uniform distribution theory

Uniform Distribution Theory 2 (2007), no.1, 79–92

uniform distribution theory

REMARKS ON ONE TYPE OF UNIFORM DISTRIBUTION ´ka Milan Paˇ st e ABSTRACT. A sequence {xn } of numbers in [0, 1] is called Buck’s uniformly distributed if for every interval I ⊂ [0, 1] the set {n ∈ N, xn ∈ I} is Buck’s measurable and its Buck’s measure density is equal to the length of I. Some examples of sequences are given. An analogy of Weyl’s criterion is proved.

Communicated by Ladislav Miˇs´ık Dedicated to Professor Robert F. Tichy on the occasion of his 50th birthday

Introduction The notion of uniformly distributed sequence was introduced in 1916 by H. Weyl in his paper [Wey]. Let {xn } be a sequence of real numbers from [0, 1). This sequence is called uniformly distributed if and only if for every subinterval I ⊆ [0, 1] it holds 1 lim card{n ≤ N ; xn ∈ I} = |I|, (1) N →∞ N where |I| is the length of I. If N is the set of positive integers and A ⊆ N then the value (if it exists): 1 d(A) := lim card(A ∩ [0, N ]) N →∞ N is called the asymptotic density of the set A. Thus a sequence {xn }, xn ∈ [0, 1), is uniformly distributed if and only if for every subinterval I ⊆ [0, 1) the set {n ∈ N; xn ∈ I} has the asymptotic density and d({n ∈ N; xn ∈ I}) = |I| .

(2)

2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 11K06, 11B05. K e y w o r d s: Uniform distribution, Buck’s measure density. This research was supported by the VEGA Grant 2/7138/27.

79

ˇ EKA ´ MILAN PAST

The aim of this paper is to study another type of uniform distribution motivated by (2). In the monograph [S-P, p. 6] there is considered the uniform distribution with respect to a given submeasure on a system of some subsets of the set of positive integers. In 1946 R. C. Buck introduced and studied the notion of measure density, motivated from one side by asymptotic density and by measurability in sense of Carath´eodory from other side. Let m ∈ N, a ∈ N ∪ {0}. Denote by mN + a = {a+k.m; k = 0, . . . }, the arithmetic progression with the difference m and prime 1 element a. Clearly mN + a has the asymptotic density and d(mN + a) = m . Let D0 be the class of all sets of the form m1 N + a1 ∪ · · · ∪ ms N + as . Put for S ⊂ N : µ∗ (S) = inf{d(A); S ⊂ A, A ∈ D0 }. Then the value µ∗ (S) is called the measure density of S. Put © ª Dµ = S ⊆ N; µ∗ (S) + µ∗ (N \ S) = 1 . Then Dµ is an algebra of sets and the function µ = µ∗ /Dµ is a finitely–additive probability measure on Dµ . For all above statements on measure density we refer to the Buck’s paper [BUC]. The algebra Dµ has the following properties: (i) D0 ⊂ Dµ . (ii) E ∈ Dµ ⇐⇒ ∀ε > 0 ∃E1 , E2 ∈ Dµ such that E1 ⊂ E ⊂ E2 and µ(E2 ) − µ(E1 ) < ε. (iii) Every set E ∈ Dµ has the asymptotic density and d(E) = µ(E). For (i), (ii), (iii) we refer to [BUC] and [PAS]. Remark that in 1973 M. Parnes proved that the algebra Dµ is very small from the measure theoretical point of view, (see [PA]). Let {yn } be a sequence of numbers from the interval [0, 1). Put for a set L ⊂ [0, 1) ¡ A {yn }, L) = {n ∈ N; yn ∈ L}. (3) ¡ ¢ (Shortly we can write A(L) = A {yn }, L , if {yn } is given.) The sequence {yn } is called Buck’s distributed if and only if for ¡ uniformly ¢ every subinterval I ⊆ [0, 1) the set A {yn }, I belongs to Dµ and ¡ ¡ ¢¢ µ A {yn }, I = |I|, (4) where |I| is the length of I. Instead of Buck’s uniformly distributed we shall write only B.u.d. Remark that (iii) yields that every B.u.d. sequence is uniformly distributed in the sense (2). 80

REMARKS ON ONE TYPE OF UNIFORM DISTRIBUTION

1. Some elementary statements

Proposition 1.

