The value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng

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1 The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia, OH 44663, USA. Abstract I this article we show that a certai sequece r of all ratioal umbers i the iterval 0,: r := {r} =,,,..., m +,..., k m +,..., m m + st block 3, 3 d block 4, rd block } {{ } m th block where k, m + =, is a almost coverget sequece i l, ad its value of Baach limits are Lr = / for all L l. Keywords: Baach limits, almost coverget sequeces, upper weight, lower weight, weight.,..., Let l be the Baach space of bouded sequeces x := {x} = of real umbers with orm x = sup x. A Baach limit L is a liear ad bouded fuctioal o l, which satisfies the three coditios: a If x := {x} = l ad x 0, =,,..., the Lx 0; b If x := {x} = l, ad T is the traslatio operator: T x := {x, x3,...}, the Lx = LT x; c L =, where :={,,...}. We kow [4, p.30] that there are ifiitely may Baach limits i l, the dual space of l. Thus, it does t make sese to speak of fidig a particular value for Baach limits of a sequece x := {x} = l, because ormally the differet Baach limits are differet fuctioals. It is, however, iterestig that there are some elemets i l for which the values of all Baach limits are the same. For example, Lx = lim x for ay Baach limit L, if x is a elemet of c, where c is the Baach space of coverget sequeces of real umbers with the superior orm. Furthermore, this pheomeo ca 000 Mathematical Subect Classificatio. Primary 40G05, 46A35, 46B5, 54A0, Secodary A5. address bfeg@tusc.ket.edu

2 happe o some elemets of l \ c. Let a := {, 0,..., 0,, 0,..., 0,,... }. m times m times Property b of Baach limits implies that for ay Baach limit L, La = LT a = = LT m a; so by liearity ad property c of Baach limits we have that La + LT a + + LT m a = L =. Hece Moreover, if the La = LT a = = LT m a = m. b := {,...,, k times 0,..., 0, m k times,...,, k times 0,..., 0, m k times... }, Lb = La + LT m a + + LT m k+ a = k m. I [], G. G. Loretz called a sequece x := {x} = l almost coverget, if its all Baach limit values Lx are the same for L l. I this case we call Lx the F limit of x. I his paper, G. G. Loretz proved the followig mai result: Theorem Loretz, [, Theorem ] A sequece x := {x} = l is almost coverget with F- limit Lx if ad oly if uiformly i i. i+ lim xt = Lx This Loretz theorem offers a way to fid values of Baach limits for almost coverget sequeces i l. Based o Loretz [] ad Suchesto [4], we give aother way [] to fid the value of Baach limits of x, whe x is a almost coverget sequeces i l. Recallig some cocepts we created i []. Defiitio A real umber a is said to be a sub limit of the sequece x := {x} = l, if there exists a subsequece {x k } k= of x with limit a. The set of all sub-limits of x is deoted by Sx ad the set of all limit poits of Sx is deoted by S x.

3 Defiitio Suppose a Sx for elemet x := {x} = l. A subsequece {x k } k= of x is called a essetial subsequece of a if it coverges to a, ad for ay subsequece {xm t } t= of x with limit a, except fiite etries, all its etries are etries of {x k } k=. Defiitio 3 ad Let x := {x} = l ad let {x k } k= be a subsequece of x. Defie w u A{k : i k i + } {x k } = lim sup sup i w l {x k } = lim if if i A{k : i k i + }, where AE is the umber of elemets of set E. w u {x k } ad w l {x k } are called the upper ad lower weight of the subsequece {x k } k= respectively. If wu {x k } = w l {x k }, the the subsequece {x k } k= is said to be weightable ad the weight of {x k } k= is deoted by w{x k}, ad w{x k } = w u {x k } = w l {x k }. We verified [, Theorem ] that all essetial subsequeces of a, a Sx, have the same upper weight ad lower weight respectively. They are called the upper ad lower weights of a, ad deoted by w u a ad w l a respectively. The weight of a is deoted by wa, if w u a = w l a. We have the followig mai results. Theorem [, Theorem 4] Suppose x := {x} = l ad the set of all sub-limits of x, Sx = {a, a,..., a m } is fiite, where a i a, if i. If wa t exists for each t, t m, the x is almost coverget ad for ay Baach limit L l, Lx = m a t wa t. t= We see that for almost coverget sequece, the value of Baach limits oly depedet o the sub-limit poits ad their weights. From theorem we obtai a familiar formula. Corollary [, Corollary ] For a give positive iteger m, let x = {x,..., x m, x,..., x m,..., x,..., x m,...}, where for each t, lim x t = a t, t m, the x is almost coverget ad for ay Baach limit L l, Lx = a + a + + a m. m 3

