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UDC 517.941.92
ASYMPTOTIC BEHAVIOR OF INTEGRALS OF QUASIPERIODIC FUNCTIONS
V. G. SPRINDZHUK
1. Introduction. In the well-known work of N. P. Erugin [1], a general theory of reducibility of linear systems of differential equations was constructed. As one of the examples, N. P. Erugin considers the question of the reducibility of the system
\[ \frac{dX}{dt} = X \begin{bmatrix} b+\cos \alpha_1 t & 0\\ \sin \alpha_3 t & b+\cos \alpha_2 t \end{bmatrix}, \tag{1} \]
where \(X\) is a matrix of the second order; \(b,\alpha_1,\alpha_2,\alpha_3\) are constants. It turns out that, for the reducibility of system (1), it is necessary and sufficient that the function
\[ J(t)=\int_0^t \exp\left(\frac{\sin \alpha_2 t}{\alpha_2} - \frac{\sin \alpha_1 t}{\alpha_1}\right) \sin \alpha_3 t\,dt \tag{2} \]
be bounded on the whole half-line \(t>0\), and that this is fulfilled if \(\alpha_1,\alpha_2\) are commensurable, but \(a_1\alpha_1+a_2\alpha_2+\alpha_3\ne 0\) for any integers \(a_1,a_2\) (see [1], pp. 39–41).
In this article we consider the question of the asymptotic behavior of integrals
\[ J(t)=\int_0^t F(x,\omega_1 x,\ldots,\omega_n x)\,dx, \]
where \(F(x_0,\ldots,x_n)\) is a real function, 1-periodic in each variable \(x_j\) \((j=0,1,\ldots,n)\), satisfying certain smoothness conditions that are explicitly indicated below; \(\omega_1,\omega_2,\ldots,\omega_n\) are certain real numbers. Obviously, the assertion on the boundedness of the integral (2) is a relation of the form
\[ J(t)=c_F t+O(1), \tag{3} \]
where \(c_F\) is some number, for example the mean value of the function \(F(x_0,x_1,\ldots,x_n)\) on the unit \((n+1)\)-dimensional cube, i.e.
\[ c_F=\int_0^1\cdots\int_0^1 F(x_0,x_1,\ldots,x_n)\,dx_0\,dx_1\cdots dx_n. \tag{4} \]
For a given function \(F\), we indicate conditions that it is sufficient to impose on the numbers \(\omega_1,\omega_2,\ldots,\omega_n\) in order that (3) hold (see Theorems 1 and 2). As the simplest consequences, new conditions are obtained for the reducibility of system (1).
The aim of this paper is not to obtain the most general and definitive results, but only to point out the deep arithmetic nature of the question of the fulfillment of equality (3), and hence of the question of reducibility of linear systems with quasiperiodic coefficients. The competent reader will notice that the arguments used allow considerable further development and can be used to solve other problems in stability theory.
In what follows, along with the symbol \(O\) we use I. M. Vinogradov’s symbol \(\ll\) in the following sense: \(X \ll Y\) for positive variables \(X, Y\) means that \(X < cY\), where \(c > 0\) is a quantity independent of \(X, Y\).
2. Boundedness conditions. First suppose that the numbers \(1, \omega_1, \ldots, \omega_n\) are linearly independent over the field of rational numbers and satisfy the following condition, quantitatively expressing their linear independence.
Let \(w(\bar\omega)=w(\omega_1,\ldots,\omega_n)\) be the exact upper bound of those \(w>0\) for which there exists an infinite number of solutions of the inequality
\[ |a_0+a_1\omega_1+\cdots+a_n\omega_n|<h_a^{-w},\qquad h_a=\max_{0\le i\le n}|a_i| \tag{5} \]
in integers \(a_0,a_1,\ldots,a_n\). If we denote
\[ w(\bar\omega,H)=\min |a_0+a_1\omega_1+\cdots+a_n\omega_n|, \tag{6} \]
where the minimum is taken over all integral \(a=(a_0,a_1,\ldots,a_n)\) with the restriction \(0\ne h_a\le H\), then it is easy to see that
\[ w(\bar\omega)=\lim_{H\to\infty} \frac{\ln \dfrac{1}{w(\bar\omega,H)}}{\ln H}. \tag{7} \]
We shall assume that the numbers \(\omega_1,\ldots,\omega_n\) satisfy the condition
\[ w(\bar\omega)=w(\omega_1,\ldots,\omega_n)<\infty . \tag{8} \]
Theorem 1. Let the 1-periodic function \(F(x_0,\ldots,x_n)\) have continuous partial derivatives \(\dfrac{\partial^s}{\partial x_j^s}F\) \((j=0,1,\ldots,n)\), and let the integrals
\[ I_j(F)=\int_0^1\cdots\int_0^1 \left| \frac{\partial^{s+1}}{\partial x_j^{s+1}}F(x_0,\ldots,x_n) \right|\,dx_0\cdots dx_n<\infty, \tag{9} \]
where \(s\ge n+1\) is a fixed number. Then for any system of numbers \(\omega_1,\ldots,\omega_n\) satisfying the condition \([w(\omega_1,\ldots,\omega_n)]\le s\), (3) holds with the constant \(c_F\) determined by equality (4).
