MATHEMATICS

A. M. POLISUEV

ON THE UNIFORM DISTRIBUTION OF A SYSTEM OF FUNCTIONS THAT IS A SOLUTION OF A SYSTEM OF LINEAR FINITE-DIFFERENCE EQUATIONS OF THE FIRST ORDER

(Presented by Academician I. M. Vinogradov on 28 VI 1958)

Let a system of functions \(\varphi_1(x), \ldots, \varphi_s(x)\) be a solution of the following system \(S\) of linear finite-difference equations of the first order with integer coefficients

\[ \begin{aligned} \Phi_1(x+1)&=a_{11}\Phi_1(x)+\ldots+a_{s1}\Phi_s(x),\\ &\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\\ \Phi_s(x+1)&=a_{1s}\Phi_1(x)+\ldots+a_{ss}\Phi_s(x). \end{aligned} \tag{1} \]

We assume that the determinant of the matrix

\[ \begin{pmatrix} a_{11}\ \cdot\ \cdot\ a_{s1}\\ \cdot\ \cdot\ \cdot\ \cdot\\ a_{1s}\ \cdot\ \cdot\ a_{ss} \end{pmatrix} \tag{2} \]

of the system (1) is different from zero.

Consider, in the unit hypercube,

\[ 0\leq x_1\leq 1,\ldots,\quad 0\leq x_s\leq 1 \tag{3} \]

an arbitrary domain \(v\), whose volume we denote by \(|v|\).

Let \(N_p(v)\) be the number of points \(M(\{\varphi_1(x)\},\ldots,\{\varphi_s(x)\})\) that fall into the domain \(v\) for \(x=1,2,\ldots,p\). The system of functions \(\varphi_1(x),\ldots,\varphi_s(x)\) is called uniformly distributed in \(s\)-dimensional space if

\[ \lim_{p\to\infty}\frac{N_p(v)}{p}=|v| \]

(see (¹)).

Theorem 1. The system of functions \(\varphi_1(x),\ldots,\varphi_s(x)\), which is a solution of the system (1), is uniformly distributed in \(s\)-dimensional space if none of the roots of the characteristic polynomial of the matrix (2) is equal in modulus to one and if, for every hypercube of volume \(|v|\) with sides parallel to the coordinate axes and lying entirely inside the unit hypercube (3), the relation

\[ \lim_{p\to\infty}\frac{N_p(v)}{p}<c|v|, \]

holds, where \(c\) is a certain constant.

For an exponential function, a similar theorem was previously proved by I. I. Shapiro-Pyateckii (²) and A. G. Postnikov (³, ⁴).

Using Theorem 1, one can prove the theorem formulated below, Theorem 2.

Let \(X_1,\ldots,X_s\) run through all integers. Then the points whose coordinates are equal to the corresponding coordinates of the vectors

\[ (X_1,\ldots,X_s) \begin{pmatrix} a_{11}\cdot\ldots\cdot a_{1s}\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ a_{s1}\cdot\ldots\cdot a_{ss} \end{pmatrix}^{-k}, \]

where \(k\) is a natural number, form in \(s\)-dimensional space a lattice of parallelepipeds. We shall call this lattice the lattice of rank \(k\). Let \(\Delta^{(k_1)}\) be one of the parallelepipeds of the lattice of rank \(k_1\). Define the mapping of the hypercube (3) onto \(\Delta^{(k_1)}\) as follows:

\[ f_{\Delta^{(k_1)}}(x_1,\ldots,x_s)= \]

\[ =(X_1^{(k_1)},\ldots,X_s^{(k_1)}) \begin{pmatrix} a_{11}\cdot\ldots\cdot a_{1s}\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ a_{s1}\cdot\ldots\cdot a_{ss} \end{pmatrix}^{-k_1} + (x_1,\ldots,x_s) \begin{pmatrix} a_{11}\cdot\ldots\cdot a_{1s}\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ a_{s1}\cdot\ldots\cdot a_{ss} \end{pmatrix}^{-k_1}, \]

where \(X_1^{(k_1)},\ldots,X_s^{(k_1)}\) are integers determined by the parallelepiped \(\Delta^{(k_1)}\).

The image of some parallelepiped \(\Delta^{(k_2)}\) of the lattice of rank \(k_2\), lying wholly inside the hypercube (3), will be denoted by \(\Delta^{(k_1)}\Delta^{(k_2)}\). It is clear that
\[ \Delta^{(k_1)}\Delta^{(k_2)}\subset \Delta^{(k_1)}. \]
Next we define inductively

\[ \Delta^{(k_1)}\ldots\Delta^{(k_n)} = (\Delta^{(k_1)}\ldots\Delta^{(k_{n-1})})\Delta^{(k_n)}, \qquad n=3,4,\ldots \]

Now take the parallelepipeds of ranks \(1,2,\ldots\) lying wholly inside the hypercube (3), and number them arbitrarily within each rank:

\[ \Delta_1^{(1)},\ldots,\Delta_i^{(1)} \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ \Delta_1^{(r)},\ldots,\Delta_{i_r}^{(r)} \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

Construct the sequence of closed sets nested one inside another

\[ \Delta_1^{(1)},\ \Delta_1^{(1)}\Delta_2^{(1)},\ldots,\Delta_1^{(1)}\Delta_2^{(1)}\ldots\Delta_i^{(1)},\ \Delta_1^{(1)}\Delta_2^{(1)}\ldots\Delta_i^{(1)}\Delta_1^{(2)},\ldots \]

\[ \ldots,\Delta_1^{(1)}\Delta_2^{(1)}\ldots\Delta_i^{(1)}\Delta_1^{(2)}\ldots\Delta_{i_2}^{(2)},\ldots \tag{4} \]

Theorem 2. Let all roots of the characteristic polynomial of the matrix (2) be, in modulus, greater than one. Then there exists a unique point \(N(\mu_1,\ldots,\mu_s)\) belonging to all the sets of the sequence (4); the system of functions \(\varphi_1(x),\ldots,\varphi_s(x)\), which is a solution of the system (1) with initial values \(\varphi_1(1)=\alpha_1,\ldots,\varphi_s(1)=\alpha_s\), where \(\alpha_1=\mu_1a_{11}+\ldots+\mu_sa_{s1},\ldots,\alpha_s=\mu_1a_{1s}+\ldots+\mu_sa_{ss}\), is uniformly distributed in \(s\)-dimensional space.

We prove both theorems by the method of the paper \((^4)\).

Remark. From Theorem 2 it follows, in particular, that the system of functions \(\alpha_1 q_1^x,\ldots,\alpha_s q_s^x\), where \(q_1,\ldots,q_s\) are integers greater than one, is uniformly distributed. This result was obtained earlier in the paper \((^5)\) by another method.

From Theorem 2 there also follows the uniform distribution of the exponential function in a complex domain considered in the paper \((^4)\).

Moscow Power Engineering Institute

Received
18 VI 1958

REFERENCES

\({}^1\) H. Weyl, Math. Ann., 77, 313 (1916).
\({}^2\) I. I. Shapiro-Pyatetskii, Izv. AN SSSR, ser. matem., 15, 47 (1951).
\({}^3\) A. G. Postnikov, DAN, 86, 473 (1952).
\({}^4\) A. G. Postnikov, Vestn. LGU, No. 13, issue 3, 81 (1957).
\({}^5\) N. M. Korobov, DAN, 84, 13 (1952).