UNIFORMLY MALDISTRIBUTED SEQUENCES IN A

UNIFORMLY MALDISTRIBUTED SEQUENCES IN A STRICT SENSE OTO STRAUCH Abstract. It is shown that the following three limits lim

N →∞

lim inf N →∞

1 N2

N X

|xm − xn | = 0,

m,n=1

N N 1 X 1 X xn = 0, lim sup xn = 1 N n=1 N →∞ N n=1

are a necessary and sufficient condition for the given sequence ω = (xn )∞ n=1 ⊂ [0, 1] to have its only distribution functions be all one-jump functions. As an application, such sequences can also be used in deriving estimates of max f for continuous functions f defined on [0, 1].

I. Introduction (xn )∞ n=1

⊂ [0, 1] be a given sequence. Let ω = 1. ω is said to be uniformly maldistributed if for every nonempty proper subinterval I ⊂ [0, 1] we have both lim inf N →∞

1 1 #{n ≤ N ; xn ∈ I} = 0 and lim sup #{n ≤ N ; xn ∈ I} = 1. N N →∞ N

The definition of uniform maldistribution is due to G. Myerson [3] , who mentioned that the first condition is superfluous, and showed the following three properties: 2. The sequence ω = ({log log n})∞ n=2 of fractional parts of the iterated logarithm is uniformly maldistributed . 3. Let (Mk )∞ k=1 be a sequence of natural numbers with lim (M1 +· · ·+Mk )/Mk = 1, k→∞

by dense in [0, 1]. Finally, let ω = (xn )∞ and let (yn )∞ n=1 ⊂ [0, 1] be defined by Pn=1 Pk k−1 xn = yk if i=1 Mi ≤ n < i=1 Mi . Then ω is uniformly maldistributed . 4. The sequence ω is uniformly maldistributed if and only if lim inf N →∞

N N 1 X 1 X f (xn ) = min f, lim sup f (xn ) = max f N n=1 N →∞ N n=1

for every function f continuous on [0, 1]. The purpose of this note is to illustrate the application of an abstract method deriving a general criterion for a class of sequences. The basic idea is to determine the set of all distribution functions of sequences from a given class, and then to use a set of solutions of the corresponding generalized moment problem to estimate this set by inclusion. 1991 Mathematics Subject Classification. Primary 11K06, 11K31. Key words and phrases. Sequences, Distribution functions. This research was supported by the Slovak Academy of Sciences Grant 363. 1

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OTO STRAUCH

We shall introduce a class of some special uniformly maldistributed sequences and then we apply our method to the study of this class. Moreover, we shall give in 15 another characterization of the uniformly maldistributed sequence. The chief tools in applying the above method are the Helly selection principle, the Helly-Bray lemma (see [2, ?]), 1 and Riemann-Stieltjes integration. It should be noted that the study of the set of all distribution functions of a given sequence was initiated by van der Corput [5], but there is as yet no general theory for finding such sets. We begin with definitions which we shall need later for the formulation of results. II. Definitions and Notation In what follows 5. A distribution function in [0, 1] will be any g : [0, 1] → [0, 1] such that (i) g(0) = 0, g(1) = 1, (ii) g is nondecreasing, and (iii) g is left continuous on (0, 1). 6. For a sequence ω = (xn )∞ n=1 ⊂ [0, 1], and for every positive integer N , we define the following distribution function 1 FN (x) = #{n ≤ N ; xn < x} for all x ∈ [0, 1), and FN (1) = 1. N Now, let g : [0, 1] → [0, 1] be a given distribution function. If there exists an increasing sequence of natural numbers N1 , N2 , . . . such that the relation lim FNk (x) = g(x)

k→∞

holds for every point x ∈ [0, 1] of continuity of g, then g(x) is called a distribution function of ω. (FNk )∞ k=1 is said to converge weakly to g. Let G(ω) be the set of all distribution functions of the sequence ω = (xn )∞ n=1 ⊂ [0, 1]. 7. Let us consider the following moment problem Z 1Z 1 F (x, y)dg(x)dg(y) = 0 0

