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Double lacunary statistically convergentsequences in topological groups

Ekrem Savas-

_Istanbul Commerce University, Department of Mathematics, Uskudar, _Istanbul, Turkey

Received 13 October 2011; received in revised form 29 April 2012; accepted 8 June 2012

Abstract

Recently, Patterson and Savas [11] defined the lacunary statistical analogue for double sequences

x¼ ðxðk,lÞÞ as follows: a real double sequences x¼ ðxðk,lÞÞ is said to be P-lacunary statistically

convergent to L provided that for each E40

P�limr,s

1

hr,sjfðk,lÞ 2 Ir,s : jxðk,lÞ�LjZEgj ¼ 0:

In this case write st2y�limx¼L or xðk,lÞ-Lðst2yÞ.

In this paper we introduce and study lacunary statistical convergence for double sequences in

topological groups and we shall also present some inclusion theorems.

& 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

We begin this section by giving some preliminaries.The notion of statistical convergence, which is an extension of the idea of usual

convergence, was introduced by Fast [4] and Schoenberg [22] and its topologicalconsequences were studied first by Fridy [5] and Sal�at [13]. Di Maio and Kocinac [7]introduced the concept of statistical convergence in topological spaces and statisticalCauchy condition in uniform spaces and established the topological nature of thisconvergence. Recently a lot of interesting developments have occurred in double statisticalconvergence and related topics (see [6,9,14,15,18]).

2.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.org/10.1016/j.jfranklin.2012.06.003

dress: [email protected]

this article as: E. Savas-Double lacunary statistically convergent sequences in topological groups,

the Franklin Institute. (2012), http://dx.doi.org/10.1016/j.jfranklin.2012.06.003


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E. Savas- / Journal of the Franklin Institute ] (]]]]) ]]]–]]]2

By X, we will denote an abelian topological Hausdorff group, written additively, whichsatisfies the first axiom of countability. For a subset A of X, s(A) will denote the set of allsequences ðxnÞ such that xn is in A for n¼ 1; 2, . . .. In [1], a sequence ðxnÞ in X is called to bestatistically convergent to an element ‘ of X if for each neighborhood U of 0

limn-1

1

njfkrn : xk�‘=2Ugj ¼ 0,

where the vertical bars indicate the number of elements in the enclosed set and is calledstatistically Cauchy in X if for each neighborhood U of 0 there exists a positive integern0ðUÞ, depending on the neighborhood U, such that

limn-1

1

njfkrn : xk�xn0ðUÞ=2Ugj ¼ 0:

The set of all statistically convergent sequences in X is denoted by st(X) and the set of allstatistically Cauchy sequences in X is denoted by stC(X). It is known that stCðX Þ ¼ stðX Þ

when X is complete.By a lacunary sequence, we mean an increasing sequence y¼ ðkrÞ of positive integers

such that k0 ¼ 0 and hr : kr�kr�1-1 as r-1. Throughout this paper, the intervalsdetermined by y will be denoted by Ir ¼ ðkr�1,kr�, and the ratio ðkrÞðkr�1Þ

�1 will beabbreviated by qr. Cakalli [2] defined lacunary statistical convergence in topological groupsas follows: a sequence ðxðkÞÞ is said to be Sy-convergent to l (or lacunary statisticallyconvergent to l) if for each neighborhood U of 0, limr-1ðhrÞ

�1jk 2 Ir : xðkÞ�l=2Ugj ¼ 0. In

this case, we write

Sy� limk-1

xðkÞ ¼ l or xðkÞ-lðSyÞ

and define

SyðX Þ ¼ ðxðkÞÞ : for some l,Sy� limk-1

xðkÞ ¼ l

� �:

2. Definitions and notations

By the convergence of a double sequence we mean the convergence in Pringsheim’s sense(see [12]). A double sequence x¼ ðxðk,lÞÞ is said to be convergent in the Pringsheim’s senseif for every E40 there exists N 2 N such that jxðk,lÞ�LjoE whenever k,lZN. L is calledthe Pringsheim limit of x. We shall describe such an x more briefly as ‘‘P-convergent’’.A double sequence x¼ ðxðk,lÞÞ is said to be Cauchy sequence if for every E40 there

exists N 2 N such that jxðp,qÞÞ�xðk,lÞjoE for all pZjZN and qZkZN.Let KDN �N and Kðm,nÞ ¼ fðk,lÞ : krm,lrng. Then the double natural density of K

is defined by

d2ðKÞ ¼ P�limm,n

jKðm,nÞj

mn

if it exists, where the vertical bars denotes the cardinality of the set Kðm,nÞ.The concept of double statistical convergence, for complex case, was introduced by