An analogous assertion is also valid in the case of linearly dependent \(1,\omega_1,\ldots,\omega_n\).
Let \(r\) be the largest number of linearly independent numbers over the field of rational numbers from the sequence \(1,\omega_1,\ldots,\omega_n\) (the “rank” of the system of numbers \(1,\omega_1,\ldots,\omega_n\)). Obviously, \(1\le r\le n\), and if \(r=1\), then all the numbers \(\omega_j\) are rational. Without loss of generality, we may assume that \(1,\omega_1,\ldots,\omega_{r-1}\) are linearly independent. We suppose that these numbers satisfy condition (8), where \(n=r-1\), and put
\[ w(\bar\omega)=w(\omega_1,\ldots,\omega_n)=w(\omega_1,\ldots,\omega_{r-1}), \]
where the quantity \(w(\omega_1,\ldots,\omega_{r-1})\) is determined by means of (5) or (7) for \(n=r-1\). In other words, if \(w(\bar\omega,H)\) is defined by the equality (6), where the minimum is taken over all integers \(a=(a_1,\ldots,a_n)\) with the condition \(a_0+a_1\omega_1+\cdots+a_n\omega_n\ne0\), then the quantity \(w(\bar\omega)\), defined by equality (7), must be finite.
Theorem 2. Let the function \(F(x_0,x_1,\ldots,x_n)\) satisfy the conditions of the preceding theorem. Then for any real numbers \(\omega_1,\ldots,\omega_n\), for which the rank of the system of numbers \(1,\omega_1,\ldots,\omega_n\) is equal to \(r\) \((1\le r\le n)\) and \([w(\omega_1,\ldots,\omega_n)]\le s+r-n-2\), (3) holds, where \(c_F\) is some number (in general, different from the integral (4)).
We proceed to the proof of Theorems 1, 2.
Lemma 1. Let \(F(x_0,\ldots,x_n)\) be a 1-periodic function whose Fourier coefficients
\[ c(\bar v)=c(v_0,\ldots,v_n)=\int_0^1\cdots\int_0^1 F(x_0,\ldots,x_n)\times \]
\[ {}\times \exp\bigl(-2\pi i(x_0v_0+\cdots+x_nv_n)\bigr)\,dx_0\cdots dx_n \tag{10} \]
satisfy the conditions
\[ \sum_{\bar v} |c(\bar v)|<\infty, \tag{11} \]
\[ \sum_{\bar v\ne(0)} \frac{|c(\bar v)|}{|(\bar v,\bar\omega')|}<\infty, \tag{12} \]
where \((\bar v,\bar\omega')=v_0+v_1\omega_1+\cdots+v_n\omega_n\), and the numbers \(1,\omega_1,\ldots,\omega_n\) are linearly independent. Then (3) and (4) are valid.
Proof. By virtue of (11), the Fourier series
\[ \sum_{\bar v} c(\bar v)\exp(2\pi i(\bar v,\bar x))\sim F(\bar x) \]
converges uniformly and everywhere represents the continuous function \(F(\bar x)=F(x_0,\ldots,x_n)\). Therefore we find
\[ J(t)=\int_0^t F(x,\omega_1x,\ldots,\omega_nx)\,dx= \]
\[ =\sum_{\bar v} c(\bar v)\int_0^t \exp\bigl((2\pi i(\bar v,\bar\omega'))x\bigr)\,dx = c(\bar 0)t+R_F(t), \]
where, by virtue of (12), we have
\[ |R_F(t)|\le \frac{1}{2\pi}\sum_{\bar v\ne(0)} |c(\bar v)| \frac{\left|\exp(2\pi i(\bar v,\bar\omega')t)-1\right|}{|(\bar v,\bar\omega')|}\le \]
\[ \le \frac{1}{\pi}\sum_{\bar v\ne(0)} \frac{|c(\bar v)|}{|(\bar v,\bar\omega')|}<\infty. \]
Lemma 2. Let \(h_{\bar \nu}=\max(|\nu_0|,\ldots,|\nu_n|)\), let the numbers \(1,\omega_1,\ldots,\omega_n\) be linearly independent, and let \(\sigma>0\),
\[ G(\sigma)=\sum_{\bar \nu\ne(0)} h_{\bar \nu}^{-\sigma}\, |(\bar \nu,\bar \omega')|^{-1}. \tag{13} \]
Then the series (13) converges for \(\sigma>w(\omega_1,\ldots,\omega_n)\) and diverges for \(\sigma<w(\omega_1,\ldots,\omega_n)\).
