N(\frac pq,\frac hq)\right|^2\, =\,
2q\,\sum\limits_M^N\,|\omega_n|^2+|\omega_{-n}|^2\,,
\tag 2.10a
$$
if $\,q\ge N-M+1$,
and
$$
\sum\limits_{h=1-(M-1)p}^{2q-(M-1)p}\,
\left| S_M^N(\frac pq,\frac hq) \right|^2\,=\,
2q\,\sum\limits_{k=1}^{2q}\,
\left| \sum\limits_{m=0}^{[{N-M+1\over 2q}]}\,
\omega_{2mq+k+M-1}\,+\,
\omega_{-2mq-k-M+1} \right|^2\,,
\tag 2.10b
$$
if $\,q < N-M+1$.
Since
$$
|\sum\limits_M^N \omega_n+\omega_{-n}\,|^2\le
2 (N-M)\,\sum\limits_M^N |\omega_n|^2+|\omega_{-n}|^2,
$$
(2.10a) yields
$$
\sum\limits_{h=1-(M-1)p}^{2q-(M-1)p}\,
\left|S_M^N(\frac pq,\frac hq)\right|^2\,\ge\,
q\,
{|\sum\limits_M^N \omega_n+\omega_{-n}\,|^2
\over N-M}\,.
$$
Hence,
$$
\left|S_M^N(\frac pq,\frac hq)\right|^2\,\ge\,
{|\sum\limits_M^N \omega_n+\omega_{-n}\,|^2
\over 2(N-M)},
$$
for at least one $\,h$, and (2.8a) follows.
%%%%%%%%%%%%%%%%%%%%%%
Similarly, the inequality
$$\multline
|\sum\limits_M^N \omega_n+\omega_{-n}\,|^2=
\left| \sum\limits_{k=1}^{2q}
\sum\limits_{m=0}^{[{N-M+1\over 2q}]}\,
\omega_{2mq+k+M-1}\,+\,
\omega_{-2mq-k-M+1} \right|^2\,\le \\
2q\,\sum\limits_{k=1}^{2q}\,
\left| \sum\limits_{m=0}^{[{N-M+1\over 2q}]}\,
\omega_{2mq+k+M-1}\,+\,
\omega_{-2mq-k-M+1} \right|^2\,,
\endmultline
$$
and (2.10b), show that for at least one integer $\,h$
we have (2.8b) with $\,x=\frac hq$.
%%%%%%%%%%%%%%%%%%%%%%%
The estimates (2.9) follow from (2.8) if we choose
$\,\omega_n\,$ to be $\,1\,$ for $\,n\in [-N,-M]\cup [M,N]$,
and $\,0\,$ otherwise.
\qed
\enddemo
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\proclaim{Theorem 2.6} Assume that $\,t\,$ and $\,t_1\,$ are such that
$$
|t-t_1|< {K\over N^2}\,,\tag 2.11
$$
with some constant $\,K>0$. Then
$$
\lceil c\rceil_1\,\max_x\,|\,\lceil S\rceil_M^N(t_1,x)\,|\,
\le\,\max_x\,|\,\lceil S\rceil_M^N(t,x)\,|\,\le\,
\lceil c\rceil_2\,\max_x\,|\,\lceil S\rceil_M^N(t_1,x)\,|\,,\tag 2.12a
$$
and
$$
c_1\,\max_x\,|\, S_M^N (t_1,x)\,|\,
\le\,\max_x\,|\, S_M^N (t,x)\,|\,\le\,
c_2\,\max_x\,|\,S_M^N (t_1,x)\,|\,,\tag 2.12b
$$
for some constants
$\,c_1$, $\,c_2$, $\,\lceil c\rceil_1$, and $\,\lceil c\rceil_2$,
which depend only on $\,K$.
\endproclaim
\demo{Proof} The proof follows the same line as the proof of Theorem 3.3
of \cite{Montgomery}.
