%%%%%%%%%%%%%%%%%%%%%%%%% AMSTeX file %%%%%%%% %%%%% %%%%% 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 %%%%%%%%%%%%%%%%%% BODY %\pageno=1 \magnification \magstep1 \input amstex \documentstyle{amsppt} \NoRunningHeads %\NoPageNumbers %\TagsOnRight \baselineskip 15pt %\hfuzz=10pt \pagewidth{6.4 truein} \pageheight{8.6 truein} %\hcorrection{} %\vcorrection{} \define\vt{\vartheta} \define\ve{\varepsilon} \define\dv{\setbox2=\hbox{{.}\kern-.27em\raise.6ex\hbox{.} \kern-.615em\raise1.2ex\hbox{.}}% \copy2} \define\ndv{\setbox3=\hbox{\dv\kern-.40em \hbox{/}}% \copy3} \define\be{\bold e\,} \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 $\,v$:\ $\;-\infty0. \endsplit \tag 1.3b $$The norm in \,\lceil B\rceil^s_{u,v} is defined as in (1.4), with \,K_j\, replaced by \,\lceil K\rceil_j. It is known that if \,10}\,B^{s+\epsilon}_{\infty}. A similar convention will be in force for the spaces \,\lceil B\rceil^s_{\infty} and \,H^s. %%%%%%%%%%%%%%%% Closely related to the persistence of regularity of solutions of (0.1a) in \,H^s\, is the following fact. \proclaim{Remark 1.1} For all \,t, the distribution \,E(t,\cdot)\, belongs to \,B^{-1/2}_{2,\infty}\, sharp. \endproclaim \proclaim{Remark 1.2} That \,E(t,\cdot)\, belongs to the Besov space \,\lceil B\rceil^{-\alpha}_{\infty}\, is equivalent to the existence of a constant \,\lceil C\rceil\, such that$$ \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 \centerline{REFERENCES} \vglue 1pc \ref\by J. Bergh and J. L\"ofstr\"om \book Interpolation spaces, an introduction \publ Springer \publaddr Berlin \yr 1976 \endref \ref\by M.V. 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