Proof. Put
\[ S(h)=\sum_{0\ne h_{\bar \nu}\le h} |(\bar \nu,\bar \omega')|^{-1}. \tag{14} \]
Let \(0<\rho_1\le \rho_2\le\cdots\le \rho_N\) be the values of the quantities \(|(\bar \nu,\bar \omega')|\), where \(\bar \nu\ne(0)\), arranged in increasing order. By the definition of the quantity
\(w(\omega_1,\ldots,\omega_n)=w_0\), we have \(\rho_1>c h^{-w_0-\varepsilon}\), where \(c=c(\varepsilon)>0\) is a number independent of \(h\). Similarly, since the difference \(\rho_k-\rho_{k-1}\) has the form
\(a_0+a_1\omega_1+\cdots+a_n\omega_n\), where
\(0\ne\max(|a_0|,\ldots,|a_n|)\le 2h\), we must have
\[ \rho_k-\rho_{k-1}>c_1 h^{-w_0-\varepsilon}\qquad (k=2,3,\ldots,N). \]
Consequently, \(\rho_k>c_2 k h^{-w_0-\varepsilon}\) \((k=2,3,\ldots,N)\), and since \(N\ll h^{n+1}\), for the sum (14) we obtain
\[ S(h)=\sum_{k=1}^N \rho_k^{-1} < c_2^{-1} h^{w_0+\varepsilon}\sum_{k=1}^N k^{-1} \ll h^{w_0+\varepsilon}\ln h. \]
Applying Abel’s transformation and using the estimate obtained, we find for the series (13)
\[ G(\sigma)=\sum_{h=1}^{\infty} h^{-\sigma} \sum_{\substack{h_{\bar \nu}=h\\ \bar \nu}} |(\bar \nu,\bar \omega')|^{-1} \ll \sum_{h=1}^{\infty} S(h)h^{-\sigma-1} \ll \sum_{h=1}^{\infty} h^{w_0+\varepsilon-\sigma-1}\ln h, \]
so that \(G(\sigma)\) is finite if \(\sigma>w_0+\varepsilon\).
Since \(\varepsilon>0\) is arbitrary, for convergence of the series (13) it is enough to take \(\sigma>w_0\).
Conversely, if \(\sigma<w_0-\varepsilon\), then for an infinite number of systems of integers \(\nu_0,\ldots,\nu_n\)
\[ h_{\bar \nu}^{-\sigma}|(\bar \nu,\bar \omega')|^{-1} > h_{\bar \nu}^{-\sigma+w_0-\varepsilon} \ge 1 \]
by virtue of the definition \(w_0=w(\omega_1,\ldots,\omega_n)\). Therefore in the sum (13) there are infinitely many terms greater than 1. By the arbitrariness of \(\varepsilon\) we conclude that, for \(\sigma<w_0\), the series (13) diverges.
Proof of Theorem 1. Integrating by parts, for the Fourier coefficients (10) we find
\[ \pm c(\bar \nu)= \frac{1}{(2\pi i\nu_j)^{s+1}} \int_0^1\cdots\int_0^1 \exp(-2\pi i(\bar \nu,\bar x))\, \frac{\partial^{s+1}}{\partial x_j^{s+1}}F(\bar x)\,d\bar x, \]
since all partial derivatives up to \(\dfrac{\partial^s}{\partial x_j^s}F\) are continuous and 1-periodic. Consequently, by (9)
\[ |c(\bar{\nu})| \ll \frac{1}{(2\pi h_{\bar{\nu}})^{s+1}}\max_{0\leq j<n} I_j(F) \ll h_{\bar{\nu}}^{-s-1}, \tag{15} \]
if \(\nu_j\) is chosen so that \(|\nu_j|=\max(|\nu_0|,\ldots,|\nu_n|)=h_{\bar{\nu}}\ne 0\). Since, by the hypothesis of the theorem, \(s\geq n+1\), we have
\[ \sum_{\bar{\nu}\ne(0)^*} |c(\bar{\nu})| \ll \sum_{\bar{\nu}\ne(0)} h_{\bar{\nu}}^{-n-2} \ll \sum_{h=1}^{\infty} h^{-n-2}h^n = \sum_{h=1}^{\infty} h^{-2}<\infty . \]
Therefore (11) holds. Finally, by the hypothesis of the theorem,
\(s+1\geq [w(\omega_1,\ldots,\omega_n)]+1>w(\omega_1,\ldots,\omega_n)\). Therefore, in view of (15), by Lemma 2, (12) holds, and Lemma 1 gives the desired result.