0

in distribution functions g : [0, 1] → [0, 1], where F (x, y) is continuous on the unit square [0, 1]2 . We denote G(F ) as the set of all solutions g of this moment problem. 8. We will specify the following three types of functions. One-jump distribution function cα : [0, 1] → [0, 1], where cα (0) = 0, cα (1) = 1, and ( 0, if x ∈ [0, α), cα (x) = 1, if x ∈ (α, 1] which has a jump of size 1 at α. Constant distribution function hα : [0, 1] → [0, 1], where hα (0) = 0, hα (1) = 1, and hα (x) = α if x ∈ (0, 1). Triangular function fα : [0, 1] → [0, 1], where ( x/α, if x ∈ [0, α), fα (x) = (1 − x)/(1 − α), if x ∈ [α, 1]. 9. Let us state the following definition: The sequence ω = (xn )∞ n=1 ⊂ [0, 1] is said to be uniformly maldistributed in the strict sense if G(ω) = {cα (x); α ∈ [0, 1]}. 1Sometimes these are referred to as the first and second theorem of Helly [6, ?].

UNIFORMLY MALDISTRIBUTED SEQUENCES IN A STRICT SENSE

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The kinship of the sets G(ω) to G(F ) raises the following criteria. III. Results First of all, it is easily seen that 10. The sequence ω is uniformly maldistributed if and only if {cα (x); α ∈ [0, 1]} ⊂ G(ω). Thus, in the theory of uniform maldistribution we need not consider distribution functions other than cα (x). This suggests the definition 9. We shall begin with establishing necessary and sufficient conditions for the set inclusion G(ω) ⊂ G(F ). 11. Proposition. Let F (x, y) be continuous on [0, 1]2 . For any ω = (xn )∞ n=1 ⊂ [0, 1] we have N 1 X F (xm , xn ) = 0. N →∞ N 2 m,n=1

G(ω) ⊂ G(F ) ⇐⇒ lim

Proof. Using the definition of the Riemann-Stieltjes integral, we have Z 1Z 1 N 1 X F (x, y)dFN (x)dFN (y) = 2 F (xm , xn ). N m,n=1 0 0 Suppose that lim FNk (x) = g(x) for all continuity points x of g. Then, applying k→∞

the Helly-Bray lemma, we find Z 1Z 1 Z lim F (x, y)dFNk (x)dFNk (y) = k→∞

0

0

0

1

Z

1

F (x, y)dg(x)dg(y), 0

and the implication ⇐= follows immediately. In order to show the implication =⇒, assume Nk 1 X F (xm , xn ) = β > 0. k→∞ Nk 2 m,n=1

lim

∞ By the Helly selection principle, there exists a subsequence (Nk0 )∞ k=1 of (Nk )k=1 such that lim FNk0 (x) = g(x) ∈ G(ω). Again, by Helly-Bray lemma we find k→∞ R1R1 F (x, y)dg(x)dg(y) = β. We conclude g ∈ / G(F ). ¤ 0 0

It should be remarked that G(ω) and G(F ) are weakly sequentially closed. The closure of G(ω) was shown by van der Corput [5, ?]. Now we have the following result. 12. Theorem. For any sequence ω = (xn )∞ n=1 ⊂ [0, 1] we have N 1 X |xm − xn | = 0. N →∞ N 2 m,n=1

G(ω) ⊂ {cα (x); α ∈ [0, 1]} ⇐⇒ lim

1

Moreover, if G(ω) ⊂ {cα (x); α ∈ [0, 1]}, then G(ω) = {cα (x); α ∈ I}, where I is a closed subinterval of [0, 1] which can be found as " # N N 1 X 1 X I = lim inf xn , lim sup xn , 2 N →∞ N N →∞ N n=1 n=1

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OTO STRAUCH

and the length |I| of I can also be found as M N 1 XX |I| = lim sup |xm − xn |. M,N →∞ M N m=1 n=1

(3)

Proof. First observe that G(|x − y|) = {cα (x); α ∈ [0, 1]}, and (1) follows from 11. Suppose, in order to show the connectivity of I, that lim FMk (x) = cα (x), k→∞

lim FNk (x) = cβ (x), and α < β. If α < γ < β, then we can construct a sequence

k→∞

(Kk )∞ k=1 such that lim FKk (γ) = 1/2. By the Helly selection principle there exists k→∞

a subsequence (Kk0 )∞ k=1 such that lim FKk0 (x) = cγ (x). k→∞

In order to find the length of subinterval of such α, one observes that Z 1Z 1 |x − y|dcα (x)dcβ (y) = |α − β| 0

0

implies (3) in the usual way.

