FACULTY OF NATURAL SCIENCES
CONSTANTINE THE PHILOSOPHER UNIVERSITY NITRA
ACTA MATHEMATICA 12
REMARKS ON DISTRIBUTION FUNCTIONS OF CERTAIN
BLOCK SEQUENCES
JÓZSEF BUKOR
ABSTRACT. In this paper we study the sequence
integers for which
xn 1  xn  2.
functions of the sequence of blocks
x1  x2  ...  xn  ...
In this case we prove the set
x x
x 
X n   1 , 2 ,..., n 
xn 
 xn xn
of positive
G( X n ) of distribution
can be infinite.
Introduction
Denote by  and   the set of all positive integers and positive real numbers,
respectively. Let X  x1, x2 ,...´where xn  xn 1 are positive integers. Denote by
R( X )  x / y; x, y  X  the ratio set of X and say that a set X is (R ) -dense if R( X ) is
topologically dense in the set   . The concept of (R ) -density of positive integers was
defined and first studied by Šalát in papers [11] and [12]. The density of R( X ) is
equivalent to the everywhere density in [0,1] of the sequence
x1 x1 x2 x1 x2 x3
x x
x
, , , , , ,..., 1 , 2 ,..., n ,...
x1 x2 x2 x3 x3 x3
xn xn
xn
derived from X and it is composed by blocks
x x
x 
n  1,2,...
X n   1 , 2 ,..., n ,
xn 
 xn xn
and X n is called the n -th block. Let D( X n ) be the maximum distance between two
consecutive terms in the n -th block. The properties of the following characteristics, called
the dispersion of the sequence X
D ( X )  lim inf n  D( X n ) .
Its relations to the (R)-density and its further properties were studied in several papers [1],
[2], [4], [14] by Filip, Mišík, Tóth, Csiba and the author.
If the distribution functions of X n are increasing, then the set X is (R ) -dense. This was a
motivation for the study of G( X n ) , the set of all distribution functions of X n , cf. [9], [10].
The case, when the set of all distribution functions of X n contains c0 , the greatest
possible distribution function was studied in [5].
Grekos and Strauch proposed as an open question to prove (or disprove) that
xn 1
 1 implies that G( X n ) is a singleton, see [6, p 76, Q.2] or in [Problem 1.9.2,
xn
Supported by VEGA grant no. 1/4006/07
JÓZSEF BUKOR
Unsolved Problem section (eds. O. Strauch and R. Nair), placed on the home page
http://udt.mat.savba.sk of the journal Uniform Distribution Theory].
This open question was solved in [3]. In this short note we give a simpler and more general
counterexample showing that in general xn 1  xn  2 does not imply that G( X n ) is a
singleton.
Definitions
In the follows we use standard notations and definitions from [8].
 By distribution function we mean any function g : [0,1]  [0,1] such
that g (0)  0, g (1)  1 and g is nondecreasing in [0,1].

For the block sequence X n define the counting function


x
A( X n , x) # i  n : i  x
xn


and step distribution function
F ( X n , x) 
A( X n , x)
n
for x  [0,1) and F ( X n ,1)  1.

Denote G( X n ) the set of all distribution functions g (x ) for which there exists an
increasing sequence of indices nk , k  1,2,... such that F ( X nk , x)  g ( x) for k  
for all points x  [0,1] of continuity of g (x ), i.e. this is equivalent to weak convergence.

For a singleton G( X n )  g ( x), the distribution function g (x ) is also called
asymptotic distribution function of X n .

On the set of all distribution functions the L2 metric is defined by
1/ 2
1

 ( g1 , g 2 )    ( g1 ( x)  g 2 ( x)) 2 dx  .
0

It is known that G ( X n ) nonempty, closed [15], but it is not connected in general [6].
Counterexample
Let m1  m2  ... be an increasing integer sequence with the property
mk
 0 for
mk 1
k   . Let E  2,4,6,8,10,... (the set of all even positive integers) and define the set

X  E  [m2 k , m2 k 1 )   .
k 1
Then xn 1  xn  2 (therefore clearly
xn 1
 1 ) and the set of all distribution
xn
functions of the sequence of blocks X n is G X n   G1  G2 , where G1 consists of the
functions
REMARKS ON DISTRIBUTION FUNCTIONS ...
t
1
x for x 
and
2t  1
t
and G2 consists of the functions
2t
1
g 2 ( x) 
x for x 
and
t
t 1
where t  1 is an arbitrary parameter.
g1 ( x) 
g1 ( x) 
2t.x  1
1
for  x  1
2t  1
t
g 2 ( x) 
t.x  1
1
for  x  1
t 1
t
Proof. First, we consider the case xn [m2 k , m2 k 1 ) . Write xn in the form xn  t.m2 k
m
for some t  1. Note, n  2 k  (t  1).m2 k for k   ( n is depending on k , for
2
simplicity we omit the indexes in n k ). We distinguish two subcases:
m
1
a.) x  2 k  .
xn
t
We write the counting function A( X n , x) as

A( X n , x) # i; xi  m2 k 1,

 
xi
 x# i; m2 k 1  xi  m2 k ,
xn
 
Therefore
F ( X n , x)  o(1) 

xi
 x.
xn

#  j; m2 k 1  2 j  x.xn 
n
and we have
x.xn  m2 k 1
x x
x
t.m2 k
t
2
lim F ( X n , x)  lim
 lim . n  lim .

.x .
n 
n 
n  2 n
k  2 m
n
2
t

1
2k
 (t  1).m2 k
2
m
1
b.) x  2 k  .
xn
t
x
Similarly, we count we count the number of xi for which i  x piecewise in the
xn
intervals (0, m2 k 1 ), [ m2 k 1, m2 k ) and [ m2 k , m2 k 1 ) . In this case