Mursaleen and Edely [8] while the idea of statistical convergence of single sequences wasfirst studied by Fast [4]. Mursaleen and Edely has given main definition as follows:

Please cite this article as: E. Savas-Double lacunary statistically convergent sequences in topological groups,

Journal of the Franklin Institute. (2012), http://dx.doi.org/10.1016/j.jfranklin.2012.06.003




E. Savas- / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 3

Definition 2.1 (Mursaleen and Edely [8]). A double sequences x¼ ðxðk,lÞÞ is said to be P-statistically convergent to L provided that for each E40

P�limm,n

1

mnfnumber of ðk,lÞ : kom and lon,jxðk,lÞ�LjZEg ¼ 0,

In this case we write st2�limk,lxðk,lÞ ¼ L and we denote the set of all statistical
convergent double sequences by S2.
Clearly, a convergent double sequence is also st2-convergent but the inverse is not true,in general. Also note that a st2-convergence need not be bounded. For example, thesequence x¼ ðxk,lÞ defined by

xðk,lÞ ¼kl if k and l are square

1 otherwise

�

is st2-convergent. Nevertheless it neither convergent nor bounded.The double sequence y¼ fðkr,lsÞg is called double lacunary if there exist two increasing of

integers such that

k0 ¼ 0, hr ¼ kr�kk�1-1 as r-1

and

l0 ¼ 0, hs ¼ ls�ls�1-1 as s-1:

Notations: kr,s ¼ krls, hr,s ¼ hrhs, y is determine by Ir ¼ fðkÞ : kr�1okrkrg, Is ¼ fðlÞ :ls�1olrlsg, Ir,s ¼ fðk,lÞ : kr�1okrkrls�1olrlsg, qr ¼ kr=ðkr�1Þ, qs ¼ ls=ðls�1Þ, andqr,s ¼ qrqs. We will denote the set of all double lacunary sequences by Nyr,s

.Let KDN �N has double lacunary density dy2ðKÞ if

P�limr,s

1

hr,sjfðk,lÞ 2 Ir,s : ðk,lÞ 2 Kgj

exists.

Example 1. Let y¼ fð2r�1,3s�1Þg and K ¼ fðk,2lÞ : k,l 2 N �Ng. Then dy2ðKÞ ¼ 0. But itis obvious that d2ðKÞ ¼ 1=2.

In 2005, Patterson and Savas [11] studied double lacunary statistically convergence bygiving the definition for complex sequences as follows:

Definition 2.2. Let y be a double lacunary sequence; the double number sequence x isst2y�convergent to L provided that for every E40

P�limr,s

1

hr,sjfðk,lÞ 2 Ir,s : jxðk,lÞ�LjZEgj ¼ 0:

In this case write st2y�lim x¼L or xðk,lÞ-LðS2yÞ.

More investigation in this direction and more applications of double lacunary anddouble sequences can be found in [16,17,19–21].

The purpose of this paper is to study double lacunary statistical convergence of doublesequences in topological groups and to give some important inclusion theorems.







Quite recently, Cakalli and Savas [3] defined the statistical convergence of doublesequences x¼ ðxðk,lÞÞ of points in a topological group as follows.In a topological group X, double sequence x¼ ðxðk,lÞÞ is called statistically convergent

to a point L of X if for each neighborhood U of 0 the set

fðk,lÞ,krn; and; lrm : xðk,lÞ�L=2Ug

has double natural density zero. In this case we write S2�limk,lxðk,lÞ ¼ L and we denotethe set of all statistically convergent double sequences by S2ðX Þ.Now we are ready to give the definition of double lacunary statistical convergence in

topological groups as follows:

Definition 2.3. A sequence ðxðk,lÞÞ is said to be S2y-convergent to L (or double lacunary

statistically convergent to L) if for each neighborhood U of 0, P�limr,s-1ðhrsÞ�1

jðk,lÞ 2 Irs : xðk,lÞ�L=2Ugj ¼ 0.

In this case, we write

S2y� lim

k,l-1xðk,lÞ ¼ L or xðk,lÞ-LðS2

yÞ

and define

S2yðX Þ ¼ ðxðk,lÞÞ : for some L,S2

y� limk,l-1

xðk,lÞ ¼L

� �

and in particular

S2yðX Þ0 ¼ ðxðk,lÞÞ : S2

y� limk,l-1

xðk,lÞ ¼ 0

� �:

It is obvious that every double lacunary statistically convergent sequence has only onelimit, that is, if a sequence is double lacunary statistically convergent to L1 and L2 thenL1 ¼L2.