¤

The following assertion is immediately evident from the preceding. 13. The sequence ω = (xn )∞ n=1 ⊂ [0, 1] is uniformly maldistributed in the strict sense if and only if N M N 1 X 1 XX |x − x | = 0 and lim sup |xm − xn | = 1, m n N →∞ N 2 M,N →∞ M N m=1 n=1 m,n=1

lim

or alternatively N N 1 X 1 X lim sup xn − lim inf xn = 1. N →∞ N N →∞ N n=1 n=1

We shall now analyze the criterion 4. function 14. Consider a given sequence ω = (xn )∞ n=1 ⊂ [0, 1].³For every continuous ´∞ PN 1 coincides f : [0, 1] → R the set of limit points of the sequence N n=1 f (xn ) N =1 with ½Z 1 ¾ f (x)dg(x); g ∈ G(ω) . 0

Moreover, this set constitutes a subinterval of [min f, max f ]. Since Z 1 Z max f = f (x)dg(x) ⇐⇒ 1.dg(x) = 0, 0

[0,1]\f −1 (max f )

we can find information about G(ω) whenever max f ∈ Similarly for min f . Using this we can rewrite the criterion 4 in the form

nR 1 0

o f (x)dg(x); g ∈ G(ω) .

{cα (x); α ∈ [0, 1]} ⊂ G(ω) ⇐⇒ ½Z

1

¾ f (x)dg(x); g ∈ G(ω) = [min f, max f ] for every continuous f : [0, 1] → R.

0

Note that we not need consider all continuous f , but only such f with f −1 (max f ) consisting of only one point. This suggests the following result.


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15. The sequence ω = (xn )∞ n=1 ⊂ [0, 1] is uniformly maldistributed if and only if lim sup N →∞

N 1 X fα (xn ) = 1 N n=1

for every α ∈ (0, 1). Here fα are triangular functions defined in 8. IV. Examples 16. Let {x} be the fractional part of x. It can be shown [2, ?] that, for ω = ({log n})∞ n=1 , ½ ¾ ¡ ¢ ex − 1 G(ω) = e−α + e−α emin(x,α) − 1 ; α ∈ [0, 1] . e−1 ∞ Starting with ω = ({log log n})∞ n=2 all the sequences ({log log . . . log n})n=n0 have

G(ω) = {cα (x); α ∈ [0, 1]} ∪ {hα (x); α ∈ [0, 1]} and thus are uniformly maldistributed . Proof. For the first iterated logarithm we chose an index-sequence (Nk )∞ k=1 with Nk = [exp exp(k + α)]. Then we have lim FNk (x) = cα (x). For Nk = [exp exp(k + k→∞

εk )], where εk → 0 such that (exp exp(k+εk ))/(exp exp k) → β, we have lim FNk (x) = k→∞

hα (x), where α = (β − 1)/β. On the other hand, let lim FNn (x) = g(x). Then Nn = exp exp(kn + εn ), n→∞

where kn = [log log Nn ], εn = {log log Nn }, and the sequence (εn )∞ n=1 cannot have different limit points. The same distribution function are of course obtained if we replace log log t by log . . . log t and exp exp t by exp . . . exp t in the above limits. ¤ 17. From [5, ?] follows the fact that any everywhere dense sequence can be rearranged to a sequence ω for which G(ω) is the set of all distribution functions g : [0, 1] → [0, 1]. Application of 10 to ω gives immediately that this ω is uniformly maldistributed . 18. Let ω be defined as ½q ¾¾¶∞ µ½ £q √ £p ¤ ω= 1 + (−1) [ log2 n]] log2 n , n=1

were [x] denotes the integral part and {x} the fractional part of x. Then G(ω) = {cα (x); α ∈ [0, 1]}. Proof. The line of construction of ω is the same as in 3, but we need an additional assumption about the differences yk+1 − yk . More precisely, let ω = (xn )∞ n=1 be the sequence constructed as follows. Assume that we have a dense sequence σ = (yn )∞ n=1 ⊂ [0, 1] with (i) lim (yk+1 − k→∞

yk ) = 0, and an increasing sequence of positive integers Ω = (Mk )∞ k=1 with (ii) Pk−1 lim ( i=1 Mi )/Mk = 0. Now form the sequence ω = (xn )∞ by setting xn = yk if n=1 k→∞ P Pk k−1 cα (x), α ∈ [0, 1], i=1 Mi ≤ n < i=1 Mi . We shall use 13 which gives that ω hasP k−1 as its distribution functions. For a detailed proof, first take N = i=1 Mi + θk Mk , where 0 ≤ θk < 1. It is not difficult to calculate ¶ µ N 1 X Mk−1 θk Mk |yk − yk−1 | . |xm − xn | = O N 2 m,n=1 (Mk−1 + θk Mk )2

