On the best estimations for dispersions of special ratio block sequences

Abstract

Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík and F. Filip [Math. Slovaca 54, 453–464 (2004; Zbl 1108.11017)]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence.

Full text

Annales Mathematicae et Informaticae
37 (2010) pp. 85–93
http://ami.ektf.hu
On the best estimations for dispersions of
special ratio block sequences∗
Ferdinánd Filipa, Kálmán Liptaib
Ferenc Mátyásb, János T. Tótha
aDepartment of Mathematics, J. Selye University
bInstitute of Mathematics and Informatics, Eszterházy Károly College
Submitted 4 October 2010; Accepted 24 November 2010
Dedicated to professor Béla Pelle on his 80th birthday
Abstract
Properties of dispersion of block sequences were investigated by J. T. Tóth,
L. Mišík, F. Filip [20]. The present paper is a continuation of the study of
relations between the density of the block sequence and so called dispersion
of the block sequence.
Keywords: dispersion, block sequence, (R)-density.
MSC: Primary 11B05.
1. Introduction
In this part we recall some basic definitions. Denote by Nand R+the set of all
positive integers and positive real numbers, respectively. For X⊂Nlet X(n) =
#{x∈X;x6n}. In the whole paper we will assume that Xis infinite. Denote by
R(X) = {x
y;x∈X, y ∈X}the ratio set of Xand say that a set Xis (R)-dense
if R(X)is (topologically) dense in the set R+. Let us notice that the concept of
(R)-density was defined and first studied in papers [17] and [18].
Now let X={x1, x2,...}where xn< xn+1 are positive integers. The sequence
x1
x1
,x1
x2
,x2
x2
,x1
x3
,x2
x3
,x3
x3
, . . . , x1
xn
,x2
xn
,..., xn
xn
,... (1.1)
∗Supported by grants APVV SK-HU-0009-08 and VEGA Grant no. 1/0753/10.
85

86 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
of finite sequences derived from Xis called ratio block sequence of the set X. Thus
the block sequence is formed by blocks X1, X2,...,Xn,...where
Xn=x1
xn
,x2
xn
,..., xn
xn;n= 1,2,....
This kind of block sequences were studied in papers, [1] , [3] , [4] , [16] and [20]. Also
other kinds of block sequences were studied by several authors, see [2], [6], [8], [12]
and [19]. Let Y= (yn)be an increasing sequence of positive integers. A sequence
of blocks of type
Yn=1
yn
,2
yn
,..., yn
yn
was invetigated in [11] which extends a result of [5]. Authors obtained a complete
theory for the uniform distribution of the related block sequence (Yn).
For every n∈Nlet
D(Xn) = max x1
xn
,x2−x1
xn
,..., xi+1 −xi
xn
,..., xn−xn−1
xn,
the maximum distance between two consecutive terms in the n-th block.
In this paper we will consider the characteristics (see [20])
D(X) = lim inf
n→∞ D(Xn),
called the dispersion of the block sequence (1.1) derived from X, and its relations
to the previously mentioned asymptotic density of the original set X.
At the end of this section, let us mention the concept of a dispersion of a general
sequence of numbers in the interval h0,1i. Let (xn)∞
n=0 be a sequence in h0,1i. For
every N∈Nlet xi16xi26... 6xiNbe reordering of its first Nterms into a
nondecreasing sequence and denote
dN=1
2max max{xij+1 −xij;j= 1,2,...N −1}, xi1,1−xiN
the dispersion of the finite sequence x0, x1, x2,...xN. Properties of this concept
can be found for example in [10] where it is also proved that
lim sup
N→∞
NdN>1
log 4
holds for every one-to-one infinite sequence xn∈ h0,1i. Also notice that the density
of the whole sequence (xn)∞
n=0 is equivalent to lim
N→∞ dN= 0. Also notice that the
analogy of this property for the concept of dispersion of block sequences defined in
the present paper does not hold.
Much more on these and related topics can be found in monograph [13].

On the best estimations for dispersions of special ratio block sequences 87
2. Results
When calculating the value D(X), the following theorems are often useful (See [20],
Theorem 1, Corollary 1, respectively).
(A1) Let
X={x1, x2,...}=∞
[
n=1
(cn, dni ∩ N,
where xn< xn+1 and let cn< dn< cn+1, for n∈N, be positive integers. Then
D(X) = lim inf
n→∞
max{ci+1 −di:i= 1,2,...,n}
dn+1
.
(A2) Let Xbe identical to the form of Xin (A1). Suppose that there exists a
positive integer n0such that for all integers n > n0
cn+1 −dn6cn+2 −dn+1 .
Then
D(X) = lim inf
n→∞
cn+1 −dn
dn+1
.
The basic properties of the dispersion D(X)and the relations between dispersion
and (R)-density are investigated in the paper [TMF]. The next theorem states the
upper bound for dispersions D(X)of (R)-dense sets where 16a= lim
n→∞
dn
cn<∞
(See [20], Theorem 10).
(A3) Let X=S∞
n=1 cn, dnE∩Nbe an (R)-dense set where cn< dn< cn+1 for all
n∈Nand suppose that the limit lim
n→∞
dn
cn=aexists. Then
D(X)6min 1
a+ 1 ,max a−1
a2,1
a2,
more precisely,
D(X)6




1
1+aif a∈ h1,1+√5
2)
1
a2if a∈ h1+√5
2,2)
a−1
a2if a∈ h2,∞).
The following theorem shows that in the third case (if a>2), that the dispersion
D(X)can be any number in the interval 0,a−1
a2, where X=S∞
n=1 cn, dnE∩Nis
(R)-dense and lim
n→∞
dn
cn=a. Thus the upper bound for D(X)is the best possible
in the case a>2(See [4], Theorem 2).