First, we define an auxiliary, period 2 function $\,f(\theta)$:
$$
f(\theta)=\left\{
\aligned
\,&\be\big({t-t_1\over 2}\,(2N\theta)^2\big)\,,
\quad\text{for}\quad 0\le\theta\le\frac 12,\\
\,&\be\big({t-t_1\over 2}\,N^2\big)
\qquad\quad\text{for}\quad \frac 12\le\theta\le 1\,,
\endaligned
\right.
$$
and $\,f(\theta)=f(-\theta)$, for $\,-1\le\theta\le 0$. Taking into
account (2.11), it is easy to
check that the total variation of the first derivative of
$\,f\,$ is bounded, $\,\text{Var}\,f^{\prime}\le \gamma_1$,
and the Fourier coefficients of $\,f\,$ decay as $\,m^{-2}$,
$\,|\hat f(m)|\le \gamma_2/m^2$. Note, that
the constants $\,\gamma_1\,$ and $\,\gamma_2\,$ do not depend on
$\,t$, $\,t_1$, and $\,N$.
%%%%%%%%%
Choose an $\,x\,$ so that
$\,|S_M^N(t,x)|\,=\,\max_y\,|S_M^N(t,y)|$. Then we have
$$\multline
|S_M^N(t,x)|=|\sum_{n=M}^{N}\,\omega_n\,\be\,({n^2t\over 2}\,+\,n x)\,
+ \omega_{-n}\,\be\,({n^2t\over 2}\,-\,n x)|=\\
|\sum_{n=M}^{N}\,\omega_n\,f({n\over 2N})\be\,({n^2t_1\over 2}\,+\,n x)\,
+ \omega_{-n}\,f({n\over 2N})\be\,({n^2t_1\over 2}\,-\,n x)|=\\
|\sum_m\,\hat f(m)\,\big(
\sum_{n=M}^{N}\,\omega_n\,\be\,({n^2t_1\over 2}\,+\,n (x+{mn\over 2N}))\,
+ \omega_{-n}\,\be\,({n^2t_1\over 2}\,-\,n (x-{mn\over 2N}))|=\\
(\;\text{note, that}\; \hat f(m)=\hat f(-m)\;)\\
|\sum_m\,\hat f(m)\,\big(
\sum_{n=M}^{N}\,\omega_n\,\be\,({n^2t_1\over 2}\,+\,n (x+{mn\over 2N}))\,
+ \omega_{-n}\,\be\,({n^2t_1\over 2}\,-\,n (x+{mn\over 2N}))|=\\
|\sum_m \hat f(m)\,S_M^N(t_1,{x+mn\over 2N})|\le
\max_y\,|S_M^N(t_1,y)|\cdot \sum_m |\hat f(m)|\le\,
c_2\,\max_y\,|S_M^N(t_1,y)|\,.
\endmultline
$$
The opposite inequality follows by reversing the r\^oles of $\,t\,$
and $\,t_1$. Also, (2.12a) is a particular case of (2.12b).
\qed
\enddemo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Section 3 %%%%%%%%%%%%%%%%%%%%%%%%%%
\head {\bf 3\/}\ \ Proofs of Main Results \endhead
\medskip
\heading{Rational times}\endheading
The fundamental solution $\,E(t,x)=\sum_n\,\be({n^2t\over2}+nx)\,$
is intimately related to the Jacobi theta function
$$
\vartheta (\tau, x,y)=
\sum_n\,\be({\tau\over 2}(n-y)^2+nx-\frac 12 xy)\,.\tag 3.1
$$
Indeed, it is easy to see that
$$
\vartheta (\tau, x,y)=\be(-\frac 12 y(x-\tau y))\,E(\tau,x)\,.\tag 3.2
$$
Recall, that by the well-known transformation property of $\,\vartheta$,
\cite{Eichler}, for any unimodular matrix
$\,g=\pmatrix a & b\\
c & d\endpmatrix\in SL(2,\Bbb Z)$, we have
$$\multline
\vartheta (\tau, x,y)=\\
\varkappa(g) |c\tau+d|^{-\frac 12}
\be(\frac 14 \eta(ax+by)-\xi(cx+dy))\,
\vartheta ({a\tau+b\over c\tau+d},ax+by-\frac 12\xi, cx+dy-\frac 12\eta),
\endmultline
\tag 3.3
$$
where $\,\xi = ab\,\mod 2$, $\,\eta=cd\,\mod 2$, and $\,\varkappa(g)$
is an eighth root of $\,1$; $\,\varkappa(g)$ depends on the matrix
$\,g$ and the choice of $\,\xi$ and $\,\eta$.
