On the best estimations for dispersions of special ratio block sequences

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(2010) pp. 85–93

http://ami.ektf.hu

On the best estimations for dispersions of special ratio block sequences ^∗

Ferdinánd Filip

^a

, Kálmán Liptai

^b

Ferenc Mátyás

^b

, János T. Tóth

^a

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 80^th 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 N and 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 thatX is infinite. Denote by R(X) ={^x_y;x∈X, y∈X} theratio set of X and say that a setX is(R)-dense if R(X)is (topologically) dense in the setR⁺. Let us notice that the concept of (R)-density was defined and first studied in papers [17] and [18].

Now letX ={x1, x2, . . .}wherexn< xn+1are 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

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of finite sequences derived fromX is calledratio block sequenceof the setX. Thus the block sequence is formed by blocksX1, 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 everyn∈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 then-th block.

In this paper we will consider the characteristics (see [20]) D(X) = lim inf

n→∞ D(Xn),

called the dispersionof the block sequence (1.1) derived fromX, and its relations to the previously mentioned asymptotic density of the original setX.

At the end of this section, let us mention the concept of a dispersion of a general sequence of numbers in the intervalh0,1i. Let(xn)^∞_n=0be a sequence inh0,1i. For every N ∈ N letxi1 6 xi2 6. . . 6 xi_N be reordering of its firstN terms 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→∞

N dN > 1 log 4

holds for every one-to-one infinite sequencexn ∈ 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].

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2. Results

When calculating the valueD(X), the following theorems are often useful (See [20], Theorem 1, Corollary 1, respectively).

(A1) Let

X ={x1, x2, . . .}= [∞

n=1

(cn, dni ∩N,

wherexn< xn+1 and letcn< dn< cn+1, forn∈N, be positive integers. Then D(X) = lim inf

n→∞

max{ci+1−di:i= 1,2, . . . , n}

dn+1

.

(A2) Let X be identical to the form of X in (A1). Suppose that there exists a positive integern0 such that for all integersn > n0

cn+1−dn6cn+2−dn+1. Then

D(X) = lim inf

n→∞

cn+1−dn

dn+1

.

The basic properties of the dispersionD(X)and the relations between dispersion and(R)-density are investigated in the paper [TMF]. The next theorem states the upper bound for dispersionsD(X)of(R)-dense sets where16a= lim

n→∞

d_n c_n <∞ (See [20], Theorem 10).

(A3) LetX =S∞ n=1

cn, dn

E∩Nbe an(R)-dense set wherecn< dn < cn+1 for all n∈Nand suppose that the limit lim

n→∞

dn

c_n =aexists. Then D(X)6min

1 a+ 1,max

a−1 a² , 1

a²

,

more precisely,

D(X)6







1

1+a ifa∈ h1,¹⁺₂^√⁵)

1

a² ifa∈ h¹⁺₂^√⁵,2)

a−1

a² ifa∈ h2,∞).

The following theorem shows that in the third case (ifa>2), that the dispersion D(X)can be any number in the interval

0,^a_a⁻2¹

, whereX =S_∞

n=1

cn, dn

E∩Nis (R)-dense and lim

n→∞

d_n

c_n =a. Thus the upper bound forD(X)is the best possible in the casea>2(See [4], Theorem 2).

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(A4) Leta>1 be a real number andk be an arbitrary natural number. Then for everyα∈ h0,^a^k_a2k⁻¹ithere exists an(R)-dense set

X =

∞

[

n=1

(cn, dni ∩N

wherecn< dn< cn+1are positive integers for everyn∈N, such that lim

n→∞

d_n cn =a andD(X) =α.

In this paper we prove that in the second case ifa∈D

1+√ 5 2 ,2

, the dispersion D(X) can be any number in the interval

0,_a¹2

, whereX = ^∞S

n=1

cn, dn

E∩N is (R)-dense and lim

n→∞

dn

cn =a. Thus the upper bound forD(X)is the best possible in the casea∈D

1+√ 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→∞

cm_k+1−dm_k

dm_k+1

.

Proof. Letn∈Nbe an arbitrary integer such thatn>m1. Then there is unique k∈Nwithmk6n < mk+1. >From the definition of the setM(X)we obtain

max{ci+1−di:i= 1,2, . . . , n}

dn+1

=cm_k+1−dm_k

dn+1

>cm_k+1−dm_k

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

dm_k+1

.