Let {yn } be a sequence of numbers from [0, 1). Then {yn } is B.u.d. if and only if ¡ ¢ ¡ ¡ ¢¢ a) ∀x ∈ (0, 1); A [0, x) ∈ Dµ ∧ µ A [0, x) = x. ¡ ¢ ¡ ¡ ¢¢ b) ∀x ∈ (0, 1); A [0, x] ∈ Dµ ∧ µ A [0, x] = x.

¡ ¢ P ¡r o o f.¢ Suppose that a) ¡ ¢ holds. For x ∈ [0,¡1) and ε > 0 we have A ¡ [0, x) ¢ ⊂ A [0, x] ⊂ A [0, x + ε) . Thus (ii)¡yields A ¢ [0, x]) ∈ Dµ and x ≤ µ A[0, x] ≤ x + ε and so for ε → 0+ we have µ A[0, x] = x. Now for an arbitrary interval I (suppose for instance that I = [a, b]) we have ¡ ¢ ¡ ¢ A(I) = A [0, b] \ A [0, a) ∈ Dµ ¡ ¢ and µ A(I) = b −¡a. Analogously ¢ ¡ ¢we can ¡ prove ¢ b). If we consider that for x ∈ [0, 1) we have A {x} = A [0, x] − A [0, x) we obtain it immediately. ¤ ¡ ¡ ¢¢ Corollary 1. If yn is B.u.d. then, ∀x ∈ [0, 1), we have µ A {x} = 0 (and ¡ ¢ A {x} ∈ Dµ ).

Proposition 2. Let {y¡n } be¢ a sequence of numbers from [0, 1). Then {yn } is B.u.d. if and only if µ∗ A(I) = |I| for every subinterval I ⊂ [0, 1). trivial. Implication P r o o f. One implication ¡ ¡ is ¢¢ ¡ ¡ ¢¢ ⇐= . Consider¡ I =¢ [0, x), J = [x, 1). Then µ∗ A [0, x) = x, µ∗ A [x, 1) = 1 −x and so A [0, x) ∈ Dµ and the assertion follows now from Proposition 1 a). ¤

Proposition 3. Let E ⊂ [0, 1) be a dense set and {yn } a sequence of numbers from [0, 1). The sequence {yn } is B.u.d. if and only if ¡ ¢ ¡ ¡ ¢¢ a) ∀x ∈ E; A [0, x) ∈ Dµ ∧ µ A [0, x) = x. ¡ ¡ ¡ ¢¢ b) ∀x ∈ E; A [0, x]) ∈ Dµ ∧ µ A [0, x] = x. c) For every ¡interval ¢ I ⊂ [0, 1), whose endpoints belong to E, we have A(I) ∈ Dµ and µ A(I) = |I|. P r o o f. Let x0 be an arbitrary number from [0, 1). Then there exist ¡x1 , x2 ∈ ¢ E such that x < x < x and x − x < ε (for ε > 0). And so A [0, x ) ⊆ 1 0 2 2 1 1 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ A [0, x0 ) ¡⊆ A [0,¢x2 ) and A [0, x1 ] ⊆ A [0, x0 ) ⊆ A [0, x2 ] . Thus from (ii) we have A [0, x0 ) ∈ Dµ and the assertion follows in the cases a) and b). c) is trivial. ¤ 81


2. The sequence of van der Corput Let q0 = 1, q1 , q2 , . . . be a sequence of positive integers such that qj > 1 for j > 1. Put Qn = qo . . . qn , n = 0, 1, 2, .... (5) It is well known that every positive integer a can be uniquely expressed in the form a = c0 Q0 + c1 Q1 + · · · + cn Qn , (6) where 0 ≤ cj < qj+1 , j = 0, . . . , n − 1. Similarly every α ∈ [0, 1) can be uniquely expressed in the form ∞ X cn α= , 0 ≤ cn < qn+1 (7) Qn+1 n=0 and cn < qn+1 − 1 for infinitely many n. If a ∈ N has the form (6), put c0 cn γ(a) = + ··· + . (8) Q1 Qn+1 If q1 = q2 = . . . b, b for some b ∈ N, b > 1, we obtain the classical van der Corput sequence with the base b (see [D-T, pp. 41, 368]; [S-P, p. 2–102]). Remark that this sequence plays an important role in the theory of discrepancy.