%%%%%%%%%%%%%%
Let $\,t$ be a rational number in $\,(0,2)$, $\,t= p/q$,
$\,p\,$ and $\,q\,$ are co-prime. As it was explained in Section 1,
we choose the finite continued fraction representation
$\, p/q=[a_0,a_1,\dots,a_{n-1},a_n]\,$ with {\it even\/} $\,n$.
If $\,{p_{n-1}/q_{n-1}}=[a_0,a_1,\dots,a_{n-1}]$, then
$\,p q_{n-1}-q p_{n-1}=1$, so, the matrix
$\,g=\pmatrix q & -p\\
q_{n-1} & -p_{n-1}\endpmatrix\,
$
is in $\,SL(2,\Bbb Z)$. Using (3.3) with this $\,g$, and (3.2),
we obtain
$$\multline
E(t,x)=\vartheta (t,x,0)=\varkappa(g)\sqrt{q}
\be(\frac 14x(q\eta-q_{n-1}\xi))
\vartheta (0,qx-\frac 12\xi, q_{n-1}x-\frac 12\eta)=\\
\varkappa(g)\sqrt{q}
\be(\frac 14 (qx-\frac 12\xi)(q_{n-1}x-
\frac 12\eta)+\frac 14 x(q\eta-q_{n-1}\xi)) E(0, qx-\frac 12\xi)=\\
\hfill
\varkappa(g)\sqrt{q}
\be(-\frac 12q q_{n-1} x^2+\frac 12q\eta x-\frac 18\xi\eta)
\sum_n\delta(n-qx+\frac 12\xi).
\endmultline
$$
This proves part 1) of Theorem {\rom I\/}.
%%%%%%%%%%%%%%
To prove part 2), we need an auxiliary result, which will
be used in the proofs of other results as well.
\proclaim{Lemma 3.1} Let $\,\chi(\cdot)\,$ be a continuously
differentiable, non-negative function on $\,\Bbb R_+\,$ with support
in the interval $\,[1/2,\, 2]$. Then, first,
$$
\sum\limits_{n=0}^{\infty}\chi(2^{-j}n)\,=\,
2^{j}\,\int_0^{\infty}\chi(y)\,dy\,
+\,O(1),\quad\text{as}\quad j\to\infty\,,\tag 3.4
$$
and second,
$$
\sum\limits_{n=2^{j-1}}^{2^{j+1}}
\big |\chi(2^{-j}(n+1)) - \chi(2^{-j}n)\big |\le \varkappa, \tag 3.5
$$
for all $\,j\ge 0$, with some constant $\,\varkappa$.
\endproclaim
\demo{Proof} The proof of (3.4)
is by direct application of the well-known
Euler-Maclaurin summation formula (see, e.g., \cite{Edwards, Section 6.2}),
$$
\sum\limits_{n=M}^{N} f(n) = \int_M^N f(y)\,dy +
\frac 12\,\left( f(M)+f(N)\right) +
\int_M^N (y-[y]-\frac 12) f^{\prime}(y)\,dy,
$$
which is valid for any continuously
differentiable function $\,f\,$ on the interval $\,[M,N]$.
To prove (3.5), apply the Newton-Leibniz formula.
\qed
\enddemo
\medskip
Now, with the help of Lemma 3.1 and Remark 1.2,
part 2) of Theorem {\rom I} follows
from Corollary 2.2, Corollary 2.4 and Theorem 2.5.