A( X n , x) # i; xi  m2 k 1,

 
xi
 x# i; m2 k 1  xi  m2 k ,
xn
 

xi
 x 
xn


# i; m2 k  xi  m2 k 1,

= o(n)  #  j; m2 k 1  2 j  m2 k  + #  j; m2 k  j  x.xn .

xi
 x 
xn

JÓZSEF BUKOR
In order to find the distribution function we have to calculate the limit
m2 k  m2 k 1
x.xn  m2 k
2
lim F ( X n , x)  0  lim
 lim
.
n 
n 
n 
n
n
Substituting xn  t.m2 k into the previous limit and after some tedious calculation we
2t.x  1
lim F ( X n , x) 
.
obtain
n 
2t  1
The case xn [m2 k 1, m2 k  2 ) can be solved by analogous way. In the same manner we can
find asymptotic distribution functions in the set G2 . The details are left to the reader.
Open problems
Let M ( a, b) stands for certain type of means of positive real numbers a, b . The
infinite set X  x1, x2 ,... is said to be of type M iff xn  M ( xn 1 , xn 1 ) for each natural
number n  2, see [12]. It is easy to see, if the set X is defined by the power mean of
1
 x p  xnp1  p
f ( xn 1 )  f ( xn 1 )
 (n  2) then f ( xn ) 
parameter p , i.e. xn   n 1
2
2


n  2,3,... holds for the function f ( x)  x p . Further, f ( x1 )  f ( x2 ) and
f ( xn )  f ( x1 )  (n  1).(( f ( x2 )  f ( x1 )) ,
n  1,2,3,...
for
(see [12])
Using this fact, it is not hard to show that if X is defined by the power mean of parameter
p , then


x
# i  n : i  x 
xn
  xp
F ( X n , x)  
n
for
n  .
It means that the set of all distribution functions of the related block sequence
(derived from X ) is a singleton.
There are other well-known types of means of positive numbers. Let a, b be positive real
numbers. The identric mean I (a, b)  (1 / e)(bb / a a )1 /(b  a ) , for a  b , I ( a, a )  a; while
REMARKS ON DISTRIBUTION FUNCTIONS ...
the logarithmic mean L(a, b)  (b  a ) /(log b  log a ), for a  b , L(a, a)  a (see, e.g.
[7]). The problem is, what can we say about the behavior of the distribution functions of
the block sequence X n , if the related set X was defined using the identric mean or
logarithmic mean.
REFERENCES
[1]
Bukor, J. – Csiba, P.: On estimations of dispersion of ratio block sequences, Math.
Slovaca, 59 (2009), 283-290.
[2]
Filip, F. – Tóth, J. T.: On estimations of dispersion of certain dense block sequences,
Tatra Mt. Math. Publ., 31 (2005), 65-74.
[3]
Filip, F. – Mišík, L. - Tóth, J. T.: On distribution functions of certain block
sequences, Uniform Distribution Theory 2 (2007), 115-126.
[4]
Filip, F. – Mišík, L. - Tóth, J. T.: Dispersion of ratio block sequences and
asymptotic density, Acta Arith. 131 (2008), 183-191.
[5]
Filip, F. – Mišík, L. - Tóth, J. T.: On ratio block sequences with extreme
distribution function, Math. Slovaca 59 (2009), 275-282.
[6]
Grekos, G. - Strauch, O.: Distribution functions of ratio sequences, II, Uniform
Distribution Theory 2 (2007), 53-77.
[7]
Sándor, J: On the identric and logarithmic means, Aequationes Mathematicae 40
(1990), 261-270.
[8]
Strauch, O. – Porubský, Š.: Distribution of sequences: A Sampler, Peter Lang,
Frankfurt am Main, 2005.
[9]
Strauch, O. - Tóth, J. T.: Asymptotic density of A  N and density of ratio set R(A),
Acta Arith. 87 (1998), 67-78.
[10] Strauch, O. - Tóth, J. T.: Distribution function of ratio sequences, Publ. Math.
Debrecen 58 (2001), 751-778.
[11] Šalát, T.: On ratio set of natural numbers, Acta Arith. 15 (1969), 273-278,
Corrigendum: Acta Arith. 16 (1969), 103.
[12] Šalát, T. – Bukor, J. – Tóth, J. – Zsilinszky, L.: Means of positive numbers and
certain types of series, Acta Mathematica et Informatica, Nitra 1 (1992), 49-57.
[13] Šalát, T.: Quotientbasen und (R)-dichte Mengen, Acta Arith. 19 (1971), 63-78.
[14] Tóth, J. T. – Mišík, L. – Filip, F.: On some properties of dispersion of block
sequences of positive integers, Math. Slovaca 54 (2004), 453-464.
[15] Winkler, R.: On the distribution behaviour sequences, Math. Nachr. 186 (1997),
303-312.
JÓZSEF BUKOR
RNDr. József Bukor, PhD.
Katedra hospodárskej matematiky
Ekonomická fakulta
Univerzita J. Selyeho
P.O.Box 54
SK – 945 01 Komárno
e-mail: bukorj@selyeuni.sk