Theorem 1.

If {γ(k)} is given by (8), then {γ(k)} is B.u.d. sequence.

P r o o f. For the proof we shall use some elementary properties of Cantor’s expansions. Consider the intervals ¶ · j+1 j j , , j = 0, . . . , Qk+1 − 1, k = 0, . . . . Ik+1 = Qk+1 Qk+1 Clearly the endpoints of these intervals form a dense set in [0, 1). Thus Proposition 3 c) guaranties that it suffices to prove ¡ ¡ j ¢¢ 1 µ∗ A Ik+1 = . (9) Qk+1 j To interval Ik+1 we can uniquely associate a finite sequence c0 , . . . , ck , ∞ P j cn 0 ≤ cs < qs+1 , s = 0, . . . , k such that α ∈ Ik+1 if and only if α = Qn+1 , n=0

where c0 = c0 , . . . , ck = ck . Similarly there exists an arithmetic progression r + Qk+1 N such that a ∈ r + Qk+1 N if and only if a = c0 Q0 + · · · + ck Qk + . . . , where c0 = c0 , . . . , ck = ck . Thus it follows j γ(a) ∈ Ik+1 ⇐⇒ a ∈ r + Qk+1 N j and so A(Ik+1 ) = r + Qk+1 N which implies (9).

82

¤


Let α be an irrational number. It is a well known statement that the sequence of fractional parts {{nα}} is uniformly distributed. We prove that this sequence is not B.u.d.

Theorem 2.

Let α be an irrational number. Put yn = {nα}, n = 1, 2, . . . . Then for every x ∈ (0, 1) we have ¡ ¡ ¢¢ ¡ ¡ ¢¢ µ∗ A [0, x) = 1 = µ∗ A [x, 1] .

P r o o f. In 1984 O. Strauch [S] proved that for every proper subinterval I the set A(I) does not contain an infinite arithmetic progression. Therefore the complement N \ A(I) has¢ a nonempty intersection with every arithmetic progression. Thus µ∗ (N \ A(I) = 1 (see [PAS]). For I = [0, x), I = [x, 1] we obtain the assertion. ¤

3. Buck’s measurable sequences Suppose that {xn } is a sequence of numbers from [0, 1). This sequence is called Buck’s ¡ ¢ measurable if and only if for every subinterval I ⊆ [0, 1) the set A {xn }, I belongs to the algebra Dµ . Remark that similar generalization of Weyl’s u.d. sequence was made by I. Schoenberg [Sch] in 1929. Thus {xn } is B.u.d. if it is Buck’s measurable and ¡ ¡ ¢¢ µ A {xn }, I = |I|. In this part we derive some analogies of the well known Weyl’s criterium. We shall use one statement from [PAS]. Let S ⊂ N, B ∈ N. Denote S (mod B) = {s (mod B); s ∈ S} and R(S : B) = |S (mod B)|. Then it holds.

Theorem A.

Let {Bn } be a sequence of positive integers fulfilling the condition ∀ d ∈ N ∃ n0 ∀ n ≥ n0 ; d|Bn .

(c)

Then for every subset S ⊂ N we have µ∗ (S) = lim

n→∞

R(S : Bn ) Bn

(see [PAS]). 83


Proposition 4.

Let {xn } be a sequence of numbers from [0, 1). Then {xn } is Buck’s measurable if and only if for every interval I ⊆ [0, 1) there exists a value C(I) such that Bn ¢ ¡ 1 X χI xk(n) = C(I) n→∞ Bn j j=1

(10)

lim

(n)

(n)

holds for arbitrary system of complete remainders systems Rn = {k1 , . . . , kBn } modulo Bn , n = 1, 2, . . . . Here {Bn } is a sequence from Theorem A and ¡χI (·) ¢is the indicator function of I. In addition in this case there holds C(I) = µ A(I) . ¡ ¢ P r o o f. The value R A(I) : Bn is the maximal number of elements from A(I) incongruent modulo Bn . Thus ¡ ¢ Bn ¡ ¢ R A(I) : Bn 1 X χI xk(n) ≤ j Bn j=1 Bn and so lim sup n→∞