\qed
\medskip
Let $\,t\in (0,2)\,$ be an irrational number and let
$\,[a_0,\,a_1,\,a_2,\,a_3,\dots]\,$ be its continued fraction expansion,
$$
t\,=\,a_0\,+\,
\cfrac1\\
a_1+\cfrac1\\
a_2+\cfrac1\\
\dots
\endcfrac
$$
The integers $\,a_0,\,a_1,\,a_2,\dots\,$ can be found from the
recurrent relations
$$
a_{k+1}=\big[{1\over t_k}\big],\qquad t_{k+1}={1\over t_k} - a_{k+1}\,,\tag 3.6a
$$
and the initial conditions
$$
a_0=\big[{ t}\big],\qquad t_0 = t-a_0\,.\tag 3.6b
$$
As usual, $\,\big[r\big]\,$ denotes the largest integer not exceeding $\,r$.
The finite parts $\,[a_0,\,\,a_1,\,a_2,\dots,\,a_n]$, $\,n=1,\,2,\dots$,
of our infinite continued fraction,
sum up to the rational numbers $\,p_n/q_n\,$ -- convergents of $\,t$. The numerators and denominators of the convergents
can be found using the relations
$$
p_{k+1} = a_{k+1} p_k + p_{k-1},\qquad q_{k+1} = a_{k+1} q_k + q_{k-1},
\tag 3.7a
$$
and the initial conditions
$$
p_{-1}=1,\quad q_{-1}=0, \quad p_0=a_0,\qquad q_0= 1.
\tag 3.7b
$$
Of course, $\,{p_n/ q_n}\to t$. Also, we have
$$
{1\over q_n (q_n+q_{n+1})}\,<\,|\,t - {p_n\over q_n}\, |\,<\,
{1\over q_n q_{n+1}}<{1\over q_n^2}. \tag 3.8
$$
For our purposes, it is important to know how fast the denominators
$\,q_n\,$ grow. The celebrated result of Khinchin and L\'evy
(see \cite{Khinchin, 1936}, \cite{L\'evy, 1937}) answers
this question in the following sense: {\it for almost all\/} $\,t$,
$$
{\ln q_n \over n}\,{\underset{n\to\infty}\to\to}\,\ln \rho_*, \tag 3.9a
$$
where $\,\rho_*\,$ is given by
$$
\ln\,\rho_*\,=\,{\pi^2\over 12\,\ln\,2}\,.\tag 3.9b
$$
This means, in particular, that for any $\,t\,$ from the set defined
by the Khinchin-L\'evy theorem and for all sufficiently large $\,j$,
there exists $\,{p_n\over q_n}$, the $\,n^{th}\,$ convergent to $\,t$,
with $\,q_n = 2^j\,\cdot\,2^{j\epsilon_j}$, where
$\,n = [j\log_{\rho_*} 2]\,$ and $\,\epsilon_j\to 0\,$
as $\,j\to\infty$. Hence, part (i) of Theorem {\rom{II}} follows
from Remark 1.2 and Corollaries 2.2 and 2.4.
\medskip
The equations (3.7) show that irrationals with bounded
quotients belong to $\,\Cal I(0)$.
Thus, the second statement of Theorem {\rom{II}}
will follow from Theorem {\rom{III}}. Since inequality (1.9)
with $\,\sigma=0\,$ and $\,c_t=1\,$ holds true for every
irrational $\,t$, as (3.7a) shows,
the third statement of Theorem {\rom{II}} follows from part (ii)
of Theorem {\rom{III}}.
We, therefore, turn to the proof of Theorem {\rom{III}}.
\medskip
\proclaim{Proposition 3.2} If $\,t\in\Cal I(\le\sigma)\,$
for some $\,\sigma\ge 0$, then
$$
E(t,\cdot)\in B^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\,\cap\,
\lceil B\rceil^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}\,. \tag 3.10
$$
\endproclaim
\demo{Proof} Recall, that $\,t\in\Cal I(\le\sigma)\,$ means that
the denominators $\,q_n\,$ of the convergents to $\,t\,$
satisfy the inequality
$$
q_{n+1}\le C_t\,q_n^{1+\sigma}\,,\tag 3.11
$$
for all sufficiently large $\,n$. In order to prove (3.10), we
have to obtain the appropriate
estimates (1.5) for truncated exponential sums. We achieve this by applying
Corollaries 2.2 and 2.4. The sums in (1.5a) and (1.5b) will be treated
similarly, so we shall work only with the one that is involved in the
definition of the space
$\,\lceil B\rceil^{{}^{-{1+\sigma\over 2+\sigma}}}_{\infty}$.
