Best bounds for dispersion of ratio block sequences for certain subsets of integers

Transcription

1 Aales Mathematicae et Iformaticae 49 (08 pp doi: /ami Best bouds for dispersio of ratio block sequeces for certai subsets of itegers József Bukor Peter Csiba Departmet of Mathematics ad Iformatics J. Selye Uiversity Komáro Slovakia Submitted Jauary 5 08 Accepted May 8 08 Abstract I this paper we study the behavior of dispersio of special types of sequeces which block sequece is dese. Keywords: block sequece dispersio (R-desity MSC: B05. Itroductio Deote by N ad R + the set of all positive itegers ad positive real umbers respectively. Let X = {x < x < x 3 < } be a ifiite subset of N. Deote by R(X = { xi x j : i j N} the ratio set of X ad say that a set X is (R-dese if R(X is (topologically dese i the set R +. The cocept of (R-desity was itroduced by T. Šalát [7]. The followig sequece of fiite sequeces derived from X x x x x x x x x 3 x x 3 x 3 x 3... x x (. is called the block sequece of the sequece X. It is formed by the blocks X X... X... where ( x X = x... x =... 55

2 56 J. Bukor P. Csiba is called the -th block. This kid of block sequeces was itroduced by O. Strauch ad J. T. Tóth [9]. For each N cosider the step distributio fuctio xi #{i ; x F (X x = < x} ad defie the set of distributio fuctios of the ratio block sequece G(X = { lim k F (X k x }. The set of distributio fuctios of ratio block sequeces was studied i [ ]. For every N let D(X = max { x x x... x i+ x i... x } the maximum distace betwee two cosecutive terms i the -th block. We will cosider the quatity D(X = lim if D(X (see [0] called the dispersio of the block sequece (. derived from X. Relatios betwee asymptotic desity ad dispersio were studied i []. The aim of this paper is to study the behavior of dispersio of the block sequece derived from X uder the assumptio that X = = (c d N is (R-dese ad the limit lim d c = s exists. I this case s+ if s D(X s if s s s if s (see [0 Theorem 0]. This upper boud for D(X is the best possible if s (see [4] ad i the case + 5 s (see [3]. We prove that the above upper boud for D(X is also optimal i the remaidig case s < + 5 i.e. D(X ca be ay umber i the iterval 0 s+.. Results First we show that there is a coectio betwee the dispersio ad the distributio fuctios of a ratio block sequece. Theorem.. Let X N ad assume that the dispersio D(X of the related block sequece is positive. Let g G(X. The g is costat o a iterval of legth D(X.

3 Best bouds for dispersio of ratio block sequeces for certai subsets of itegers 57 Proof. Let ε < D(X be a arbitrary positive real umber. By the defiitio of dispersio it follows that for sufficietly large the step distributio fuctio F (X x is costat o some iterval ( x i xi+ of legth D(X ε. A simple compactess argumet yields that there exist real umbers γ δ 0 such that δ γ D(X ε a icreasig sequece ( k ad a sequece (m k of positive itegers such that m k < k x mk x mk + lim = γ lim = δ ad lim k k k x F (X k x = g(x a.e. o 0. k k Hece g is costat o the iterval (γ δ of legth D(X ε. Sice ε ca be chose arbitrary small ad the assertio of the theorem follows. The ext lemma is useful for the determiatio of the value of the dispersio D(X (see [0 Theorem ]. Lemma.. Let X = (c d N = ad for N let c < d < c + be positive itegers. The D(X = lim if max{c i+ d i : i =... } d +. For the proof of (R-desity we shall use the followig lemma. Lemma.3. Deote by (p (q (u (v (w ad (z be strictly icreasig sequeces of positive itegers satisfyig Further let p < q < u < v ad w < z ( = ( q p ( u p ( v u coverge to real umbers greater tha moreover z u lim lim. w q ( w ( z ad u w The the ratio set of ( (p q (u v (w z N is dese o the iterval lim w z lim. v p

4 58 J. Bukor P. Csiba The proof is elemetary ad we leave it to the reader. Let us suppose that k N is a costat. Note that the assertio of the lemma remais still true if oe removes k elemets from the sets (u v N for all sufficietly large. The mai result of this paper is the followig. Theorem.4. Let s ( + 5 be a arbitrary real umber. The for ay α 0 s+ there is a (R-dese set X = (c d N = d where c < d < c + are positive itegers for that lim c = s ad D(X = α. Proof. It was show i [4 Theorem ] that the dispersio D(X ca take ay umber i the iterval 0 s s. I what follows we suppose s s < α s+. Let us cosider the fuctio f(x = x sx. Clearly this fuctio is cotiuous ad icreasig o the iterval. Moreover f(s = s s ad f(s + = s +. Thus there exists a real umber t (s s + with the property t st = α. (. Write α i the form sk+δ where k is a iteger ad 0 δ <. The lower boud k follows from the facts that s + α ad s + s wheever < s + 5. Defie the set X N by where Put a = ad A = X = ( A B N = k (a i b i ad B = i= (c j d j. j= b i = [s.a i ] for N ad i =... k d! for i = a i = [s δ.b ] + for N i = b i + for N i = 3... k { [t.b k ] + for N j = c j = [t.d j ] + for N j =...

5 Best bouds for dispersio of ratio block sequeces for certai subsets of itegers 59 d j = [s.c j ] for N ad j =.... First we prove that D(X = α. For sufficietly large by the defiitio of the set X we have the iequalities a + d > c d > c d > > c 3 d > c d > c b k (. further ad a d < c b k (.3 a b < a d. (.4 Observe that iequality (.3 holds if α (t >. equivalet with st > which evidetly holds. As s +δ s < s s I virtue of (. this is ad s s is egative for s ( + 5 iequality (.4 follows. Now we use Lemma. From the iequalities (..4 oe ca see that it is sufficiet to study the quotiets I case a we see i case b a a d b k b c b k d c c k d k d k. lim if lim if a d b k c b k d = lim if = lim if a α a = α tb k b k = t = α stb k st ad the remaiig case c is aalogous to case b. It remais to prove that the set X is (R-dece. Usig Lemma we show that the ratio set of the set X is dese o the itervals (for p = a q = b u = w = a v = z = b k α t i s i α si t i (for p = a q = b u = a v = b k w = c i z = d i. Hece by α s + ad t < s + we have α i= t i s i α si t i = ad therefore the (R-desity of the set X follows.

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