Bn ¢ ¡ ¢ 1 X χI (xk(n) ≤ µ∗ A(I) . j Bn j=1

Suppose now that A(I) ∈ Dµ and lim inf n→∞

Bn ¢ ¡ ¢ 1 X χI (xk(n) < µ∗ A(I) . j Bn j=1

¡ ¢ Then there exists α < µ∗ A(I) such that Bn ¢ 1 Xi ¡ χI xk(ni ) → α , j Bni j=1

i→∞

for suitable subsequence {ni }. But 1=

Bn Bn ¢ ¡ ¢ 1 Xi ¡ 1 Xi χI xk(ni ) + χ¬I xk(ni ) , j j Bni j=1 Bni j=1

where ¬I denotes the complement of I. For i → ∞ we obtain from (11) ¡ ¢ ¡ ¢ ¡ ¢ 1 ≤ α + µ∗ A(¬I) < µ∗ A(I) + µ∗ A(¬I) = 1 – a contradiction. 84

(11)


¡ ¢ Suppose now that (10) holds. Denote q(n) = R A(I) : Bn . Then A(I) (n) (n) contains the numbers k1 , . . . , kq(n) incongruent modulo Bn . These numbers we can complete to a complete remainder system modulo Bn . © (n) (n) ª , kq(n) , . . . , kBn . Rn = k1 , . . . , kq(n) (n) (n)+1 Nearly (10) yields Bn ¡ (n) ¢ q(n) 1 X = χ I kj → C(I) Bn Bn j=1 ¡ ¢ and so C(I) = µ∗ A(I) . If we consider that 1 = χI + χ¬I , then we obtain

¡ ¢ ¡ ¢ 1 = C(I) + C(¬I) = µ∗ A(I) + µ∗ A(¬I) and the assertion follows.

¤

As an immediate consequence we have

¢ The sequence {xn }, xn ∈ [0, 1 is B.u.d. if and only if for every subinterval I ⊂ [0, 1) there holds

Corollary 2.

Bn ¡ ¢ 1 X χI xk(n) = |I|, n→∞ Bn j j=1

lim

using the same notation as in Proposition 4. Corollary 2 implies that {xn } is B.u.d. if and only if Z 1 Bn ¡ ¢ 1 X lim g xk(n) = g(x) dx n→∞ Bn j 0 j=1 P holds for every step function g = ci χIi . i

For every Riemannian integrable function f : [0, 1] → (∞, ∞) and ε > 0 there exist two steps functions g1 , g2 that Z g1 < f < g2 ,

1¡

¢ g2 (x) − g1 (x) dx < ε .

(12)

0

Thus we get

Theorem 3.

and only if

Suppose the notation from Proposition 4. Then {xn } is B.u.d. if Z 1 Bn ¡ ¢ 1 X lim f (x) dx f xk(n) = n→∞ Bn j 0 j=1

(13) 85


for arbitrary Riemannian integrable function f : [0, 1] → (−∞, ∞). Every Riemannian integrable function can be approximated in the sense (12) by two continuous functions g1 , g2 and so we get

Theorem 3'.

Suppose the notation from Proposition 4. Then {xn } is B.u.d. if

and only if

Z 1 Bn ¢ ¡ 1 X f (x) dx f xk(n) = n→∞ Bn j 0 j=1 lim

for every continuous function f : [0, 1] → (−∞, ∞). Every continuous function can be uniformly approximated by trigonometric polynomials and so we have analogy to Weyl’s criterion:

Theorem 4.

Suppose the notation from Proposition 4. Then {xn } is B.u.d. if

and only if

Bn 1 X exp 2π i hxk(n) = 0 n→∞ Bn j j=0

lim

for every h ∈ Z, h 6= 0. This yields

Corollary 3. If {xn } is B.u.d., then the sequence of fractional parts {{kxn }} is B.u.d. for k ∈ Z, k 6= 0. A sequence {xn } is called polyadically continuous if and only if ∀ε > 0 there exists B ∈ N such that n ≡ m (mod B) =⇒ |xn − xm | < ε.

Theorem 5.

(14)

Every u.d. sequence which is polyadically continuous is B.u.d.