Proof of Theorem 2. Let the numbers \(1,\omega_1,\ldots,\omega_{r-1}\) be linearly independent over the field of rational numbers. Since the rank of the system of numbers \(1,\omega_1,\ldots,\omega_n\) is \(r\), the numbers \(\omega_r,\omega_{r+1},\ldots,\omega_n\) are expressed linearly in terms of \(1,\omega_1,\ldots,\omega_{r-1}\), say
\[ \omega_j=l_{0j}+l_{1j}\omega_1+\cdots+l_{r-1,j}\omega_{r-1} \quad (j=r,r+1,\ldots,n), \tag{16} \]
where the \(l_{ij}\) are rational numbers. Consequently,
\[ (\bar{\nu},\bar{\omega}')=\nu_0+\nu_1\omega_1+\cdots+\nu_n\omega_n =\mu_0+\mu_1\omega_1+\cdots+\mu_{r-1}\omega_{r-1}, \tag{17} \]
where
\[ \mu_k=\nu_k+\sum_{j=r}^{n}\nu_j l_{kj} \quad (k=0,1,\ldots,r-1). \tag{18} \]
We divide all integer systems \(\bar{\nu}=(\nu_0,\nu_1,\ldots,\nu_n)\) into two classes: to the first class \(K_1\) belong all those \(\bar{\nu}\) for which the quantities \(\mu_k\), defined by (18), vanish; to the second class \(K_2\)—all the remaining \(\bar{\nu}\) (i.e., those for which one of the numbers \(\mu_k\ne 0\)).
Arguing as in the proof of Lemma 1, we find
\[ J(t)=\sum_{\bar{\nu}} c(\bar{\nu})\int_0^t \exp(2\pi i(\bar{\nu},\bar{\omega}')x)\,dx = \sum_{\bar{\nu}\in K_1}+\sum_{\bar{\nu}\in K_2} = t\sum_{\bar{\nu}\in K_1} c(\bar{\nu})+R_F(t), \]
where, in view of (15), we have
\[ |R_F(t)|\leq \frac{1}{\pi}\sum_{\bar{\nu}\in K_2} \frac{|c(\bar{\nu})|}{|(\bar{\nu},\bar{\omega}')|} \ll \sum_{\bar{\nu}\in K_2} \frac{h_{\bar{\nu}}^{-s-1}}{|(\bar{\nu},\bar{\omega}')|}. \tag{19} \]
By (17), the last sum can be written as
\[ \sum_{\bar{\nu}\in K_2} h_{\bar{\nu}}^{-s-1} \left|\mu_0+\mu_1\omega_1+\cdots+\mu_{r-1}\omega_{r-1}\right|^{-1}, \tag{20} \]
where the quantities \(\mu_k\) are defined by (18). From (18) we find that the numbers \(\mu_k\) are rational and their denominators divide the least common multiple \(q\) of the denominators of the numbers \(l_{kj}\).