4 We actually proved a more geeral result as below. Theorem 3 [, Theorem 6] Suppose x := {x} = l ad Sx is ifiite but coutable ad S x, the set of limit poits of Sx, is a oempty fiite set. If wa exists for all a Sx, the x is almost coverget ad for ay Baach limit L l, Lx = awa. a Sx Natural questio ca be asked that does there exist a almost coverget sequece x l for which S x is a ifiite set? The aswer is Yes. We proved more that S x ca be ucoutable ifiite i ext theorem 4 of this article. We eed some prelimiary kowledge. Suppose m is a positive iteger. We defie ϕm to be the umber of itegers k, k < m, such that k, m =, which deotes that k ad m are relative prime. The fuctio ϕ is called the Euler phi fuctio [3, p.54]. It is well kow i umber theory that if m = p α p α... p αt t, where the p s are distict primes ad α i, i t, are positive itegers, the [3, p.58, Theorem.7] ϕm = t p p α =. We itroduce the followig lemmas that we were uable to fid i the literatures. Lemma For ay positive iteger m, m k,m=, k<m k = ϕm. Proof Suppose that m = p α p α p αt t. Note that there are p α p α p αt t umbers cotaiig the factor of p amog the set of {,, 3,..., m }: p, p, 3p,..., p α p α p αt t p, there are p α p α p αt t umbers cotaiig the factor of p amog the set of {,, 3,..., m }: p, p, 3p,..., p α p α p αt t p,..., ad there are p α p α... p αt t umbers cotaiig the factor of p t amog the set of {,, 3,..., m }: The sums of these t groups are pα p α p αt p t, p t, 3p t,..., p α p α p αt t p t. t p α p α p αt t = mpα p α p αt t, 4

5 pα p α p αt t p α p α p αt t = mpα p α p αt t,. pα p α p αt t p α p α p αt t = mpα p α p αt t respectively. Similarly, there are p α p α p αt t umbers cotaiig the factor of p p amog the set of {,, 3,..., m }:, ad there are p α p α t t the set of {,, 3,..., m }: The sums of these t groups are pα p α pα p α p p, p p, 3p p,..., p α p α p αt t p p, pt αt umbers cotaiig the factor of p t p t amog p t p t, p t p t, 3p t p t,..., p α p α t t p αt t p t p t. p αt respectively, ad so o. t p αt t p α p α p α p αt t = mpα p α p αt t, p α. p αt t = mpα p α t t p αt t Fially, there are p α p α p αt t umbers cotaiig the factor of p p p t amog the set of {,, 3,..., m }: The sum of this group is p p p t, p p p t,..., p α p α p αt t p p p t. pα p α p αt t p α p αt t = mpα p αt t. 5

6 Thus, k,m=, k<m k t p α = m m m = i + + t m t p α = i [ t = m t p α = = t + + t = p α i p α p α i i p α i i i p α i i + m p α i i i< t + t m + i< t + t t = p α i i t = p α k i, p α p α i i p α p α k i, ] p α k k p α k k = m = m = m [ t = p α + t t = t = p α p α t = t = p α p α i [ t p = i p α i i p α i i + + t t p i + = i t p = m ϕm. = i< t t = p α i i p α k i, p α t ] i< t k i, p α k k + p k + + t ] t p + t = Hece m k,m=, k<m k = ϕm. Lemma Suppose that m, l are positive itegers with l >. Let = m+l =m+ ϕ, the lim l l = 0. Proof We kow that [3, p. 8, Theorem 6.] m = ϕ = 3m π 6 + Om l m,

