Arch. Math., Vol. 43, 530-534 (1984) 0003-889 X/84/4306-0530 $ 2.50/0 �9 1984 Birkhauser Verlag, Basel

A general rearrangement theorem for sequences

By

HARALD NIEDERREITER

1. Introduction. By a rearrangement theorem for sequences we mean a result which says that, under suitable hypotheses, a sequence will attain a desired property after a suitable rearrangement of terms. Such results have been of particular interest in the theory of uniform distribution of sequences (see Kuipers and Niederreiter [7]). The classical result of this type is the rearrangement theorem of yon Neumann [10], according to which any dense sequence in the interval [0, 1] can be rearranged into a sequence that is uniformly distributed in [0, 1]. This was later refined in a quantitative way by van der Corput [9] and Hlawka [6]. Rearrangement theorems in the general setting of compact metrizable spaces were first considered by Hlawka [4], and then by Hlawka [5], Descovich [2], and the author [8], among others.

We will show that all the standard rearrangement theorems in the theory of uniform distribution are simple consequences of a general rearrangement theorem. This theorem establishes the precise condition under which a sequence can be rearranged so as to approximate another given sequence. For a sequence co = (z,), n = 1, 2 . . . . , of ele- ments of the compact metric space (X, d) let A (co) denote the set of accumulation points of co, i.e.

A (co) = {x E X: for every open neighborhood U of x there exist infinitely many n ~ N with z, e U},

where N is the set of positive integers. From the topological conditions it follows that A (co) is a nonempty compact subset of X.

Theorem. Let col = (x.) and ~ = (Y.) be sequences of elements of the compact metric space (X, d). Then the following two conditions are equivalent:

(i) there exists a permutation z of N such that

lim d(x. , y~.)) = 0; n~r

(ii) A(col) = A(co2).

The proof of this theorem is given in Section 2. In Section 3 we show that the standard rearrangement theorems in the theory of uniform distribution can be derived easily from this result. The author notes that this paper originated from discussions with

Vol. 43, 1984 Rearrangement theorem 531

H. Furs tenberg of the Hebrew University, Jerusalem, and R. F. Tichy of the Techno- logical Universi ty of Vienna.

In the sequel we will simply write X for the compact metric space (X, d). We use B(x; ~) to denote the open ball with center x ~ X and radius ~ > 0.

2. Proof of the theorem. The fact that (i) implies (ii) can be verified in a s traightforward manner. F o r the converse we need two auxil iary results.

Lemma 1. For any sequence co = (z,) o f elements o f X there exists a sequence (w,) o f elements o f A (co) with lim d (z,, w.) = 0.

n--* oo

P r o o f. Since A = A (~o) is compact, we can define the function

f ( x ) = min d(x, a) for all x ~ X . a e A

The function f is cont inuous on X. F o r any n ~ N there exists w, c A with f ( z , ) = d (z,, w,). To prove lira f(z~) = 0, we assume by way of contradict ion that there

n ~ o o

exists e > 0 with f ( z , ) > e for infinitely many n. Then, since S = {x ~ X: f ( x ) _>_ e} is compact, ~o must have an accumulat ion point z ~ S. But also z ~ A by the definition of A, hence f ( z ) = 0, and the desired contradic t ion is obtained.

Lemma 2. Let (9 a = (v,) and ( ~ = (w,) be sequences o f elements o f X with A(a~3) = A (a~) and v, ~ A (co3), w, ~ A (o94)for all n ~ N . Then for every e > 0 there exists afinite collection { C1 . . . . . Cq} o f subsets o f X with the following properties:

(i) each v n and each w, lies in exactly one o f the Cfi (ii) for each Cj there exist infinitely many n ~ N with v n ~ C i and infinitely many n ~ N

with w n ~ C j; (iii) the diameter o f each Cj is at most e.

P r o o f. Put A = A (a~3) = A (o94), and let co -- (u,) be the "mixed" sequence vl, wl, rE, w2, . . . . Then A (w) = A. Since A is compact , we can cover A by finitely many open balls B(al; e/4) . . . . , B(ak; e/4) with al . . . . . a k ~ A. The distances d(ah, u,) with 1 _< h _< k and n e ]N at tain only countably many values, thus there exists 6 with e/4 _< 6 -< e/2 and d(ah, Un) =[= 8 for 1 ~ h _< k and n e N. Let P be the set of all nonempty subsets of {1, 2 . . . . . k}. F o r I e P define C(I) = {x e X: d(ah, x) < 8 for all h ~ I, d(ah, x) > 8 for all h ~ IC}, where I c is the complement of I in {1, 2 . . . . , k}. Let Q be the set of all I e P for which C (I)c~ A is nonempty. We will show that the collection {C(I): I e Q} satisfies the desired properties.

