Reports of the Academy of Sciences of the USSR

1963, Vol. 148, No. 3

MATHEMATICS

B. KARIMOV

ON LINEAR DIOPHANTINE APPROXIMATIONS

(Presented by Academician I. M. Vinogradov on 26 VII 1962)

From Minkowski’s lemma on linear forms (see (¹), p. 36) the following assertion obviously follows:

Lemma. Whatever three real numbers \(\omega_1,\omega_2,\omega_3\) we take, there exists an infinite number of solutions in integers of the system of inequalities

\[ |\omega_1x-y|\leq \frac{1}{\sqrt{x}},\qquad |\omega_2x+\omega_3y-z|\leq \frac{1}{\sqrt{x}}. \]

With the help of the device set forth in the book (²), p. 24, the assertion of the lemma can be strengthened.

Lemma. Whatever three real numbers \(\omega_1,\omega_2,\omega_3\) we take, there exists an infinite number of solutions in integers of the system of inequalities

\[ |\omega_1x-y|\leq \frac{2}{3\sqrt{x}},\qquad |\omega_2x+\omega_3y-z|\leq \frac{2}{3\sqrt{x}}. \]

We shall call a system of three real numbers \(\omega_1,\omega_2,\omega_3\) \(K\)-supernormal if, for every \(\varepsilon>0\), the system of inequalities

\[ |x\omega_1-y|<\frac{\varepsilon}{\sqrt{x}},\qquad |x\omega_2+y\omega_3-z|<\frac{\varepsilon}{\sqrt{x}} \]

has an infinite number of solutions in integers. We shall call a system of numbers \(\omega_1,\omega_2,\omega_3\) \(K\)-normal if it is not \(K\)-supernormal.

Theorem 1. If for a system of three numbers \(\omega_1,\omega_2,\omega_3\) one can find three integers \(x_1^{(0)}\ne 0,\ x_2^{(0)},\ x_3^{(0)}\) such that

\[ x_1^{(0)}(\omega_1\omega_3+\omega_2)+x_2^{(0)}\omega_1+x_3^{(0)}=0, \]

then the system of numbers \(\omega_1,\omega_2,\omega_3\) is \(K\)-supernormal.

Theorem 2. Let \(\theta\) be a real irrationality of degree three, \(\omega_1=\theta\), \(\omega_3\) any real number, \(\omega_2=\theta^2-\omega_3\theta\). The system of numbers \(\omega_1,\omega_2,\omega_3\) is a \(K\)-normal system of numbers.

In this circle of ideas there also lies the following metric theorem.

Theorem 3. Consider the system of Diophantine inequalities

\[ |\omega_1x-y|\leq \frac{c}{\sqrt{x}\ln^\gamma x},\qquad |\omega_2x+\omega_3y-z|\leq \frac{c}{\sqrt{x}\ln^\gamma x}. \tag{1} \]

If \(\gamma>1/2\) (strictly), then the measure of the set of points \((\omega_1,\omega_2,\omega_3)\) of the unit cube for which inequality (1) has an infinite number of solutions in integers is equal to zero. If \(0<\gamma\leq 1/2\), then the measure of the set of points \((\omega_1,\omega_2,\omega_3)\) of the unit cube for which inequality (1) has an infinite number of solutions in integers is equal to one.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
19 VII 1962

CITED LITERATURE

1. B. A. Venkov, Elementary Number Theory, 1937.
2. J. W. S. Cassels, An Introduction to Diophantine Approximation, Moscow, 1961.