Distribution functions of ratio sequences, IV
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DISTRIBUTION FUNCTIONS OF RATIO SEQUENCES, IV ´ˇ Vladimir Bala z1 , Ladislav Miˇ s´ık2 , Oto Strauch3 ´nos T. To ´ th4 and Ja 1 Slovak Technical University Radlinsk´eho 12, SK-812 37 Bratislava, Slovakia E-mail: [email protected] 2
Department of Mathematics University of Ostrava 30. dubna 22, 701 03 Ostrava 1, Czech Republic E-mail: [email protected] 3 Mathematical Institute Slovak Academy of Sciences ˇ anikova 49, SK-814 73 Bratislava, Slovakia Stef´ E-mail: [email protected] 4
Department of Mathematics, University of J. Selye Rol’n´ıckej ˇskoly 1519, SK-945 01 Kom´ arno, Slovakia E-mail: [email protected]
(Received March 22, 2010; Accepted March 22, 2012) [Communicated by Attila Peth˝ o]
Abstract In this paper we continue our study of distribution functions g(x) of n the sequence of blocks Xn = ( xxn1 , xxn2 , . . . , x xn ), n = 1, 2, . . ., where xn is an increasing sequence of positive integers. Applying a special algorithm we find a lower bound of g(x) also for xn with lower asymptotic density d = 0. This extends the lower bound of g(x) for xn with d > 0 found in the previous part III. We also prove that for an arbitrary real sequence yn ∈ [0, 1] there exists an increasing sequence xn of positive integers such that any distribution function of yn is also a distribution function of Xn .
1. Introduction Mathematics subject classification numbers: 11K31, 11K38. Key words and phrases: block sequence, distribution function, asymptotic density. Supported by APVV Project SK-CZ-0075-11 and VEGA Project 1/1022/12 and by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070). 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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´ Z, ˇ L. MIS ˇ´IK, O. STRAUCH and J. T. TOTH ´ V. BALA
Let xn , n = 1, 2, . . . , be an increasing sequence of positive integers (by “increasing” we mean strictly increasing). The double sequence xm /xn , m, n = 1, 2, . . . ˇ at [Sa]. He studied its is called the ratio sequence of xn . It was introduced by T. Sal´ everywhere density. For further study of ratio sequences, O. Strauch and J.T. T´oth [ST] introduced a sequence Xn of blocks x x xn 1 2 Xn = , n = 1, 2, . . . , , ,..., xn xn xn and they studied the set G(Xn ) of its distribution functions. The motivation is that the existence of strictly increasing g(x) ∈ G(Xn ) implies everywhere density ˇ at [Sa]. Further motivation is that the block of xm /xn , the basic problem by Sal´ sequences are tools for the study of distribution functions of sequences, see [SN, p. 12, 1.9]. Some examples of G(Xn ) can be found in [ST], [FMT], [GS] and unsolved problems concerning G(Xn ) in [SN]. In the previous part III of this series, we found optimal boundaries for g(x) ∈ G(Xn ) for xn with positive lower asymptotic density d > 0. We also constructed xn having some piecewise linear g(x). General characterization of G(Xn ) is an open problem. In this part IV we study the case d = 0, and also we find some structural properties
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