P r o o f. Let {xn } be a u.d. sequence. Then for every continuous function f there holds Z 1 N 1 X ¡ lim f xn ) = f (x) dx. N →∞ N 0 n=1 Suppose that a sequence {Bn } fulfils the condition (c) of Theorem A. Clearly Z 1 Bn ¡ 1 X f (x) dx = lim f xj ). n→∞ Bn j=1 0 (n)

(n)

If Rn = {k1 , . . . , kBn } is a complete remainder system modulo Bn , we can sup(n)

pose that j ≡ kj (mod Bn ), j = 1, . . . , Bn . If {xn } is polyadically continuous, 86


then {f (xn )} is also polyadically continuous, because f is uniformly continuous. Let ε > 0 and B be from (14). Then from (c) we have B|Bn , n ≥ n0 for suitable ¢ n0 , hence |f (xj ) − f (xk(n) | < ε. And so j

¯ ¯ Bn Bn ¯ 1 X ¢¯ 1 X ¯ f (xj ) − f (xk(n) ¯¯< ε ¯ Bn j B n j=1 j=1 for n ≥ n0 . For n → ∞ we have ¯Z 1 ¯ Bn ¯ ¢¯ 1 X ¯ ¯≤ε f (x) dx − lim sup f (x (n) ¯ ¯ kj n→∞ Bn 0 j=1 ¯Z ¯ ¯ ¯

0

1

¯ Bn ¢¯ 1 X f (xk(n) ¯¯ ≤ ε f (x) dx − lim inf n→∞ Bn j j=1

and the assertion follows.

¤

If we consider the sequence of van der Corput {γ(a)} given by (8), then for a1 ≡ ¯ a2 (mod Qn )¯ we have that a1 , a2 have the same first n digits in (6) and so ¯γ(a1 ) − γ(a2 )¯ ≤ Q−1 n . Therefore {γ(a)} is an example of polyadically continuous sequence. If {xn } is a Buck’s measurable sequence, then the function ¡ ¡ ¢ g(x) = µ A {xn }, [0, x)) (15) is called the Buck’s distribution function of {xn }. Remark that for the asymptotic density in 1929 ¡Schoenberg introduced the notion of asymptotic distribution ¡ ¢ function as g(x) = d A {xn }, [0, x) (see [Sch], [K-N], [D-T], [S-P]). Let g : [0, 1] → [0, 1], g(0) = 0, g(1) = 1 be an increasing and continuous function. Then g −1 : [0, 1] → [0, 1] is also increasing ¡ and¢ continuous. Consider the van der Corput sequence {γ(n)}. Put xn = g −1 γ(n) , n = 1, 2, . . . . Then {xn } is polyadically continuous, because {γ(n)} is polyadically continuous. Moreover, for x ∈ [0, 1) we have xn < x ⇐⇒ γ(n) < g(x), thus g and {xn } fulfil (15). We proved

Proposition 5.

If g : [0, 1] → [0, 1], g(0) = 0, g(1) ¡ = ¢1 is an increasing and continuous function, then {xn }, where xn = g −1 γ(n) , is a polyadically continuous sequence with Buck’s distribution function g. For the case of the asymptotic density there hold stronger results (see [K-N], [D-T]). 87


Proposition 4 yields that if {xn } is Buck’s measurable, then Z 1 Bn ¡ ¢ 1 X lim h xk(n) = h(x) dg(x) n→∞ Bn j 0 j=1 ¡ ¢ for every step function h, where g(x) = µ A[0, x) . As every continuous function f can be approximated in the sense (12) (instead of Riemann integral we consider ¡ ¢ Riemann-Stieltjes integral with respect to g(x) = µ A([0, x)) we obtain

Theorem 6.

Suppose the notation from Proposition 4. a)If {xn } is Buck’s measurable then, for every continuous function f : [0, 1] → (−∞, ∞) we have Z 1 Bn ¢ ¡ 1 X lim f xk(n) = f (x) dg(x), (e) n→∞ Bn j 0 j=1 where g(x) is the Buck’s distribution function of {xn }. b) Let g(x) be a continuous, non decreasing function defined on [0, 1] and g(0) = 0, g(1) = 1. Then g(x) is the Buck’s distribution function of {xn } if and only if (e) holds. Part b) follows from the fact that in the case that g(x) is continuous, every step function can be approximated by two continuous functions in the sense (12) (See [K-N, p. 54], proof of the criterium for asymptotic distribution functions). Analogously to Theorem 5, we can prove

Corollary 4.

