MATHEMATICS

A. G. POSTNIKOV

A CRITERION FOR A COMPLETELY UNIFORMLY DISTRIBUTED SEQUENCE

(Presented by Academician I. M. Vinogradov on 17 I 1958)

In the present paper a criterion is given for a sequence of real numbers \(\alpha\), taken from the interval \([0,1]\):
\(\alpha=\alpha_1,\alpha_2,\alpha_3,\ldots\), to be completely uniformly distributed.* Our arguments will be analogous to the arguments of § 4 of paper \((^2)\).

By \(\Delta_s=(\delta_1,\delta_2,\ldots,\delta_s)\) we shall denote a parallelepiped lying in the \(s\)-dimensional unit cube, defined by the condition that the \(i\)-th coordinate of its points belongs to the interval \(\delta_i\); by \(|\Delta_s|\) we shall denote the volume of the parallelepiped \(\Delta_s\).

Theorem. Let the sequence \(\alpha\) be such that there exists a constant \(c\) such that, for any \(s \geqslant 1\) and any \(\Delta_s\),

\[ \lim_{P\to\infty}\frac{N_P(\Delta_s)}{P}<c|\Delta_s|. \]

Then the sequence \(\alpha\) is completely uniformly distributed.

Lemma 1. Let \(r\geqslant 1\) be an integer. Let \(\delta\) be some interval in \([0,1]\), and let \(|\delta|\) be its length. Consider the unit cube in \(l\)-dimensional space. The Lebesgue measure of those points of this cube for which the number of coordinates falling in the interval \(\delta\) (denote this number by \(A_\delta^l\)) satisfies the inequality

\[ \left|A_\delta^l-l|\delta|\right|\geqslant \frac{l}{r}, \]

does not exceed \(r^4/4l^2\).

Proof. There are \(C_l^k\) ways by which one can distribute \(k\) marks “fell in \(\delta\)” among \(l\) positions. Let \(\delta\) be the interval \([a,b]\). The volume of the region defined by the condition that \(k\) specified coordinates satisfy the inequalities \(a\leqslant \xi\leqslant b\), and the remaining \(l-k\) the inequalities either \(0\leqslant \xi\leqslant a\), or \(b<\xi\leqslant 1\), is equal to \((b-a)^k(1-(b-a))^{l-k}=|\delta|^k(1-|\delta|)^{l-k}\). Therefore the volume of the region of points \(k\) of whose coordinates have fallen in \(\delta\), and the remaining ones outside \(\delta\), is \(C_l^k|\delta|^k(1-|\delta|)^{l-k}\). Hence the measure sought in the lemma is equal to

\[ L=\sum_{|k-l|\delta||\geqslant l/r} C_l^k|\delta|^k(1-|\delta|)^{l-k}. \]

Carrying out the usual estimate of this expression, we obtain the lemma.

* For the definition of a completely uniformly distributed sequence, see paper \((^1)\).

Let \(s \geqslant 1\) and \(l \geqslant 1\) be natural numbers. Consider the unit cube in \(ls\)-dimensional space. Let \(\Delta_s=(\delta_1,\ldots,\delta_s)\) be a fixed parallelepiped in the unit cube of \(s\)-dimensional space; \(|\Delta_s|=|\delta_1|\cdots|\delta_s|\) is the volume of this parallelepiped. Let some point of the cube have coordinates
\[ (a_1,a_2,\ldots,a_s,a_{s+1},\ldots,a_{2s},\ldots,a_{(l-1)s+1},\ldots,a_{ls}). \]
Group the coordinates \(s\) at a time, \((B_1\ldots B_l)\), where
\[ B_k=(a_{(k-1)s+1}\ldots a_{ks}),\qquad k=1,2,\ldots,l, \]
points of the unit cube of \(s\)-dimensional space. Denote by \(A_{\Delta_s}^{(l)}\) the number of these points that fall into \(\Delta_s\).

Lemma 2. Let \(r \geqslant 1\) be an integer. The Lebesgue measure of the points of the unit cube of \(ls\)-dimensional space for which
\[ \bigl|A_{\Delta_s}^{(l)}-l|\Delta_s|\bigr| \geqslant \frac{l}{r} \]
does not exceed \(r^4/4l^2\).

Proof. We compute the Lebesgue measure of the set of those points of the \(ls\)-dimensional cube for which, in the representation \((B_1,\ldots,B_l)\), at certain \(x\) places there is a hit in \(\Delta_s\), and at the remaining places there is not. Let \(E_s\) be the unit cube of \(s\)-dimensional space. Obviously, this measure is equal to
\[ \underbrace{\int_{\Delta_s}\cdots\int_{\Delta_s}}_{x\ \text{times}} \underbrace{\int_{E_s/\Delta_s}\cdots\int_{E_s/\Delta_s}}_{l-x\ \text{times}} dx_1\cdots dx_{ls} = |\Delta_s|^x(1-|\Delta_s|)^{l-x} \]
(each integral is \(s\)-fold).

Since the \(x\) hits can be arranged among the \(l\) places in \(C_l^x\) ways, the measure sought in the lemma is equal to
\[ L=\sum_{\substack{x=0\\ |x-l|\Delta_s||\geqslant l/r}}^{l} C_l^x|\Delta_s|^x(1-|\Delta_s|)^{l-x}. \]

Carrying out the usual estimate, we obtain that \(L\leqslant r^4/4l^2\).