On the other hand, the sequence ^c^mk_d⁺¹⁻^d^mk

mk+1

∞

k=1 is a subsequence of the sequence _max

{ci+1−di:i=1,2,...,n} d_n+1

n∈N, hence D(X) = lim inf

n→∞

max{ci+1−di :i= 1,2, . . . , n}

dn+1

6lim inf

k→∞

cm_k+1−dm_k

dm_k+1

. The last two inequalities imply

D(X) = lim inf

k→∞

cm_k+1−dm_k

dmk+1

.

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Theorem 2.2. Let a ∈ 1+√ 5 2 ,2

be an arbitrary real number. Then for every α∈ h0,_a¹2ithere 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 andD(X) =α.

Proof. Let a ∈ h¹⁺₂^√⁵,2). According to (A4), it is sufficient to prove Theorem 2.2 for ^a_a⁻2¹ < α 6 ¹

a². Define function f(b) = ^b_ab⁻¹. Clearlyf is continuous and increasing on the interval ha,∞). Moreover

f(a) = a−1

a² and f(a²) = a²−1 a³ . We have ^a²_a⁻3¹ >_a¹2 ifa> ¹⁺

√5

2 . Thus there exists a real numbera < b6a²such

that b−1

ab =α.

Define a setX ⊂Nby

X =

∞

[

n=1

An∪Bn

∩N,

where for everyn∈N

An = (an,1, bn,1i ∪(an,2, bn,2i a Bn=

n

[

k=1

(cn,k, dn,ki.

Put a1,1= 1and for everyn∈Nandk= 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, andan+1,1= (n+ 1)dn,n.

Obviously for everyn∈N a < bn,1

an,1

6a+ 1

an,1

and a < bn,1

an,1

6a+ 1

an,1

, and fork= 1,2, . . . , n

a < dn,k

cn,k

6a+ 1

an,1

.

First we prove thatD(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

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The inequalitya²(a−1)>afollows froma> 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 everyk= 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 ∈ N the 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)^a^n+1,1_b_n+1,2⁻^d^n,n, b)^c^n,1_d⁻^b^n,2

n,1 ,

c)^c^n,k⁻_d_n,k^d^n,k−¹ fork= 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

6 b−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 thatD(X) =α.

It remains to prove that the setXis(R)-dense. We have _a¹2 6¹

b and_bla¹^l+2 6 ¹

b^l+1a^l

for everyl= 1,2, . . ., hence

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1 a²,1E

∪ [∞

l=1

1 b^la^l+2, 1

b^la^l⁻¹

E= (0,1i

and it is sufficient to prove that the ratio set of the setX is dense on intervals 1

a²,1E

and 1

b^la^l+2, 1 b^la^l⁻^l

E

for everyl= 1,2, . . ..

Now we prove that the ratio set ofX is 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 _b²

n,2. Then an,1+ 1

bn,2

= an,1

bn,2

+ 1 bn,2

6 an,1

a²an,1

+ 1 bn,2

= 1 a² + 1

bn,2

. If we choosen∈Nso that_b²

n,2 < ε, then the interval(e, f)is not disjoint with (2.2), hence the ratio set of X is dense in the interval

1 a²,1E

.

Letl∈Nbe arbitrary. We prove that the ratio set ofX is dense in the interval 1

b^la^l+2,_bla¹^l−1

E. Let (e, f)⊂

1

b^la^l+2,_bla¹^l−1

E. Put ε =f −e. Choose n1 ∈N so that n1> l andan,1+ 1> ²_ε for everyn > n1. Consider the set

n bn,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 6_a ²

n,1+1. On the other hand,

nlim→∞

bn,2

cn,l+ 1 = 1

b^la^l⁻¹ and lim

n→∞

an,1+ 1 dn,l

= 1

b^la^l+2 . Then there existsn2∈N, such that for everyn > n2

bn,2

cn,l+ 1 > 1

b^la^l⁻¹ −ε and an,1+ 1 dn,l

< 1

b^la^l+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 X is dense in the interval _bl,a¹^l+2,_bl,a¹^l−¹

. This concludes

the proof.

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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

On the best estimations for dispersions of special ratio block sequences