Let u, be a fixed element of w. Then u, e A, and so u, ~ B(al; 8) for some i, 1 _< i _< k. Therefore I = {h e N : 1 <_ h <_ k, d(ah, u,) < 8} is a nonempty subset o f{ l , 2 , . . . , k}. I t is clear that u, e C(I ) and I e Q. If u, ~ C(J) for some J e Q, then it follows immediately from the definitions that J = I. Therefore proper ty (i) is satisfied. Next, each C (I) with I e Q contains a point of A and is thus an open ne ighborhood of that point. It follows then from the definition of A that (ii) holds. Finally, each C (I) with I e Q is contained in the ball B(ah; 8) for a fixed h e I, and so the diameter of C(I ) is < 28 < e. The proof of Lemma 2 is thus complete.

34*

532 H . N I E D E R R E I T E R A R C H . M A T H .

Now let (01 = (x,) and (02 = ( Y n ) be sequences of elements of X with A ((01) = A ((02)- Put A = A ((01) = A ((02). By Lemma 1 we can find sequences (03 = (v,) and (04 = (w,) of elements of A with

(1) lira d(x, , v,) = 0 and lim d(w,, y,) = O. n--+ o9 n ~ o o

It follows from (1) that A (093) = A (094) = A, and so the sequences (03 and (04 satisfy the hypotheses of Lemma 2. F o r m e N let D m be the finite collection of subsets of X constructed in Lemma 2 for e = m-1. Let D be the set of all ordered pairs (m, C) with m e N and CeDm.

We consider now the sequence (03 = (v,) and define a mapping ~ = ~3: D ~ N in the following way. We first define ~b for the pairs (1, C) with C e D1 by letting ~ (1, C) be the least n e N with v, e C. Suppose that ~b has already been defined on

E, = {(m, C): m e N , m < r, CeDm}

for some r e N . Then we define ~ for the pairs (r + 1, C) with C e D,+I by letting (r + 1, C) be the least n e N with v, e C and n r ~ (Er). The definition of 0 makes sense

by proper ty (ii) in Lemma 2. We show next that the mapping ~ is a bijection. Suppose ~ (r, C) = ~ (r', C') for some

(r, C), (r', C') e D, and let n e N be the common value. If r 4= r', we can assume w.l.o.g, that r > r'. Then (r', C') e E,_ 1, and so n e q; (E,_ 1)- I t follows then from the definition of that ~ (r, C) 4 = n, a contradiction. Thus we must have r = r'. Then v, e C and v, e C' with C, C' e D r, and so C = C' by proper ty (i) in Lemma 2. Thus ~9 is injective. Suppose ~ were not surjective, and let s be the least positive integer that is not an element of ~k (D). Since vl e C for some C e D 1, we have ~ (1, C) = 1, and so s > 2. The integers 1, 2 . . . . . s - 1 are all in ~ (D), hence in some ~k (Er). Now vs e C' for some C'e Dr+~, and so the definition of ~O yields ~ (r + 1, C') = s, a contradiction. Therefore ~ is a bijection.

In the same way as we have constructed the bijection ~3: D ~ N from the sequence (03 = (v,), we can construct a bijection q;4: D ~ N from the sequence (04 = (w,). Then z (n) = q;4(~k31 (n)) for n e N defines a permuta t ion of N. F o r given n e N put ~ 1 (n) = (m, C). Then O3(m, C) = n, and so v, e C. Also z(n) = O4(m, C), hence w,(,) e C. Since C ~ D m has diameter at most m -1 by proper ty (iii) in Lemma 2, it follows that d(v,, w~.)) <= rn-1. But m-1 becomes arbi trar i ly small for sufficiently large n, hence

(2) lim d(v,, w~(.)) = O. n--~ ~3

Now d(x, , y~,)) <= d(x,,, v,) + d(v,, w.(,,)) + d(w,(,), Y~oo),

and so (1) and (2) imply lira d(x,, y~(,)) = 0. The proof of the theorem is thus complete. n ~ o o

3. Some consequences. We show now that the classical rearrangement theorems in the theory of uniform distr ibution are easy consequences of our theorem. We recall that if # is a Borel probabi l i ty measure in X, then a sequence (z,) of elements of X is called #-uniformly distr ibuted (#-u.d.) if

1 N lim ~ ~,lf(z,) = ~xf dp

N- + ao =

Vol. 43, 1984 Rearrangement theorem 533

holds for all real-valued continuous functions f on X. Equivalently, (z,) is #-u.d. if

1 N (3) lira i n f - - Z ca (z,) > # (B)

for all open sets B in X, where cB is the characteristic function of B; compare with Kuipers and Niederreiter [7, Ch. 3]. Various versions of the following result can be found in Descovich [2], Hlawka [4], and Kuipers and Niederreiter [7, Ch. 3].

Corollary 1. The sequence co = (y,) of elements of X has a #-u.d. rearrangement if and only if A (co) contains the support of #.