88 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
(A4) Let a>1be a real number and kbe an arbitrary natural number. Then for
every α∈ h0,ak−1
a2kithere exists an (R)-dense set
X=∞
[
n=1
(cn, dni ∩ N
where cn< dn< cn+1 are positive integers for every n∈N, such that lim
n→∞
dn
cn=a
and D(X) = α.
In this paper we prove that in the second case if a∈D1+√5
2,2, the dispersion
D(X)can be any number in the interval 0,1
a2, where X=∞
S
n=1 cn, dnE∩Nis
(R)-dense and lim
n→∞
dn
cn=a. Thus the upper bound for D(X)is the best possible
in the case a∈D1+√5
2,2. The following lemma will be useful.
Lemma 2.1. Let the set
M(X) = {n∈N:cn+1 −dn= max{ci+1 −di:i= 1,2,...,n}} =
={m1< m2<···< mk< . . . }
be infinite. Then
D(X) = lim inf
k→∞
cmk+1 −dmk
dmk+1
.
Proof. Let n∈Nbe an arbitrary integer such that n>m1. Then there is unique
k∈Nwith mk6n < mk+1. >From the definition of the set M(X)we obtain
max{ci+1 −di:i= 1,2,...,n}
dn+1
=cmk+1 −dmk
dn+1
>cmk+1 −dmk
dmk+1
.
Then obviously
D(X) = lim inf
n→∞
max{ci+1 −di:i= 1,2,...,n}
dn+1
>lim inf
k→∞
cmk+1 −dmk
dmk+1
.
On the other hand, the sequence cmk+1 −dmk
dmk+1 ∞
k=1 is a subsequence of the sequence
max{ci+1−di:i=1,2,...,n}
dn+1 n∈N, hence
D(X) = lim inf
n→∞
max{ci+1 −di:i= 1,2,...,n}
dn+1
6lim inf
k→∞
cmk+1 −dmk
dmk+1
.
The last two inequalities imply
D(X) = lim inf
k→∞
cmk+1 −dmk
dmk+1
.


On the best estimations for dispersions of special ratio block sequences 89
Theorem 2.2. Let a∈1+√5
2,2be an arbitrary real number. Then for every
α∈ h0,1
a2ithere is an (R)-dense set
X=∞
[
n=1
(cn, dni ∩ N,
where cn< dn< cn+1 are positive integers for every n∈Nsuch that lim
n→∞
dn
cn=a
and D(X) = α.
Proof. Let a∈ h 1+√5
2,2). According to (A4), it is sufficient to prove Theorem
2.2 for a−1
a2< α 61
a2. Define function f(b) = b−1
ab .Clearly fis continuous and
increasing on the interval ha, ∞). Moreover
f(a) = a−1
a2and f(a2) = a2−1
a3.
We have a2−1
a3>1
a2if a>1+√5
2. Thus there exists a real number a < b 6a2such
that b−1
ab =α.
Define a set X⊂Nby
X=∞
[
n=1 An∪Bn∩N,
where for every n∈N
An= (an,1, bn,1i ∪ (an,2, bn,2iaBn=
n
[
k=1
(cn,k, dn,k i.
Put a1,1= 1 and for every n∈Nand k= 2,3,...,n
bn,1= [aan,1] + 1, an,2=bn,1+ 1, bn,2= [aan,2] + 1 ,
cn,1= [bbn,2] + 1, dn,1= [acn1] + 1, cn,k = [bdn,k−1] + 1, dn,k = [acn,k ] + 1 ,
and an+1,1= (n+ 1)dn,n.
Obviously for every n∈N
a < bn,1
an,1
6a+1
an,1
and a < bn,1
an,1
6a+1
an,1
,
and for k= 1,2,...,n
a < dn,k
cn,k
6a+1
an,1
.
First we prove that D(X) = α. We have the following inequalities:
cn+1,1−bn+1,2>bbn+1,2−bn+1,2>(b−1)bn+1,2>(b−1)a2an+1,1>
>(a−1)a2an+1,1>aan+1,1> an+1,1> an+1,1−dn,n