3. Inclusion theorems

In this section, we prove some analogues for double sequences. For single sequencessuch results have been proved by C- akalli [2].

Theorem 3.1. For any double lacunary sequence y¼ fðkr,lsÞg, S2ðX ÞDS2yðX Þ if and only if

lim inf qr41 and lim inf qs41

Proof. Sufficiency: Suppose that lim inf qr41, and lim inf qs41, lim inf qr ¼ a1 andlim inf qs ¼ a2, say. Write b1 ¼ ða1�1Þ=2 and b2 ¼ ða2�1Þ=2. Then there exist a positiveinteger r0 and s0 such that qrZ1þ b1 for rZr0 and qsZ1þ b2 for sZs0. Hence for rZr0,and sZs0

hrsðkrÞ�1ðlsÞ�1¼ ð1�ðkr�1ÞðkrÞ

�1Þð1�ðls�1ÞðlsÞ

�1Þ ¼ ð1�ðqrÞ

�1Þð1�ðqsÞ

�1Þ

Zð1�ð1þ b1Þ�1ð1�ð1þ b2Þ

�1Þ ¼ ðb1ð1þ b1Þ

�1Þðb2ð1þ b2Þ

�1Þ:

Take any ðxðk,lÞÞ 2 S2ðX Þ, and S2�limðk,lÞ-1xðk,lÞ ¼L, say. We prove thatS2y�limðk,lÞ-1xðk,lÞ ¼ L. Let us take any neighborhood U of 0. Then for rZr0, and







sZs0 we have

ðkrÞ�1ðlsÞ�1jfkrkr,lrls : xðk,lÞ�L=2UgjZðkrÞ

�1ðlsÞ�1jfðk,lÞ 2 Isr : xðk,lÞ�L=2Ugj

¼ hr,sðkrÞ�1ðlsÞ�1ðhrsÞ

�1jfðk,lÞ 2 Irs : xðklÞ�L=2Ugj

Zb1ð1þ b1Þ�1b2ð1þ b2Þ

�1ðhrsÞ

�1jfðk,lÞ 2 Irs : xðk,lÞ�L=2Ugj:

Therefore, S2y�limðk,lÞ-1xðk,lÞ ¼ L

Necessity: Suppose that lim inf rqr ¼ 1 and lim inf sqs ¼ 1. Then we can choose asubsequence fðkrðjÞ,ðlsðiÞÞ of the lacunary sequence yrs ¼ fkr,lsg such that

krðjÞðkrðjÞ�1Þ�1o1þ j�1,krðjÞ�1ðkrðj�1ÞÞ�14j

and

lsðiÞðlsðiÞ�1Þ�1o1þ i�1,lsðiÞ�1ðlsði�1Þ�14i,

where rðjÞ4rðj�1Þ þ 2 and sðiÞ4sði�1Þ þ 2. Take an element x of X different from 0.Define a sequence ðxðk,lÞÞ by xðk,lÞ ¼ x if ðk,lÞ 2 fIrðjÞ,sðiÞg for some j ¼ 1; 2, . . . ,m, . . . ,i¼ 1; 2, . . . ,n, . . ., and xðk,lÞ ¼ 0 otherwise. Then ðxðk,lÞÞ 2 S2ðX Þ (in fact ðxðk,lÞÞ 2S20ðX Þ ). To see this take any neighborhood U of 0. Then we may choose a neighborhood

W of 0 such that W � U and x=2W . On the other hand, for each m and n we can find apositive number jm and in such that krðjmÞomrkrðjmÞþ1 and lsðinÞonrlsðinÞþ1. Then

ðmnÞ�1jfkrm,lrn : xðk,lÞ=2Ugjrk�1rðjmÞl�1sðinÞjfkrm,lrn : xðk,lÞ=2W gj

rk�1rðjmÞl�1sðinÞfjfkrkrðjmÞ,lrlsðinÞ : xðk,lÞ=2W gj

þjfkrðjmÞokrm,lsðinÞolrn : xðk,lÞ=2W gjg

rk�1rðjmÞl�1sðinÞjfkrkrðjmÞ,lrlsðinÞ : xðk,lÞ=2W gj

þk�1rðjmÞðkrðjmÞþ1�krðjmÞÞ,l

�1sðinÞðlsðinÞþ1�lsðinÞÞ

oððjm þ 1Þ�1 þ 1þ j�1m �1Þðin þ 1Þ�1 þ 1þ i�1n �1Þ

oððjm þ 1Þ�1 þ j�1m Þððin þ 1Þ�1 þ i�1n Þ

for each (m,n). Therefore ðxðk,lÞÞ 2 S20ðX Þ. Now let us see that ðxðk,lÞÞ=2S2

0ðX Þ. Since X is aHausdorff space, there exists a symmetric neighborhood V of 0 such that x=2V . Hence