P r o o f. The necessity of the condition follows immediately from (3) and the fact that #(U) > 0 for every open neighborhood U of a point in the support K of #. Conversely, suppose A (co) ___ K. F rom [7, Ch. 3, Theorems 1.3 and 2.2] it follows that there exists a #-u.d. sequence (z,) with all z, in K. Putting x, = yp for a square n = p2, p = 1, 2, . . . , and x, = z, otherwise, we get a #-u.d. sequence col = (x,) with A (coo = A (co). Our theorem yields lim d(x, , y,~,)) = 0 for a suitable permutat ion z of N, and this implies that (y~,~)

n ~ 3

is #-u.d. A sequence (Zn) of elements of X is called #-well distributed (#-w.d.) if for all real-valued

continuous functions f on X we have

1 N+h lira ~ Z f ( z , ) = [ f d # uniformly in h = 0 , 1

N--*6o l~/ n = l + h 2

See Hansel and Troallic [3] and Kuipers and Niederreiter [7, Ch. 3] for variants of the following result.

Corollary 2. The sequence co = (y,) of elements of X has a #-w.d. rearrangement if and only if A (09) contains the support of p.

P r o o f. The necessity follows from Corollary 1 since every #-w.d. sequence is #-u.d. The sufficiency is shown as in Corollary 1, with a result of Baayen and Hedrlin [1] guaranteeing the existence of a #-w.d. sequence of elements in the support of #.

A sequence (z~) of elements of X is called completely #-u.d. if for all k ~ N the sequence of k-tuples ((Zn, Z,+ 1 . . . . . Z,+k-1)), n = 1, 2, . . . , is #k-u.d., where #k is the product measure in the cartesian product X k induced by #. The following result is essentially due to Hlawka [5].

Corollary 3. The sequence co = (Yn) of elements of X has a completely p-u.d, rearrange- ment if and only if A (co) contains the support of#.

P r o o f. The necessity follows from Corollary 1. The sufficiency is shown as in Corol- lary 1, with [7, Ch. 3, Theorem 3.13 and Exercise 3.11] guaranteeing the existence of a completely #-u.d. sequence of elements in the support of #.

The essential feature of the arguments above is that each of these distribution properties is such that if a sequence (x,) satisfies the property, then any sequence (z,)

5 3 4 H . NIEDERREITER ARCH. MATH.

with lim d (x,, z,) = 0 also satisfies the property. Our rearrangement theorem can ob- n--* co

viously be applied to any other proper ty showing the same feature. Fo r instance, we can apply it to uniform distr ibution with respect to regular summat ion methods, or to sequences (z,) having a prescribed set of "accumulat ion measures", i.e. measures # for which there exists an increasing sequence N1, N2 . . . . of positive integers such that

1 N~ lim - - 5Z f (z,) = ~xf d# i~oo Ni n=l

for all real-valued continuous functions f on X. The only rearrangement theorems in the theory of uniform distr ibution that cannot be obtained from our theorem are quanti tat ive results such as those of Hlawka [6] and the author [8].

References

[1] P. C. BAAYEN and Z. HEDRLIN, The existence of well distributed sequences in compact spaces. Indag. Math. 27, 221-228 (1965).

[2] J. DESCOVlCH, Zur Theorie der Gleichverteilung auf kompakten Riiumen. Sitzungsber. Osterr. Akad. Wiss. Math.-naturw. K1. Abt. II 178, 263 283 (1969).

[3] G. HANSEL and J.-P. TROALLIC, Suites uniform~ment distribu6es et suites bien distribu6es; une approche combinatoire. In: Noncommutative structures in algebra and geometric combina- torics (Naples, 1978), pp. 101-110, Rom 1981.

[4] E. HLAWKA, Folgen auf kompakten R/lumen. Abh. Math. Sem. Univ. Hamburg 20, 223-241 (1956).

[5] E. HLAWKA, Folgen auf kompakten Rfiumen. II. Math. Nachr. 18, 188-202 (1958). [6] E. HLAWKA, Interpolation analytiseher Funktionen auf dem Einheitskreis. In: Number theory

and analysis (P. Turfin, ed.), pp. 97-118. New York-Berlin 1969. [7] L. KuIPERS and H. NIEDERREITER, Uniform distribution of sequences. New York 1974. [8] H. NIEDERRErrER, Rearrangement theorems for sequences. In: Ast6risque 24-25, 243-261,

Paris 1975. [9] J. G. VAN DER CORPUT, Verteilungsfunktionen I-VIII . Proc. Akad. Amsterdam 38, 813-821,

1058-1066 (1935); 39, 10-19, 19-26, 149-153, 339-344, 489-494, 579-590 (1936). [10] J. YON NEtnaA~, Uniformly dense sequences of numbers (Hungarian). Mat. Fiz. Lapok 32,

32-40 (1925).

Eingegangen am 10.2. 1984

Anschrift des Autors:

Harald Niederreiter Kommission fiir Mathematik Osterreicbische Akademie der Wissenschaften Dr. Ignaz-Seipel-Platz 2 A-1010 Wien ()sterreich