Proof of the theorem. Let the sequence \(\alpha=\alpha_1,\alpha_2,\ldots,\alpha_s,\ldots\) satisfy the condition of the theorem. Let \(\Delta_s=(\delta_1,\ldots,\delta_s)\) be a parallelepiped in the unit cube of \(s\)-dimensional space. Arrange the numbers of the sequence into groups of \(s\):
\[ P_1^{(s)},P_2^{(s)},\ldots, \]
where
\[ P_k^{(s)}=(\alpha_{(k-1)s+1},\ldots,\alpha_{ks}), \]
i.e. consider the sequence of points of the \(s\)-dimensional unit cube. From \(X\) terms of the sequence we obtain \([X/s]\) points. Denote by \(A_{\Delta_s}^{[X/s]}\) the number of times the points \(P_j^{(s)}\) \((j=1,\ldots,[X/s])\) fall into \(\Delta_s\). Choose arbitrarily \(l\geqslant 1\) and arrange the points \(P^{(s)}\) into groups of \(l\):
\[ \underbrace{P_1^{(s)}P_2^{(s)}\cdots P_l^{(s)}}_{l}\ \underbrace{P_{l+1}^{(s)}P_{l+2}^{(s)}\cdots P_{2l}^{(s)}}_{l}\ \cdots \]

Take a natural \(r\geqslant 1\) and call an \(l\)-term group of points “good” if the number of points falling into \(\Delta_s\) is equal to \(l(|\Delta_s|+\theta/r)\), \(|\theta|\leqslant 1\), and “bad” otherwise. Denote by \(M([X/s])\) the number of times bad groups occur in the sequence of groups formed from the first \(l\left[\frac{[X/s]}{l}\right]\) numbers of the sequence \(\alpha\).

Then the number of good groups will be
\[ \left[\frac{X}{sl}\right]-M\left(\left[\frac{X}{s}\right]\right) = \frac{X}{sl}-M\left(\left[\frac{X}{s}\right]\right)+O(1) \]
with an absolute constant in \(O\). A good group contributes \(l(|\Delta_s|+\theta/r)\) points falling into \(\Delta_s\). Therefore, among the \([X/s]\) points \(P^{(s)}\), the number falling into \(\Delta_s\) is

\[ A_{\Delta_s}^{[X/s]} = l\left(|\Delta_s|+\frac{\theta}{r}\right) \left(\frac{X}{sl}-M\left(\left[\frac{X}{s}\right]\right)\right) + l\theta_1 M\left(\left[\frac{X}{s}\right]\right)+O(l). \]

The term \(O(l)\) arises from the fact that, possibly, there is an incomplete group of points. Hence

\[ A_{\Delta_s}^{[X/s]} = \frac{X}{s}\left(|\Delta_s|+\frac{\theta}{r}\right) + O\left(M\left(\left[\frac{X}{s}\right]\right)l\right)+O(l). \]

Let \(\mathfrak M\) denote the set of those points in the \(ls\)-dimensional unit cube for which
\[ \left|A_{\Delta_s}^{(l)}-l|\Delta_s|\right|\geq l/r. \]
Each bad combination represents a point of \(\mathfrak M\). Therefore
\[ M\left(\left[\frac{X}{s}\right]\right)\leq N_{s[X/s]}(\mathfrak M). \]
But for \(X\geq X_0\), according to the hypothesis of the theorem,

\[ N_{s[X/s]}(\mathfrak M)\leq 2C\,\operatorname{mes}\mathfrak M\,X, \]
\[ A_{\Delta_s}^{[X/s]} = \frac{X}{s}\left(|\Delta_s|+\frac{\theta}{r}\right) + O\left(X\frac{r^4}{l}\right)+O(l) \]
(by Lemma 2).

Hence, letting \(X\) tend to infinity, we obtain

\[ \overline{\lim}_{X\to\infty} \left| \frac{A_{\Delta_s}^{[X/s]}}{X/s} - |\Delta_s| \right| \leq \frac{1}{r} + O\left(\frac{sr^4}{l}\right). \]

Now letting the parameter \(l\) tend to infinity, we obtain

\[ \overline{\lim}_{X\to\infty} \left| \frac{A_{\Delta_s}^{[X/s]}}{X/s} - |\Delta_s| \right| \leq \frac{1}{r}. \]

Consider the \(s\) sequences \(T^j\alpha,\ j=0,1,2,\ldots,s-1\) (among which \(T^0\alpha\) is our sequence \(\alpha\)),

\[ T^j\alpha=\alpha_{j+1},\alpha_{j+2},\alpha_{j+3},\ldots \]

Each of these sequences satisfies the condition of the criterion. Denote the quantity \(A_{\Delta_s}^{[X/s]}\), constructed for the sequence \(T^j\alpha\), by \(A_{\Delta_s}^{[X/s]}(T^j\alpha)\). We have

\[ \lim_{X\to\infty} \frac{A_{\Delta_s}^{[X/s]}(T^j\alpha)}{X/s} = |\Delta_s|, \qquad j=0,1,\ldots,s-1. \]

But, obviously,

\[ N_X(\Delta_s)=\sum_{j=0}^{s-1} A_{\Delta_s}^{[X/s]}(T^j\alpha)+O(s). \]

Hence

\[ \lim_{X\to\infty} \frac{N_X(\Delta_s)}{X} = s\frac{|\Delta_s|}{s} = |\Delta_s|. \]

Since \(s\geq 1\) and \(\Delta_s\) is an arbitrary parallelepiped, the theorem is proved.

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

Received
12 I 1958

References

1. A. G. Postnikov, Izv. AN SSSR, Ser. Mat., 22, No. 3 (1958).
2. A. G. Postnikov, I. I. Pyatetskii, Izv. AN SSSR, Ser. Mat., 21, No. 4, 501 (1957).