7 where f = Og meas that if there exists a costat c > 0 such that fx cgx for all x i the itersectio of domais of f ad g. Thus, = m+l =m+ ϕ = 3m + l 3m π + Om + l lm + l Om l m = 6ml + 3l π + Om + l lm + l Om l m, which implies = Ol. Hece lim l l = 0. Now we back to our goal of the followig result. Theorem 4 Defie a elemet of l as follows r := {r} =, 3,, 3 4, 3,..., 4 m +,..., k m +,..., m }{{ m + } st block d block 3 rd block m th block,..., where k, m+ =. The S r = Sr = [0, ] is a ucoutable set, ad r is a almost coverget sequece i l. ad also Lr = / for ay Baach limit L l. Proof It is well kow that S r = Sr = [0, ]. We ust verify the remaiig parts of the coclusio. For all positive itegers i there exists a iteger m such that m ϕ i < = m+ = ϕ. Let i = m = ϕ + q, where 0 q < ϕm +. For ay positive itegers ad m, m is determied by i above, there is a positive iteger l such that m+l ϕ i + < = m+l+ = ϕ. 7

8 Let i + = m+l = ϕ + s, where 0 s < ϕm + l +. The = = = m+l ϕ + s i + = m+l m ϕ + s ϕ + q + = m+l =m+ = ϕ + s q +, from which we have m+l =m+ ϕ = ϕm + q s. With the help of Lemma ad otice the fact of ϕm < m for ay m, we have = = = i+ rt m + m+l = ϕ t= m+ = ϕ+ rt k,m+=, k<m+ [ ϕm ] ϕm + l k + + k + l = ϕm + q s ϕm + q + ϕm + l + + m + + m + l + + = m + l + 3 = m + 3 l, k,m+l=, k<m+l m+l t=m+ ϕt k which holds for ay positive itegers ad ay fixed positive iteger i. We kow that m+3 m is fixed whe i is fixed, so lim = 0. Note that l goes to ifiity as goes to ifiity. By Lemma we have i+ lim if rt lim if m + 3 l = 8

9 for all i N. O the other had, we have i+ rt = m + = m+l+ = ϕ t= m = ϕ+ rt k,m+=, k<m+ k + + k + l + ] [ ϕm ϕm + l + = m+l+ ϕt = = t=m+ + ϕm + l + + q s + ϕm + l + s + q + m + l + + m + = + m + + l, k,m+l+=, k<m+l+ which holds for ay positive itegers ad ay fixed positive iteger i. We kow that m+ lim = 0. Usig Lemma agai we have i+ lim sup rt lim sup + m + + l = for all i N. Summarizig the discussio above we obtai that, for all i N. Thus, lim i+ i+ lim rt = rt i+ = lim rt Usig G. G. Loretz s formula 6 [, p.69], we have: i+ rt Lr lim lim i+ for all Baach limits L l. Hece Lr =, for all L l ad r is a almost coverget sequece i l. 9 = rt k

10 From the argumet of Theorem 4, we have a ew corollary of Loretz Thoerem as follows. Corollary Suppose x := {x} = l. if the followig limits exist with the same value l, i+ lim xt = l for all positive itegers i N, the x is a almost coverget sequece i l. ad Lx = l for all Baach limits L l. Refereces [] Feg, B. Q. ad Li, J. L., Some estimatios of Baach limits, J. Math. Aal. Appl No., MR60 46B45 46A45. [] Loretz, G. G., A cotributio to the theory of diverget sequeces, Acta Math., MR ,367e [3] Nathaso, M. B., Elemetary methods i umber theory, Spriger 95 GTM, 000. MR c:050 [4] Suchesto, L., Baach limits, Amer. Math. Mothly, MR #740 0