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Hence, the limit in 13 follows immediately. In order to compute the lim sup, one Pik Pjk observes, for Nik = i=1 Mi and Njk = i=1 Mi , that M N 1 XX |xm − xn | = M,N →∞ M N m=1 n=1

lim sup

= lim sup k→∞

1 Mik Mjk

Nik X

N jk X

|xm − xn | = lim sup |yik − yjk | = 1 k→∞

m=Nik −1 m=Njk −1

holds if yik → 0 and yjk → 1 as k → ∞. Applying the above construction with ¾¶∞ µ½ √ ©√ ª [ k] σ= 1 + (−1) k k=1

and

³ ´ 2 2 ∞ Ω = 2(k+1) − 2k , k=1

we thus have the desired property of ω.

¤

The following example demonstrates the importance of criterion 13 and may be compared with [3, ?]. 18’. Suppose the continuous function f which map [0, 1] onto itself has bounded derivative |f 0 | ≤ c. Let ω = (xn )∞ n=1 ⊂ [0, 1] be uniformly maldistributed in the strict sense . Then so is (f (xn ))∞ n=1 . Proof. This follows from criterion 13, since it can be easily verified that N N c X 1 X |f (x ) − f (x )| ≤ |xm − xn |, m n N 2 m,n=1 N 2 m,n=1

and

¯ Ã !¯ N N N ¯1 X ¯ 1 X c X ¯ ¯ |xm − xn |. f (x ) − f x ≤ ¯ n n ¯ ¯N ¯ N2 N n=1 m,n=1 n=1 ¤

We now discuss certain computational aspects of the foregoing classes of sequences. 1. V. Applications 19. A quasi-Monte Carlo method for the approximate evaluation of the extreme values of a function was proposed by H. Niederreiter [7]. He shows that the error between the approximate value max f (xn ) and the correct value max f (x) can 1≤n≤N

x∈[0,1]

be estimated in terms of dispersion of ωN = (xn )N n=1 ⊂ [0, 1] and the modulus of continuity of f (cf. [7, ?]). The dispersion of a finite sequence ωN = (xn )N n=1 ⊂ [0, 1] is defined as dN = max min |x − xn |; x∈[0,1] 1≤n≤N

it is a measure of the denseness of a sequence. The dispersion is related to the well-known discrepancy of ωN , which is a measure of the uniform distribution of a sequence (cf.[7, ?]). An analogous relation between uniform maldistribution in the strict sense and denseness of a sequence can be derived on the basis of criterion 13 and the following bound of dispersion.


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19’. For a given finite sequence ωN = (xn )nn=1 ⊂ (0, 1) define the numbers AN and BN by M M 1 X 1 X AN = min xn , BN = max xn . 1≤M ≤N M 1≤M ≤N M n=1 n=1 Assume that these min and max are attained in M1 and M2 , respectively, and denote CN by CN =

M 1 X |xm − xn |. min(M1 ,M2 )≤M ≤max(M1 ,M2 ) M 2 m,n=1

max

Then, dN ≤ max(AN , 1 − BN , 2CN ).

Proof. Let xi < xj be neighbouring points from ωN (i.e. (xi , xj ) ∩ ωN = ∅) and consider the mean (xi + xj )/2. We distinguish two cases. 1o We either have (xi +xj )/2 ∈ [0, AN ]∪[BN , 1] and then xj −xi ≤ 2 max(AN , 1− BN ), or 2o (xi + xj )/2 ∈ (AN , BN ). If we assume that M1 < M2 (the case M2 < M1 is completely similar), then there is an integer M 0 such that M 0 , M 0 + 1 ∈ [M1 , M2 ] and 0 M0 M +1 X 1 X xi + xj 1 x ≤ ≤ xn . n M 0 n=1 2 M 0 + 1 n=1 There are four possibilities for the situation of xi , xj , PM 0 +1 1 n=1 xn (= B), M 0 +1 (i) (ii) (iii) (iv)

1 M0

PM 0 n=1

xn (= A), and

either xi , xj ∈ [A, B], or A ∈ (xi , xj ) and B ∈ / (xi , xj ), or B ∈ (xi , xj ) and A ∈ / (xi , xj ), or A, B ∈ (xi , xj ).

