Báo cáo toán học: Some Results on Mid-Point Sets of Sets of Natural Numbers

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Báo cáo toán học: "Some Results on Mid-Point Sets of Sets of Natural Numbers" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:1 (2005) 85–89 RI 0$7+(0$7,&6 ‹ 9$67 Some Results on Mid-Point Sets of Sets of Natural Numbers D. K. Ganguly1 , Rumki Bhattacharjee1 , and Maitreyee Dasgupta2 1 Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700 019, India 2 WIB(M) 3/2, Phase II, Golf Green, Kolkata 700 095, India Received February 4, 2004Abstract. In this paper the authors study some properties of the mid-point setsof sets of natural numbers using upper (lower) asymptotic density of sets of naturalnumbers. In this connection a set has been introduced here and studied.1. IntroductionLet P and Q be two linear sets of points. The mid-point set M (P, Q) has been x+y : x ∈ P, y ∈ Q . In particular, fordefined as the set M (P, Q) = 2P = Q, we write M (P, P ) = M (P ). Again whenever A and B are two linearsets of points with positive abscissae then their ratio set R(A, B ) is defined asR(A, B ) = {(a/b) : a ∈ A, b ∈ B }. In particular, when A = B , we writeR(A, A) = R(A). With the usual notations N is the set of natural numbers andR+ is the set of positive rational numbers. One may recall here the notion of asymptotic density of a set of positive ˘integers which is extensively used by Sala t [5] in studying some properties ofratio sets of sets of natural numbers. Later, other authors viz Bukor, Kmetovaand Toth [2] worked on ratio sets of sets of natural numbers. Let A ⊂ N, A = ∅ then A(n) denotes the counting function of A given by A(n)A(n) = 1. The lower asymptotic density of A is given by lim inf = n→∞ n a∈A, a≤n A(n)d(A) and the upper asymptotic density is given by lim sup = d(A). If n n→∞86 D. K. Ganguly, Rumki Bhattacharjee, and Maitreyee Dasguptad(A) = d(A) we call the common value d(A) as the asymptotic density of A. On the other hand, mid-point sets, primarily of Cantor type sets were studiedby Randolph [4] and subsequently by Bose Majumdar [1]. Then Ganguly andMajumdar [3] proved some results on mid-point sets of two linear sets in thelight of the Lebesgue density. In the present paper the authors restrict theirinvestigations into mid-point sets of sets of natural numbers with the help of thenotion of asymptotic density.2. Main ResultsWe shall study some properties of A ⊂ N which guarantee the denseness of M (A)in [1, ∞).Theorem 2.1. Let d(A) = 1 where A ⊂ N. Then each positive rational numbercan be represented as the mid-point for infinite number of pairs (g, h), g ∈ A,h ∈ A.Proof. Assuming the theorem not to be true there must exist an r(∈ R+ ) = g+h(p/q ) = 1, (p, q ) = 1 such that r = only for a finite number of pairs 2(g, h), g ∈ A, h ∈ A. Let (gi , hi ), i = 1, 2, ..., m, be all the pairs of numbers g i + hiof A satisfying the relation r = , i = 1, 2, ..., m. Let us denote max 2(g1 , g2 , ..., gm , h1 , h2 , ..., hm ) by a and form the sequence a, a + 1, ..., n (n > a). (1)The numbers in the sequence (1) are characterized by the fact that the mid-point of any two of them is different from r. Now, to sequence (1) belong all thenumbers p + u where a < p + u ≤ n i.e. a − p < u ≤ n − p. (α)and also the numbers q − v where a < q − v ≤ n i.e. a − q < −v ≤ n − q. (β )Next we put s = max(p, q ) and s = min(p, q ). Then relation (α) leads toa − s < u ≤ n − s and (β ) yields a − s < −v ≤ n − s . Combining these twoinequalities we can state that the numbers p + i, q − i belong to sequence (1) if a − s < |i| ≤ n − s . ...