If {xn } is a polyadically continuous sequence with continuous asymptotic distribution function, then {xn } is Buck’s measurable. Now we can construct Buck’s measurable sequences, using some results about multiplicative functions. Suppose that P to a prime number p a positive value ap is associated such that the series ap converges. Then the sequence xn = Q (1 − a ) is a sequence with a continuous asymptotic distribution function p p|n (see [E, p. 185]), and it can be easily proved that {xn } is polyadically continuous. Thus Corollary 4 yields that {xn } is Buck’s measurable.

4. Some metric properties Denote by B the set of all B.u.d. sequences and by U the set of all u.d. sequences. The property (iii) yields that B ⊂ U .

Theorem 7. 88

The set B is closed with respect to the supremum metric.


P r o o f. Let {xn } ∈ B - closure of B in the supremum metric. Then for ε > 0 there exists {yn } ∈ B such that |xn − yn | < ε, n = 1, 2, . . . . If Rn is the system from Proposition 4, we have ¯ ¯ Bn Bn ¯ 1 X ¯ 1 X ¯ ¯ exp 2π i hxk(n) − exp 2π i hyk(n) ¯ ¯ j j ¯ ¯ Bn B n j=1 j=1 ≤

Bn ¯ ¯ 1 X | . . . | ≤ 2πh ¯xk(n) − yk(n) ¯ ≤ 2πhε . j j Bn j=1

For ε → 0+ we get, using Theorem 4, that Bn 1 X exp 2π i hxk(n) = 0 , j Bn j=1

h 6= 0, h ∈ Z,

and the assertion follows.

¤

It is a well known fact that λ∞ (U ) = 1, where λ∞ is a product measure on [0, 1)∞ (see [K-N]). Let A ⊂ N. If there exists limN →∞ N1 card(A ∩ [k, k + N ]) uniformly with respect to k ∈ N then its value is called the uniform density of A (= u(A)). (cf. [GLS, Theorem 1.1, Theorem¡1.2]). A sequence {yn }, yn ∈ [0, 1) is called well distributed if and only if u(A {yn }, I)) = |I| for every interval I ⊂ [0, 1] (cf. [K-N, p. 51]). Every set A ∈ Dµ has a uniform density u(A) and u(A) = µ(A). Thus λ∞ (B) = 0. Thus every B.u.d. sequence is well distributed. The product measure of the set of well distributed sequences is 0 (see [K-N, p. 201, Theorem 3.8]). Theorem 2 gives also the example of sequence which is well distributed but no B.u.d. (cf. [K-N, p. 42, Example 5.2]).

5. The theorem of John von Neumann: John von Neumann proved that every dense sequence in (0, 1) can be rearranged to a u.d. sequence. We prove this result for B.u.d. sequence. Directly from Theorem 4 we obtain:

Proposition 6. Let {yn } be a B.u.d. sequence and |xn − yn | → 0 as n → ∞. Then {xn } is B.u.d. Proposition 7.

Let {yn } be a B.u.d. sequence and {xn } be such that xn = yn , for n ∈ A, where A ∈ Dµ and µ(A) = 1. Then {xn } is B.u.d. 89


P r o o f. We have ¯ ¯ Bn Bn ¯ 1 X ¯ 1 X ¯ ¯ exp 2π i hyk(n) − exp 2πh i xk(n) ¯ ¯ j j ¯ Bn ¯ B n j=1 j=1 ≤

1 Bn

X ¯ ¯exp 2π i hx j≤Bn (n) k ∈¬A j

(n)

kj

¯ − exp 2π i hyk(n) ¯ j

2 R(¬A : Bn ) . Bn But µ(¬A) = 1 − µ(A) = 0, so we have ≤

2 R(¬A : Bn ) = 0 Bn and the assertion follows from Theorem 4. lim

n→∞

Theorem 8.

¤

Every sequence {xn } dense in [0, 1) can be rearranged to a B.u.d.

sequence.