Our goal is to show that there is a constant $\,\lceil C\rceil$
such that, for all $\,x$, the inequality
$$
\big | \sum\limits_{m=2^{j-1}+1}^{2^{j+1}} \be ({m^2t\over 2}+mx)\big |
\le \lceil C\rceil\,2^{\alpha j},
\tag 3.12
$$
holds true for all sufficiently large $\,j$, where
$$
\alpha={1+\sigma\over 2+\sigma }.\tag 3.13
$$
By Legendre's theorem,
$\,|t-(p_n/q_n)|\le 1/(2q_n^2)$, for any convergent $\,p_n/q_n$.
Hence, Corollary 2.2 provides an estimate,
$$
\big | \sum\limits_{m=2^{j-1}+1}^{2^{j+1}} \be ({m^2t\over 2}+mx)\big |
\,\le\,\tilde C \,\left({2^j\over\sqrt{q_n}}\,+\,\sqrt{q_n}\right),
\tag 3.14
$$
for an arbitrary $\,n$. It remains to show that
$$
{2^j\over\sqrt{q_n}}\,+\,\sqrt{q_n}\,\le\, C\,2^{\alpha j},\tag 3.15a
$$
for every sufficiently large $\,j$, provided $\,n\,$ is chosen appropriately.
Define the exponents
$\,s_n\,$ so that
$$
2^{s_n}\,=\,q_n\, .\tag 3.16
$$
In order that (3.15a) be true,
the following inequalities ought to be satisfied:
$\,\alpha \ge 1/2$ (which is the case when $\,\sigma\ge 0$), and
$$
{s_n\over 2\alpha}\,\le\,j\,\le {s_n\over 2(1-\alpha)}\,.\tag 3.17a
$$
We would like to have (3.15a) with $\,q_n\,$ replaced by $\,q_{n+1}\,$ as well.
To achieve this, we impose a stronger requirement, which uses
the assumption (3.11). Namely, we require that
$$
{2^j\over\sqrt{q_{n+1}}}\,+\,\big(C_t\,q_n^{1+\sigma}\big)^{1/2}\,
\le\, C\,2^{\alpha j},\tag 3.15b
$$
This inequality is satisfied provided
$$
{s_n (1+\sigma)\over 2\alpha}\,\le\,j\,
\le {s_{n+1}\over 2(1-\alpha)}\,.\tag 3.17b
$$
Since $\,\alpha = {1+\sigma\over 2+\sigma}$, the right end of the interval
(3.17a), $\,{s_n\over 2(1-\alpha)}$, coinsides
with the left end of the interval (3.17b), $\,{s_n (1+\sigma)\over 2\alpha}$.
Since the exponents $\,s_n\,$ grow to infinity,
the above argument shows that the desired
estimate (3.12) holds for all sufficiently large $\,j$. Proposition follows.
\qed
\enddemo
\proclaim{Proposition 3.3} If $\,t\in\Cal I(\ge\sigma)\,$
for some $\,\sigma\ge 0$, then
$$
E(t,\cdot)\notin
\left(
\cup_{\epsilon>0}B^{{}^{-{1+\sigma\over 2+\sigma}+\epsilon}}_{\infty}
\right)\,
\cup\,
\left(\cup_{\epsilon>0}\lceil
B\rceil^{{}^{-{1+\sigma\over 2+\sigma}+\epsilon}}_{\infty}
\right)\,.
$$
\endproclaim
\demo{Proof} Let $\,t\in\Cal I(\ge\sigma)$. Recall, that this means
that
$$
q_{n+1}\ge c_t\,q_n^{1+\sigma} \tag 3.18
$$
for an infinite number of $\,n$'s.