Let \(\bar\mu_0=\left(\dfrac{m_0}{q},\ldots,\dfrac{m_{r-1}}{q}\right)\), where the \(m_i\) are integers, not all equal to 0. Let us estimate the sum
\[ \sum_{\bar\nu,\ \bar\mu=\bar\mu_0} h_{\bar\nu}^{-s-1}, \tag{21} \]
where the summation is over those \(\bar\nu\) for which \(\bar\mu=\bar\mu_0\). It follows from (18), if we put \(\bar\mu=\bar\mu_0\), that \(h_{\bar\mu_0}=\max_{(i)}\left|\dfrac{m_i}{q}\right|\ll h_{\bar\nu}\), so that this sum does not exceed
\[ \sum_{h\gg h_{\bar\mu_0}} h^{-s-1}N_{\bar\mu_0}(h), \tag{22} \]
where \(N_{\bar\mu_0}(h)\) is the number of those \(\bar\nu\) for which \(\bar\mu=\bar\mu_0\) and \(h_{\bar\nu}=h\). Obviously, from (18) we find that \(N_{\bar\mu_0}(h)\ll h^{\,n-r+1}\), since for fixed \(\nu_r,\ldots,\nu_n\) the numbers \(\nu_0,\ldots,\nu_{r-1}\) are determined uniquely. Consequently, the sums (22), (21) will be
\[ \ll \sum_{h\gg h_{\bar\mu_0}} h^{-s+n-r+1}\ll h_{\bar\mu_0}^{-s+n-r+2}, \]
and then the sum (20)
\[ \ll \sum_{\bar\mu_0\ne(0)} h_{\bar\mu_0}^{-s-r+n+2} \left|\frac{m_0}{q}+\frac{m_1}{q}\omega_1+\cdots+\frac{m_{r-1}}{q}\omega_{r-1}\right|^{-1}\ll \]
\[ \ll \sum_{\bar m\ne(0)} h_{\bar m}^{-s-r+n+2} \left|m_0+m_1\omega_1+\cdots+m_{r-1}\omega_{r-1}\right|^{-1}. \]
By Lemma 2 this series converges if \(s+r-n-2>\mathfrak w_0=\mathfrak w(\omega_1,\ldots,\omega_{r-1})\), which holds by the hypothesis of the theorem.
Thus, the series in (19) converge, and \(R_F(t)=O(1)\), so that (3) holds with the constant \(c_F\) equal to \(\sum_{\bar\nu\in K_1} c(\bar\nu)\), where \(K_1\) is the set of those \(\bar\nu\) which satisfy equations (18) with \(\mu_k=0\) \((k=0,1,\ldots,r-1)\).
3. Conclusion. Returning to system (1) and the integral (2), we note that if the numbers \(a_1,a_2,a_3\) are linearly independent and, for some \(\mathfrak w_0>0\), the inequality
\[ |a_1\alpha_1+a_2\alpha_2+a_3\alpha_3|>ch_a^{-\mathfrak w_0},\qquad h_a=\max_{(i)}|a_i|, \tag{23} \]
holds, where \(c=c(a_1,a_2,a_3)>0\), then system (1) is reducible (Theorem 1). In fact, for reducibility of system (1) it is enough that (23) hold with \(a_3=1\), in view of the special form of the integral (2). For the same reason it suffices to require weaker restrictions in the right-hand side of (23), ensuring only the convergence of the series (12).
Analogous conclusions follow from Theorem 2.
Theorems 1 and 2 may suggest that, for any choice of the numbers \(\omega_1,\ldots,\omega_n\), (3) holds with a suitable value of \(c_F\). However, this is not so, and one can construct smooth functions \(F(x_0,x_1,\ldots,x_n)\) and choose the numbers \(\omega_1,\ldots,\omega_n\) so that (3) will not hold for any constant \(c_F\). This problem requires a more delicate number-theoretic analysis,
and we shall not touch upon it here. In particular, one can find such \(\alpha_1,\alpha_2,\alpha_3\) that the integral (2) will be unbounded and, consequently, system (1) with such numbers \(\alpha_1,\alpha_2,\alpha_3\) will be irreducible.
There are many results on systems of numbers \(\omega_1,\ldots,\omega_n\) satisfying condition (8). For example, it is fulfilled for linearly independent algebraic numbers, for values at algebraic points of exponential functions and Siegel \(E\)-functions, for powers of logarithms of algebraic numbers (see [2, 5]). It is fulfilled for almost all systems of numbers \((\omega_1,\ldots,\omega_n)\), or in some cases for almost all values of a parameter, if \(\omega_1,\ldots,\omega_n\) are functions of some parameter (see [3, 4]).
I express my heartfelt gratitude to N. P. Erugin for posing the problem and for numerous discussions on related questions.
References
- Erugin N. P. Reducible systems. Tr. MIAN, 13, 1946.
- Gelfond A. O. Transcendental and algebraic numbers. Moscow, 1952.
- Sprindzhuk V. G. Mahler’s problem in metric number theory. Minsk, 1967.
- Sprindzhuk V. G. DAN BSSR, 11, No. 1, 5–6, 1967.
- Mahler K. Philos. Trans. Roy. Soc., 245, No. 898, 371—398, 1953.
Received by the editors
25 May 1966
Institute of Mathematics
Academy of Sciences of the BSSR