limj,i-1ðhrðjÞ,sðiÞÞ

�1jfkrðjÞ�1okrkrðjÞ,lsðiÞ�1olrlsðiÞ : xðk,lÞ=2Vgj

¼ limj,i-1ðhrðjÞÞ

�1ðhsðiÞÞ

�1ðkrðjÞ�krðjÞ�1ÞðlsðiÞ�lsðiÞ�1Þ

¼ limj,i-1ðhrðjÞÞ

�1hrðjÞðhsðiÞÞ�1ðhsðiÞÞ ¼ 1

and

limr,s-1

rarðjÞ,sasðiÞ j ¼ i ¼ 1;2,...

h�1r,s jfkr�1okrkr,ls�1olrls : xðk,lÞ�x=2Vgj ¼ 1a0:

Hence, neither x nor 0 can be double lacunary statistical limit of ðxðk,lÞÞ. No other pointof X can be double lacunary statistical limit of the sequence as well. Thus ðxðk,lÞÞ=2S2

yðX Þ.This completes the proof. &







Theorem 3.2. For any lacunary sequence y¼ fðkr,lsÞg, S2yðX ÞDS2ðX Þ if and only if

lim suprqro1 and lim supsqso1.

Proof. Sufficiency: If lim suprqro1 and lim supsqso1, there exists an H40 such thatqroH and qsoH for all (r,s). Let ðxðk,lÞÞ 2 S2

yðX Þ, S2y�limk,l-1xðk,lÞ ¼L, say. Take any

neighborhood U of 0. Let e40. Write Nr,s ¼ fðk,lÞ 2 Ir,s : xðk,lÞ�L=2Ug by the definition ofdouble lacunary statistical convergence, there is a positive integer r0 and s0 such thatNr,sðhr,sÞ

�1oe for all r4r0 and s4s0. Let M ¼maxfNr,s : 1rrrr0 and 1rsrs0g and let n

and m be such that kr�1omrkr and ls�1onrls; hence we obtain the following

ðmnÞ�1jfkrm,lrn : xðk,lÞ�L=2Ugjrr0s0Mðkr�1Þ�1ðls�1Þ

�1þ eH2qrs:

Since limr,s-1krls ¼1, there exists a positive integer r1Zr0 and s1Zs0 such that

ðkr�1Þ�1ðls�1Þ

�1oð4r0s0M=eÞ�1 for r4r1 and s4s1

Hence for r4r1 and s4s1

ðmnÞ�1jfkrm,lrn : xðk,lÞ�L=2Ugjre2þ

e2¼ e:

It follows that S2�limk,l-1xðk,lÞ ¼L.Necessity: We shall assume that lim suprqr ¼1 and lim suprqr ¼1. Take an element x

of X different from 0. Construct a subsequence ðkrðjÞÞ and ðlsðiÞÞ of the lacunary sequenceyr,s ¼ ðkr,lsÞ such that krðjÞ4j and lsðiÞ4i, krðjÞ4j þ 3 and lsðiÞ4i þ 3 and define a sequenceðxðk,lÞÞ by xðk,lÞ ¼ x if krðjÞ�1okr2krðjÞ�1 and lsðiÞ�1olr2lsðiÞ�1 for some i¼ j ¼ 1; 2, . . .,and xðk,lÞ ¼ 0 otherwise. Let U be a symmetric neighborhood of 0 that does not include x.Then for j41,

ðhrðjÞ,sðiÞÞ�1jfkrkrðjÞ,lrlsðiÞ : xðk,lÞ=2UgjoðkrðjÞ�1ÞðlsðiÞ�1ÞðhrðjÞ,sðiÞÞ

�1oðj�1Þ�1ði�1Þ�1:

