Full Text
R. Kh. Mukhutdinov
A DIOPHANTINE EQUATION WITH A MATRIX EXPONENTIAL FUNCTION
(Presented by Academician I. M. Vinogradov on 20 VII 1961)
In paper (¹) M. P. Mineev gave a new proof of the Fortet–Kac theorem (²). M. P. Mineev relied on the asymptotic formula proved by him for the number of solutions of the Diophantine equation
\[ m_1 g^{x_1}+\cdots+m_k g^{x_k}=n_1 g^{y_1}+\cdots+n_k g^{y_k}, \tag{1} \]
where \(m_1,\ldots,m_k,n_1,\ldots,n_k\) are positive rational integers, \(g\) is an integer \(\ge 2\), in nonnegative integers of the interval \([0,p-1]\) as \(p\) grows.
In the present note an asymptotic formula with a remainder term is obtained for the number of solutions of the matrix Diophantine equation
\[ \widetilde m_1\psi(A^{x_1})+\cdots+\widetilde m_a\psi(A^{x_a}) = \widetilde n_1\psi(A^{y_1})+\cdots+\widetilde n_b\psi(A^{y_b}) \tag{2} \]
in nonnegative integers \(x_1,\ldots,y_b\) of the interval \([0,p-1]\), where \(A\) is a nonsingular integral matrix of order \(n\), among whose eigenvalues there is no root of unity; \(\psi(x)\) is an arbitrary polynomial with integral coefficients, not identically constant; \(\widetilde m_1,\ldots,\widetilde m_a,\widetilde n_1,\ldots,\widetilde n_b\) are \(n\)-dimensional integral vectors.
Notation. If \(\widetilde\alpha=(\alpha_1,\ldots,\alpha_n)\), then by the symbol \(\{\widetilde\alpha\}\) we shall mean the vector \((\{\alpha_1\},\ldots,\{\alpha_n\})\), where \(\{\alpha_j\}\) is the fractional part of \(\alpha_j\). \(A\) is a nonsingular integral matrix of order \(n\), among whose eigenvalues there is no root of unity. \(\psi(x)\) is an arbitrary polynomial with integral coefficients, not identically constant. \(N_p(\Delta)\) is the number of fractional parts \(\{\widetilde\alpha A^x\}\), \(x=0,1,\ldots,p-1\), that have fallen into the hypercube \(\Delta\). \(\varepsilon(t)\) is an arbitrary nonnegative function tending to zero as slowly as desired as \(t\to0\). \(|\Delta|\) is the volume of the hypercube \(\Delta\), mes is Lebesgue measure, \(\pi\) is the hypercube \((0\le x_1\le1,\ldots,0\le x_n\le1)\).
On the basis of the formula proved for the number of solutions of equation (2), the following theorems have been proved:
Theorem. If, for some \(c>0\),
\[ \overline{\lim}_{p\to\infty}\frac{N_p(\Delta)}{p}\le c|\Delta|^{1-\varepsilon(|\Delta|)} \tag{3} \]
for every \(n\)-dimensional hypercube \(\Delta\) lying in the hypercube \(\pi\), then the sequence \(\{\widetilde\alpha A^x\}\), \(x=0,1,\ldots\), is uniformly distributed in the hypercube \(\pi\).
The condition stated in the theorem cannot be improved in the sense that \(\varepsilon(t)\) cannot be replaced by a constant function different from zero in any interval \((0,t)\), as was shown in paper (³) for a special case.
A weaker theorem for matrices, in the sense of the requirement of condition (3), was proved by A. M. Polosuev (⁴) and I. Cigler (⁵).
Central limit theorem. Let \(f(\widetilde x)=f(x_1,\ldots,x_n)\) be a real function periodic in each argument
with period 1, a function with Fourier coefficients \(a_{m_1,\ldots,m_n}=a_{\widetilde m}\) such that
\[ \sum \|\widetilde m\|^\varepsilon |a_{\widetilde m}|^2 < \infty \]
for some \(\varepsilon>0\), where \(\|\widetilde m\|\) is the usual norm of the vector, and \(a_{\widetilde 0}=0\).
Then:
1) the limit
\[ \lim_{p\to\infty}\int_{\pi} \left( \frac{1}{\sqrt p}\sum_{t=0}^{p-1} f\bigl(\widetilde\alpha\psi(A^t)\bigr) \right)^2 d\widetilde\alpha \]
exists; denote it by \(\sigma^2\).