90 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
The inequality a2(a−1) >afollows from a>1+√5
2. Then
cn,2−dn,1>bdn,1−dn,1= (b−1)dn,1>(b−1)acn,1>(a−1)acn,1>cn,1> cn,1−bn,2
and for every k= 2,3,...,n−1
cn,k+1 −dn,k >bdn,k −dn,k = (b−1)dn,k >(b−1)acn,k >
>(a−1)acn,k >cn,k > cn,k −dn,k−1.
Finally
an+2,1−dn+1,n+1 = (n+ 2)dn+1,n+1 −dn+1,n+1 >
> dn+1,n+1 > cn+1,n+1 > cn+1,n+1 −dn+1,n .
From the above inequalities we have for a sufficiently large n∈Nthe following
inequalities:
1 = an,2−bn,1< an,1−dn−1,n−1< cn,1−bn,2< cn,2−dn,1< . . .
···< cn,n −dn,n−1< an+1,1−dn,n .(2.1)
Now we use Lemma 2.1. From (2.1) one can see that it is sufficient to study
the quotients:
a)an+1,1−dn,n
bn+1,2,
b)cn,1−bn,2
dn,1,
c)cn,k −dn,k−1
dn,k for k= 2,3,...,n.
In case a)
lim inf
n→∞
an+1,1−dn,n
bn+1,2
= lim inf
n→∞
(n−1)dn,n
na2dn,n
=1
a2>α,
in case b)
lim inf
n→∞
cn,1−bn,2
dn,1
= lim inf
n→∞
(b−1)bn,2
abbn,2
=b−1
ab =α
and in case c)
cn,k −dn,k−1
dn,k
6(b−1)dn,k−1+ 1
abdn,k−1
6b−1
ab +1
abdn,k−1
6α+1
abdn,1
and cn,k −dn,k−1
dn,k
>(b−1)dn,k−1
abdn,k−1+b+ 1 >
>b−1
ab −b−1
ab
b+ 1
abdn,k−1+b+ 1 >α−b2−1
dn,1
.
From this it is obvious that D(X) = α.
It remains to prove that the set Xis (R)-dense. We have 1
a261
band 1
blal+2 61
bl+1al
for every l= 1,2,..., hence

On the best estimations for dispersions of special ratio block sequences 91
1
a2,1E∪∞
[
l=1 1
blal+2 ,1
blal−1E= (0,1i
and it is sufficient to prove that the ratio set of the set Xis dense on intervals
1
a2,1Eand 1
blal+2 ,1
blal−lE
for every l= 1,2,....
Now we prove that the ratio set of Xis dense on 1
a2,1E. Let (e, f )⊂1
a2,1E.
Put ε=f−e. Consider the set
nan,1+ 1
bn,2
<an,1+ 2
bn,2
<···<bn,1
bn,2
<
<an,2+ 1
bn,2
<an,2+ 2
bn,2
<···<bn,2−1
bn,2
<bn,2
bn,2
= 1o,
(2.2)
which is obviously a subset of the ratio set of X. The largest difference between
consecutive terms of (2.2) is 2
bn,2. Then
an,1+ 1
bn,2
=an,1
bn,2
+1
bn,2
6an,1
a2an,1
+1
bn,2
=1
a2+1
bn,2
.
If we choose n∈Nso that 2
bn,2< ε, then the interval (e, f )is not disjoint with (2.2),
hence the ratio set of Xis dense in the interval 1
a2,1E.
Let l∈Nbe arbitrary. We prove that the ratio set of Xis dense in the interval
1
blal+2 ,1
blal−1E. Let (e, f )⊂1
blal+2 ,1
blal−1E. Put ε=f−e. Choose n1∈Nso
that n1> l and an,1+ 1 >2
εfor every n > n1. Consider the set
nbn,2
cn,l + 1 >bn,2−1
cn,l + 1 >··· >an,2+ 1
cn,l + 1 >bn,1
cn,l + 1 >
>bn,1−1
cn,l + 1 >··· >an,1+ 1
cn,l + 1 >an,1+ 1
cn,l + 2 >···>an,1+ 1
dn,l o,
(2.3)
which is obviously a subset of the ratio set of X. The largest difference between
consecutive terms of (2.3) is 62
an,1+1 . On the other hand,
lim
n→∞
bn,2
cn,l + 1 =1
blal−1and lim
n→∞
an,1+ 1
dn,l
=1
blal+2 .
Then there exists n2∈N, such that for every n > n2
bn,2
cn,l + 1 >1
blal−1−εand an,1+ 1
dn,l
<1
blal+2 +ε .
If we choose n > max{n1, n2}, then the interval (e, f )is not disjoint with (2.3),
hence the ratio set of Xis dense in the interval 1
bl,al+2 ,1
bl,al−1. This concludes
the proof. 

92 F. Filip, K. Liptai, F. Mátyás, J. T. Tóth
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On the best estimations for dispersions of special ratio block sequences 93
Ferdinánd Filip,János T. Tóth
Department of Mathematics
J. Selye University
Bratislavská cesta 3322
945 01 Komárno
Slovakia
e-mail: filip.ferdinand@selyeuni.sk
toth.janos@selyeuni.sk
Kálmán Liptai,Ferenc Mátyás
Institute of Mathematics and Informatics
Eszterházy Károly College
H-3300 Eger
Leányka út 4.
Hungary
e-mail: liptaik@ektf.hu
matyas@ektf.hu