There is a simple relationship between the configurations (i)–(iv) and the bound of xj − xi which we shall establish by using the following propositions. For any finite sequence ωM = (xn )M n=1 ⊂ [0, 1], we have ¯ ¯ M +1 M M +1 ¯ 1 X ¯ X X 1 2 ¯ ¯ xn − xn ¯ ≤ |xm − xn |. 4 ¯ ¯M M + 1 n=1 ¯ (M + 1)2 m,n=1 n=1 Assuming

1 M

PM

Ã

min

xn ∈ (u, v) ⊂ [0, 1] and (u, v) ∩ ωM = ∅, we have ! M M M 1 X 1 X 1 X xn − u, v − xn ≤ 2 |xm − xn |. M n=1 M n=1 M m,n=1 n=1

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It follows from the above inequalities that xj − xi ≤ 2CN for (i), and xj − xi ≤ 4CN for (ii),(iii) and (iv). Since (4) is evident, we shall begin with proving (5). Integration by parts shows that every continuous f with piecewise continuous derivative f 0 ¯ ¯ ¯Z ¯ Z 1 M ¯ 1 X ¯ ¯ 1 ¯ ¯ ¯ ¯ 0 f (xn ) − f (x)dcα (x)¯ = ¯ (FM (x) − cα (x))f (x)dx¯¯ . 6 ¯ ¯M ¯ 0 0 n=1

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Now let us take as f the following function  0,    x − u, f (x) = α−u  (v − x),    v−α 0,

if if if if

x ∈ [0, u), x ∈ [u, α), x ∈ [α, v), x ∈ [v, 1],

where α ∈ (u, v) and α − u = min(α − u, v − α). Then, after the application of Cauchy inequality, we obtain Z 1 (FM (x) − cα (x))2 dx. min(α − u, v − α) ≤ 2 0

By an easy computation, we evaluate Z 1 M M X 1 X 1 (FM (x) − cα (x))2 dx = |xn − α| − |xm − xn |. M n=1 2M 2 m,n=1 0 Setting α =

1 M

PM n=1

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xn we thus conclude (5). ¤ VI. Notes

20. Let ω = (xn )∞ n=1 be a sequence of real numbers. Then ω is said to be statistically convergent to the number α provided that for each ε > 0, 1 lim #{n ≤ N ; |xn − α| ≥ ε} = 0. N →∞ N The definition of statistical convergence was given by H. Fast [1] and I.J. Schoenberg [4], independently. As in Schoenberg [4] we see that 21. The sequence ω = (xn )∞ n=1 ⊂ [0, 1] is statistically convergent to the number α if and only if the sequence ω admits the limiting distribution cα (x). As an illustration of 12 we shall give a characterization of the statistically convergence which does not rely on knowing the statistical limit of the sequence. 22. The sequence ω = (xn )∞ n=1 ⊂ [0, 1] possesses a statistical limit if and only if M N 1 XX |xm − xn | = 0. M,N →∞ M N m=1 n=1

lim

References [1] FAST, H. Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. [2] KUIPERS, L. - NIEDERREITER, H. Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974. [3] Myerson, G. A sampler of recent developments in the distribution of sequences, in: Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991), Lecture Notes in Pure and App. Math. 147 Marcel Dekker, New York, 1993, 163–190. [4] Schoenberg, I. J. The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. [5] Van der Corput, J. G. Verteilungfunktionen I, Proc. Akad. Amsterdam 38 (1935), 813– 821. [6] Shanot, J. A.- Tamarkin, J. D. The Problem of Moments, Mathematical Surveys 1. Amer. Math. Soc. Providence, Rhode Island, 1943. [7] Niederreiter, H. A quasi-Monte Carlo method for the approximate computation of the extreme values of a function, in: Studies in Pure Mathematics (To the Memory of Paul Tur´ an) Akad´ emiai Kiad´ o-Birkh¨ auser Verlag Budapest, Basel, 1983, 523–529.


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ˇ ´ nikova 49; 8814 73 BRATISLAVA; Mathematical Institute; Slovak Academy of Sciences; Stef a Slovakia