MATHEMATICS

L. P. POSTNIKOVA

ON A THEOREM OF N. M. KOROBOV

(Presented by Academician I. M. Vinogradov on 23 IV 1960)

N. M. Korobov proved, \((^3)\), that if

\[ \alpha_1,\alpha_2,\ldots,\alpha_p,\ldots \tag{1} \]

is a completely uniformly distributed sequence, then the sequence

\[ [\alpha_1 q],\ [\alpha_2 q],\ldots,[\alpha_p q],\ldots \]

is a normal sequence of digits*.

In the present work we prove a theorem extending this theorem.

Theorem. The system of sequences

\[ \begin{gathered} \varepsilon_1,\ \varepsilon_2,\ldots,\varepsilon_p,\ldots,\\ \delta_1,\ \delta_2,\ldots,\delta_p,\ldots, \end{gathered} \tag{2} \]

where

\[ \varepsilon_j=[\alpha_j q_1],\qquad \delta_j=[q_2\{\alpha_j q_1\}], \]

is jointly normal.

Proof. Let \(s=1\). Take the one-column matrix
\[ \binom{\varepsilon}{\delta}, \]
\(0\leqslant \varepsilon \leqslant q_1-1,\ 0\leqslant \delta \leqslant q_2-1\). The occurrence of such a matrix in the system of sequences (2) in the \(j\)-th place is possible if and only if the number \(\alpha_j\) falls into the half-interval
\[ \left(\frac{\varepsilon q_2+\delta}{q_1q_2},\ \frac{\varepsilon q_2+\delta+1}{q_1q_2}\right). \]
Observe that the length of this half-interval is
\[ \frac{1}{q_1q_2}. \]

Since the sequence (1) is completely uniformly distributed, the asymptotic frequency of occurrence of the matrix
\[ \binom{\varepsilon}{\delta} \]
in the system of sequences (2) is equal to
\[ \frac{1}{q_1q_2}. \]

Similarly, it is proved that for any natural number \(s\) and any \(s\)-column matrix
\[ \begin{pmatrix} \bar{\varepsilon}_1,\ \bar{\varepsilon}_2,\ldots,\bar{\varepsilon}_s\\ \bar{\delta}_1,\ \bar{\delta}_2,\ldots,\bar{\delta}_s \end{pmatrix}, \]
where \(0\leqslant \bar{\varepsilon}_k\leqslant q_1-1,\ 0\leqslant \bar{\delta}_k\leqslant q_2-1,\ k=1,2,\ldots,s\), the asymptotic frequency of occurrence in the system of sequences (2) is equal to
\[ \frac{1}{q_2^s q_1^s}. \]
Consequently, the system of sequences (2) is jointly normal.

In the paper \((^2)\)** the following problem was solved.

* For the concepts of a normal sequence, jointly normal sequences, and a completely uniformly distributed sequence, see \((^1)\), §§ 1, 7, 12.

** For an exposition of this paper, see \((^1)\).

Let there be given a normal sequence consisting of the symbols \(0,1,\ldots,q_1-1\),

\[ \varepsilon_1,\ \varepsilon_2,\ldots,\varepsilon_P,\ldots \tag{3} \]

Construct a sequence, consisting of the symbols \(0,1,\ldots,q_1-1\),

\[ \delta_1,\ \delta_2,\ldots,\delta_P,\ldots \tag{4} \]

so that the system of sequences

\[ \begin{gathered} \varepsilon_1,\ \varepsilon_2,\ldots,\varepsilon_P,\ldots,\\ \delta_1,\ \delta_2,\ldots,\delta_P,\ldots \end{gathered} \tag{5} \]

is jointly normal.

In § 7 of paper \({}^{1}\) a method is presented for constructing, for a given sequence, any number \(l\) of sequences consisting of the symbols \(0,1,\ldots,q_1-1\), such that the system of sequences

\[ \varepsilon_1,\ \varepsilon_2,\ldots,\varepsilon_P,\ldots, \]

\[ \varepsilon^{(1)}_1,\ \varepsilon^{(1)}_2,\ldots,\varepsilon^{(1)}_P,\ldots, \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ \varepsilon^{(l)}_1,\ \varepsilon^{(l)}_2,\ldots,\varepsilon^{(l)}_P,\ldots \]

is jointly normal.

With the aid of the theorem proved in the present paper, from the sequence (3) we shall construct a sequence consisting of the symbols \(0,1,\ldots,q_2-1\),

\[ \delta_1,\ \delta_2,\ldots,\delta_P,\ldots \]

such that the system of sequences will be jointly normal.

In paper \({}^{1}\), § 15, the construction is given of a completely uniformly distributed sequence

\[ \alpha_1,\ \alpha_2,\ldots,\alpha_P,\ldots, \]

such that \(\varepsilon_j=[q_1,\alpha_j]\). Taking \(\delta_j=[q_2\{q_1\alpha_j\}]\), we obtain a sequence \(\delta_1,\delta_2,\ldots,\delta_P,\ldots\) such that the system of sequences (5) is jointly normal.

Mathematical Institute
named after V. A. Steklov
Academy of Sciences of the USSR

Received
18 IV 1960

CITED LITERATURE

\({}^{1}\) A. G. Postnikov, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, No. 57 (1960).
\({}^{2}\) L. P. Starchenko, Izv. AN SSSR, Ser. Matem., 22, 757 (1958), \({}^{3}\) N. M. Korobov, Izv. AN SSSR, Ser. Matem., 14, 215 (1950).