P r o o f. Consider the sequence {γ(n)} of van der Corput. This is a B.u.d. sequence (see ¯ ¯ Theorem 1). As {xn } is dense, for n = 1, 2 . . . there exists kn that ¯xkn − γ(n)¯ < 1 . And so Proposition 5 yields that {xkn } is a B.u.d. Let {xk0 } n n be the sequence of elements of {xn } which do not occur in {xkn }. If {xkn0 } is finite, the proof is finished. Let {xkn0 } be infinite. Let A ⊆ N be such infinite set that µ(A) = 0. For instance A = {n!; n = 1, 2, . . . }. Then B = N \ A ∈ Dµ and µ(B) = 1. Define the sequence {zn } as follows: zn = xkn ;

n ∈ B.

Let A = {s1 < s2 < . . . }, then zs2j = xkj0 ,

j = 1, 2, . . .

zs2j+1 = xksj ,

j = 1, 2, . . .

Then {zn } is a rearrangement of {xn } and Proposition 6 implies that {zn } is B.u.d. ¤

6. The B.u.d. preserving mapping In the paper [PSSa] the mapping g : [0, 1] → [0, 1], such that {g(yn )} is u.d., if {yn } u.d. is studied. Such mapping is called u.d. preserving. In this paper 90


it is proved, that if g is Riemannian integrable, then g is u.d. preserving if and only if Z 1 Z 1 ¡ ¢ h g(x) dx = h(x) dx (16) 0

0

holds for every continuous function h : [0, 1] → (−∞, ∞). A mapping g : [0, 1] → [0, 1] is called B.u.d. preserving if for every B.u.d. sequence {yn } is {g(yn )} B.u.d.

Theorem 9. A Riemannian integrable function g : [0, 1] → [0, 1] is B.u.d. preserving if and only if (16) holds for every continuous function h : [0, 1] → (−∞, ∞). P r o o f. In the third part we proved that {yn } is B.u.d. if and only if Z 1 Bn X ¡ lim h yk(n) ) = h(x) dx n→∞

j=1

j

0

(see the notation of Proposition 4). If we consider that both sequences {yn }, {g(yn )} are B.u.d., then using the standard procedure from [PSSa] we obtain the assertion. ¤

Corollary 5. If g : [0, 1] → [0, 1] is a continuous function, fulfilling (16) then the sequence {g(γ(n))} is a polyadically continuous B.u.d. sequence. REFERENCES [BUC] BUCK, R.C.: The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560–580. [D-T] DRMOTA, M.–TICHY, R.F.: Sequences, Discrepancies and Applications, Springer, Berlin, Heidelberg, 1997. [E] ELLIOTT, P.D.T.A. : Probabilistic Number Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1979. ´ ´ Z.–LAZSL ´ ´ B.–SAL ˇ AT, ´ [GLS] GALIKOV A, O, T.: Remarks on Uniform Density of Integers, Acta Acad. Paed. Agriensis Sectio Mathematicae 29 (2002), 3–13. [K-N] KUIPERS, L.–NIEDERREITER, H.: Uniform Distribution of Sequences, John Wiley and Sons, London, Sydney, Toronto, New York, 1974. [PA] PARNES, M.: On the measure of measurable sets of integers, Acta Arith. XXV (1973), 51–54. ˇ EKA, ´ [PAS] PAST M.: Some properties of Buck’s measure density, Math. Slovaca 43 (1992), no. 1, 15–32. ´ S.–STRAUCH, ˇ ˇ AT, ´ [PSSa] PORUBSKY, O.–SAL T.: Transformations that preserve uniform distributions, Acta Arith. 43 (1988), 459–479. ¨ [Sch] SCHOENBERG, I.: Uber die asymptotische Verteilung reeller Zahlen mod 1., Math. Z. 28 (1928), 171–199.

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ˇ EKA ´ MILAN PAST ´ S.: ˇ Distribution of Sequences: A Sampler, Peter Lang, STRAUCH, O.–PORUBSKY, SAV, Frankfurt am Main, 2005. [S] STRAUCH, O.: Two properties of the sequence nα (mod 1), Acta Math. Univ. Comenianae 44–45 (1984), 65–73. ¨ [Wey] WEYL, H.: Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352. [S-P]

Received March 3, 2006 Revised May 21, 2007 Accepted

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Milan Paˇ st´ eka Mathematical Institute Slovak Academy of Sciences ˇ anikova 49 Stef´ Sk-841 73 Bratislava SLOVAKIA E-mail: [email protected]