2)
\[ \operatorname*{mes}_{\widetilde\alpha\in\pi} E\left\{ \frac{1}{\sqrt p}\sum_{t=0}^{p-1} f\bigl(\widetilde\alpha\psi(A^t)\bigr)<\lambda \right\} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\lambda/\sigma} e^{-t^2/2}\,dt + O\left( c\,\frac{\sqrt{\log\log p}}{\sqrt{\log p}} \right), \]
where
\[ c= \begin{cases} 1, & \text{if } \sigma=0,\\[4pt] \dfrac{\|f\|_\varepsilon}{\sigma}, & \text{if } \sigma\ne 0, \end{cases} \qquad \|f\|_\varepsilon= \left[\sum \|\widetilde m\|^\varepsilon |a_{\widetilde m}|^2\right]^{1/2}; \]
the \(O\)-term contains quantities depending on the matrix \(A\), the polynomial \(\psi(x)\), and \(\varepsilon\).
Remark 1. If \(\sigma=0\), then by the symbol \(\lambda/\sigma\) we mean \(+\infty\) for \(\lambda>0\) and \(-\infty\) for \(\lambda<0\).
Remark 2. If \(\psi(x)\) has two nonzero coefficients, then \(\sigma=\|f\|\).
Theorem. Let \(f(\widetilde x)\) be a complex function, periodic in each argument with period 1; suppose that for some \(\varepsilon>0\)
\[ \sum \|\widetilde m\|^\varepsilon |a_{\widetilde m}|^2<\infty, \]
where \(a_{\widetilde m}\) are the Fourier coefficients of the function \(f(\widetilde x)\) on the hypercube \(\pi\), and \(a_{\widetilde 0}=0\).
Then:
1) The limits
\[ \lim_{p\to\infty}\int_{\pi} \left( \frac{1}{\sqrt p}\sum_{t=0}^{p-1} \operatorname{Re} f\bigl(\widetilde x\psi(A^t)\bigr) \right)^2 dx, \qquad \lim_{p\to\infty}\int_{\pi} \left( \frac{1}{\sqrt p}\sum_{t=0}^{p-1} \operatorname{Im} f\bigl(\widetilde x\psi(A^t)\bigr) \right)^2 d\widetilde x, \]
\[ \lim_{p\to\infty}\int_{\pi} \left( \frac{1}{\sqrt p}\sum_{t=0}^{p-1} \operatorname{Re} f\bigl(\widetilde x\psi(A^t)\bigr) \right) \left( \frac{1}{\sqrt p}\sum_{t=0}^{p-1} \operatorname{Im} f\bigl(\widetilde x\psi(A^t)\bigr) \right) d\widetilde x \]
exist; denote them, respectively, by \(\sigma_{20},\sigma_{02},\sigma_{11}\).
2)
\[ \operatorname*{mes}_{\widetilde x\in\pi} E\left\{ \frac{1}{\sqrt p} \left| \sum_{t=0}^{p-1} f\bigl(\widetilde x\psi(A^t)\bigr) \right| <\lambda \right\} \]
\[ = 1-\frac{\sqrt M}{2\pi} \int_{0}^{2\pi} e^{-\frac12\lambda^2\frac{\delta(\theta)}{M}} \frac{d\theta}{\delta(\theta)} + O\left( \frac{(\log\log p)^{3/2}}{\sqrt{\log p}} \right) \quad \text{if } M>0, \]
where
\[ M=\sigma_{20}\sigma_{02}-\sigma_{11}^2, \qquad \delta(\theta)=\sigma_{20}\cos^2\theta -2\sigma_{11}\cos\theta\sin\theta +\sigma_{02}\sin^2\theta. \]
A special case of this theorem without indicating the remainder term was obtained by M. P. Mineev.
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
17 VII 1961
REFERENCES
- M. P. Mineev, Izv. Akad. Nauk SSSR, Ser. Mat., 22, 585 (1958).
- M. Kac, Ann. of Math., 2nd ser., 47, No. 1, 33 (1946).
- I. I. Pyatetskii-Shapiro, Uch. Zap. Moscow State Pedagogical Institute named after V. I. Lenin, 108, issue 2 (1957).
- A. M. Polosuev, Vestnik Moskov. Univ., No. 5 (1960).
- J. Cigler, J. f. reine u. angew. Math., 205, Heft 1/2, 91 (1960).