Hence ðxðk,lÞÞ 2 S2yðX Þ. But ðxðk,lÞÞ=2S2ðX Þ. For

ð2krðjÞ�1Þ�1ð2lsðiÞ�1Þ

�1jfkr2krðjÞ�1,lr2lsðiÞ�1 : xðk,lÞ=2Ugj

¼ ð2krðjÞ�1Þ�1ð2lsðiÞ�1Þ

�1½krð1Þ�1 þ krð2Þ�1 þ � � � þ krðjÞ�1�

½lsð1Þ�1 þ lsð2Þ�1 þ � � � þ lsðiÞ�1�41=4,

which implies that ðxðk,lÞÞ cannot be double statistically convergent. This completes theproof. &

Corollary 3.3. Let y¼ fðkr,lsÞg be a double lacunary sequence, then S2ðX Þ ¼ S2yðX Þ iff

1olim infr

qrrlim supr

qro1

and

1olim infs

qsr lim sups

qso1:

Of course, the S2y-limit is unique as shown in Section 2. It is possible, however, for a

sequence to have different S2y-limits for different y’s.The following theorem shows that this

situation cannot occur if x 2 S2ðX Þ.







Theorem 3.4. If ðxðk,lÞÞ belongs to both S2ðX Þ and S2yðX Þ, then

S2� limk,l-1

xðk,lÞ ¼ S2y� lim

k,l-1xðk,lÞ:

Proof. Take any ðxðk,lÞÞ 2 S2ðX Þ \ S2yðX Þ and S2�limk,l-1xðk,lÞ ¼L1, S2

y�limk,l-1

xðk,lÞ ¼ L2, say. Suppose that L1aL2. Since X is a Hausdorff space, there exists asymmetric neighborhood U of 0 such that L1�L2=2U . Then we may choose a symmetricneighborhood W of 0 such that W þW � U . Then we obtain the following inequality:

ðkmÞ�1ðlnÞ�1jfkrkn,lrlm : zðk,lÞ=2Ugj

rðkmÞ�1ðlnÞ�1jfkrkm,lrln : xðk,lÞ�L1=2W gj

þðkmÞ�1ðlnÞ�1jfkrkm,lrlm : L2�xðk,lÞ=2W gj,

where zðk,lÞ ¼ L2�L1 for all k,l 2 N �N. It follows from this inequality that

1rðkmÞ�1,ðlnÞ

�1jfkrkm,lrln : xðk,lÞ�L1=2W gj

þðkmÞ�1,ðlnÞ

�1jfkrkm,lrln : L2�xðk,lÞ=2W gj:

The second term on the right side of this inequality tends to 0 as m-1. To see this write

ðkmÞ�1,ðlnÞ

�1jfkrkm,lrln : L2�xðk,lÞ=2W gj

¼ ðkmÞ�1,ðlnÞ

�1 k,l 2[m,n

r,s ¼ 1

Ir,s : L2�xðk,lÞ=2W

( )��

¼ ðkmÞ�1,ðlnÞ

�1Xm,n

r,s ¼ 1

jfk,l 2 Ir : L2�xðk,lÞ=2W gj

¼Xm,n

r,s ¼ 1

hr,s

!�1 Xm,n

r,s ¼ 1

hr,str,s

!,

where tr,s ¼ h�1r,s jfk,l 2 Ir,s : L2�xðk,lÞ=2W gj is a Pringsheim null sequence, sinceS2y�limk,l-1 xðk,lÞ ¼ L2. Therefore, the regular weighted mean transform of ðtr,sÞ also

tends to zero that is

P� limm,n-1

ðkmÞ�1,ðlnÞ

�1jfkrkm,lrln : L2�xðk,lÞ=2W gj ¼ 0: ð3:1Þ

On the other hand, since S�limk-1xðk,lÞ ¼L1

P� limm-1ðkmÞ

�1,ðlnÞ�1jfkrkm,lrln : xðk,lÞ�L1=2W gj ¼ 0: ð3:2Þ

By (3.1) and (3.2) it follows that

P� limm,n-1

ðkmÞ�1,ðlnÞ

�1jfkrkm,lrln : zðk,lÞ=2Ugj ¼ 0:

This contradiction completes the proof. &

Definition 3.5. Let y¼ ðkr,lsÞ be a double lacunary sequence; the sequence ðxðk,lÞÞ is said tobe an S2

y-Cauchy sequence if there is a subsequence ðxðk0ðrÞ,l0ðsÞÞÞ of ðxðk,lÞÞ such that







k0ðrÞ,l0ðsÞ 2 Ir,s, for each r,s, limr,s-1xðk0ðrÞ,l0ðsÞÞ ¼ L and for each neighborhood U of 0

P� limr,s-1

ðhr,sÞ�1jfðk,lÞ 2 Ir,s : xðk,lÞ�xðk0ðrÞ,l0ðsÞÞ=2Ugj ¼ 0:

It turns out that a sequence is S2y-convergent if and only if it is an S2

y-Cauchy sequencewithout the completeness assumption.