We are going to prove that
$\;E(t,\cdot)\notin\cup_{\ve>0} B^{{}^{-{\alpha}+\epsilon}}_{\infty}$, where
$\,\alpha\,$ is defined in (3.13). It is sufficient to show that
there exists an infinite number of $\,j$'s, such that
$$
\sup_x\,
\big | \sum\limits_{m=2^{j-1}}^{2^{j+1}}
\chi(2^{-j}m) \be ({m^2t\over 2}+mx)+
\chi(2^{-j}m) \be ({m^2t\over 2}-mx)
\big |
\ge
C\,2^{\alpha j}\,. \tag 3.19
$$
We first note that if
$\,p /q \,$ is such that
$$
|t-{p \over q }|\,\le\,K\,2^{-2j}\,,\tag 3.20
$$
then, by Theorem 2.6, we can replace $\,t\,$ by $\,p /q \,$
in (3.19). After this replacement, we can use Theorem 2.5 to get
the following estimate:
$$\multline
\sup_x\,
\big | \sum\limits_{m=2^{j-1}}^{2^{j+1}}
\chi(2^{-j}m) \be ({m^2 p \over 2q }+mx)+
\chi(2^{-j }m) \be ({m^2p \over 2q }-mx)
\big |\\
\hfill\ge\,\tilde c\,\min\,\{2^{j/2},\;{2^{j}\over\sqrt{q }}\}\,.
\hfill
\endmultline
\tag 3.21
$$
Note, that we have simplified the right sides of
inequalities (2.8) by observing that
$$
\sum\limits_{m=2^{j-1}}^{2^{j+1}} \omega_{m}+\omega_{-m}\,=\,
2\,\sum\limits_{m=2^{j-1}}^{2^{j+1}}
\chi(2^{-j}m)\, \ge\,\sqrt{2}\,\tilde c\;2^{j},
$$
where the last inequality is a consequence of Lemma 3.1.
Now, (3.19) will be proved if we show that
$$
\min\,\{2^{j/2},\;{2^{j}\over\sqrt{q}}\}\,\ge\,
C\,2^{\alpha j}, \tag 3.22
$$
for an infinite number of $\,j$'s, provided $\,p /q \,$
are chosen appropriately.
Our plan is to use the convergents $\,p_n/q_n\,$
in the above argument. We first check (3.20). Since
$\,|t-(p_n/q_n)|<1/(q_nq_{n+1})\,$ for all $\,n$, we have
$$
|t-{p_n\over q_n}|\,<\,{1\over c_t q_n^{2+\sigma}}\,
=\,(1/c_t)\,2^{-s_n(2+\sigma)},\tag 3.23
$$
for those $\,n$, for
which (3.18) takes place. Here and further on, $\,s_n=\log_2 q_n$.
Thus, to satisfy (3.20), it is sufficient to require that
$$
j\,\le\,\frac 12\,s_n(2+\sigma)\,.\tag 3.24a
$$
Next, in order to have (3.22), it is sufficient to ask for
the following inequality to hold:
$$
j\,\ge\,{s_n\over 2(1-\alpha)}-1\,=\,\frac 12\,s_n(2+\sigma)\,.\tag 3.24b
$$
The interval $\,[\frac 12\,s_n(2+\sigma)-1,\,\frac 12\,s_n(2+\sigma)]\,$
always contains an integer $\,j$, and, since $\,s_n\to\infty\,$ (we use
only those $\,n\,$ for which (3.18) takes place),
(3.19) is satisfied for an infinite number of $\,j$'s.
This completes the proof of the proposition.
\qed
\enddemo
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\inbook Progress in Approximation Theory, an International perspective
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\paperinfo Springer Series in Computational Mathematics, 19
\publ Springer-Verlag
\publaddr New York
\yr 1992
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\vol 34
\yr 1981
\pages 719-739
\endref
\ref\by P. Sarnak
\paper Class numbers of indefinite binary quadratic forms
\jour J. Number Theory
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\ref\by Schmidt
\book Diophantine approximation
\bookinfo Lecture Notes in Mathematics, 785
\publ Springer-Verlag
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\yr 1980
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\ref \by H. Triebel
\book Theory of function spaces
\publ Birkh\"auser
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\yr 1983
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\ref\by K. Yajima
\paper Smoothness and non-smoothness of the fundamental solution
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\jour Comm. Math. Phys.
\vol 181
\issue 3
\pages 605-629
\yr 1996
\endref
\end
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