Theorem 3.6. A sequence ðxðk,lÞÞ is S2y-convergent if and only if it is an S2

y-Cauchy sequence.

Proof. Sufficiency: Suppose that ðxðk,lÞÞ is an S2y-Cauchy sequence. Let U be any

neighborhood of 0. Then we may choose a neighborhood W of 0 such that W þWDU .Therefore

ðhr,sÞ�1jfðk,lÞ 2 Ir,s : xðk,lÞ�L=2Ugjrðhr,sÞ

�1jfðk,lÞ 2 Ir,s : xðk,lÞ�xðk0ðrÞ,l0ðsÞÞ=2W gj

þðhr,sÞ�1jfðk,lÞ 2 Ir,s : xðk0ðrÞ,l0ðsÞÞ�L=2W gj:

Since limr,s-1ðhr,sÞ�1jfðk,lÞ 2 Ir,s : xðk,lÞ�xðk0ðrÞ,l0ðsÞÞ=2Ugj ¼ 0, in Pringsheim sense,

and limr,s-1xðk0ðrÞ,l0ðsÞÞ ¼L, by the assumption, it follows from the last inequality thatS2y�limk-1xðk,lÞ ¼ L.Necessity: Suppose that S2

y�limk,l-1xðk,lÞ ¼L. Let ðUnmÞ be a nested base ofneighborhoods of 0. Write K ði,jÞ ¼ fðk,lÞ 22 N �N : xðk,lÞ�L 2 Uijg for each ði,jÞ 2N �N. Thus for each (i,j) we obtain the following K ðiþ1,jþ1ÞDK ði,jÞ andlimr,sjK

ði,jÞ \ Ir,sjðhrsÞ�1¼ 1. This implies that there exit mð1Þ and nð1Þ such that rZmð1Þ

and sZnð1Þ and ðjK ð1;1Þ \ IrsjÞ=hr,s40, that is, K ð1;1Þ \ Irsa|. We next choose mð2Þ4mð1Þand nð2Þ4nð1Þ such that r4mð2Þ and s4nð2Þ implies that K ð2;2Þ \ Irsa|. Thus, for each(r,s) satisfying mð1Þrromð2Þ and nð1Þrsonð2Þ, we choose ðk0ðrÞ,l0ðsÞÞ 2 Irs such thatðk0ðrÞ,l0ðsÞÞ 2 K ðr,sÞ \ Irs, that is xðk0ðrÞ,l0ðsÞÞ�L 2 U11. In general, we choose mðpþ

1Þ4mðpÞ,nðqþ 1Þ4nðqÞ such that r4mðpþ 1Þ and s4nðqþ 1Þ implies thatIrs \ K ðpþ1,qþ1Þa|. Then for all (r,s) satisfying mðpÞrromðpþ 1Þ and nðqÞrsonðqþ 1Þchoose ðk0ðrÞ,l0ðsÞÞ 2 Irs \ K ðp,qÞ, that is xðk0ðrÞ,l0ðsÞÞ�L 2 Upq. Hence, it follows thatP�limr,s-1xðk0ðrÞ,l0ðsÞÞ ¼L. Let U be any neighborhood of 0. Then, we may choose asymmetric neighborhood W of 0 such that W þWDU . Now we write

ðhrsÞ�1jfðk,lÞ 2 Irs : xðk,lÞ�xðk0ðrÞ,l0ðsÞÞ=2UgjrðhrsÞ

�1jfðk,lÞ 2 Irs : xðk,lÞ�L=2W gj

þðhrsÞ�1jfðk,lÞ 2 Irs : L�xðk0ðrÞ,l0ðsÞÞ=2W gj:

Since S2y�limk,l-1xðk,lÞ ¼L and P�limr,s-1xðk0ðrÞ,l0ðsÞÞ ¼ L, we find

limr,s-1

ðhrsÞ�1jfðk,lÞ 2 Irs : xðk,lÞ�xðk0ðrÞ,l0ðsÞÞ=2Ugj ¼ 0,

in Pringsheim sense. Thus the theorem is proven. &

Note that Definition 3.5 and Theorem 3.6 are given, for the complex case, in [11].

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withdrawn: double lacunary statistically convergent sequences in topological groups

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