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On some properties of dispersion of block sequences of positive integers
... (see [10]) called the dispersion of the block sequence (1.1) derived from X. Relations between asymptotic density and dispersion were studied in [11]. The aim of this paper is to study the behavior of dispersion of the block sequence derived from X under the assumption that X = ∞ n=1 (c n , d n ∩ N is (R)-dense and the limit lim n→∞ dn cn = s exists. ...
... The aim of this paper is to study the behavior of dispersion of the block sequence derived from X under the assumption that X = ∞ n=1 (c n , d n ∩ N is (R)-dense and the limit lim n→∞ dn cn = s exists. In this case [10,Theorem 10]). This upper bound for D(X) is the best possible if s ≥ 2 (see [4]) and in the case 1+ [3]). ...
... The next lemma is useful for the determination of the value of the dispersion D(X) (see [10,Theorem 1]). For the proof of (R)-density we shall use the following lemma. ...
In this paper, we study the behavior of dispersion of special types of sequences which block sequence is dense.
... Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík, F. Filip [20]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence. ...
... This kind of block sequences were studied in papers, [1] , [3] , [4] , [16] and [20]. Also other kinds of block sequences were studied by several authors, see [2], [6], [8], [12] and [19]. ...
... the maximum distance between two consecutive terms in the n-th block. In this paper we will consider the characteristics (see [20]) ...
Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík and F. Filip [Math. Slovaca 54, 453–464 (2004; Zbl 1108.11017)]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence.
... This kind of block sequences have been studied in [16], [18] and [1]. Also other kinds of block sequences have been studied by several authors (see [2], [4], [7], [12] and [17]). ...
... be the maximum distance between two consecutive terms in the nth block. We will consider the quantity (see [18]) ...
... Item (ii) of the above theorem implies that if d(X) = b = 1 then we immediately have D(X) = 0. In [18,Theorem 2] it is proved that D(X) = 0 implies density of the block sequence (1). This gives the result of [10] that d(X) = 1 implies (R)-density of X. ...
... 7 0 . For the function h 1,g (x) defined in (26), putting g(x) = g t (x), we have: ...
... (83) 26 8. Now, assume that F (X N , x) → g(x) for some sequence of N ∈ [n k , n k+1 ], i.e. g(x) ∈ G(X n ). Then we can find subsequence of N (denoting again as N ) such that n k N , N −n k abbreviated as ω = (X n ) ∞ n=1 , will be called a block sequence associated with the sequence of single blocks X n , n = 1, 2, . . . . ...
This expository paper presents known results on distribution functions g(x) of the sequence of blocks where x n is an increasing sequence of positive integers. Also presents results of the set G(X n ) of all distribution functions g(x). Specially:
- continuity of g(x);
- connectivity of G(Xn);
- singleton of G(Xn);
- one-step g(x);
- uniform distribution of Xn, n = 1, 2, . . . ;
- lower and upper bounds of g(x);
- applications to bounds of ;
- many examples, e.g., , where p n is the nth prime, is uniformly distributed.
The present results have been published by 25 papers of several authors between 2001-2013.
... For A ⊂ N we define the lower and upper densities of A with respect to the weight function w, or w-density of A as follows. To calculate densities of sets is a standard task occurring frequently in papers on density theory, see e.g. ( [1], [2], [3], [4], [5], [6], [7], [8]). Usually the sets in question are written as infinite union of consecutive blocks of positive integers and there is no general formula for densities of such sets. ...
... and so (8) follows. Indeed -take into account the first equality in (2) and (6). ...
In the paper continuous variants of densities of sets of positive integers are introduced, some of their properties are studied
and formula for their calculation is proved.
... . Moreover, D(N) = 0 and every value from the interval [0, 1] is attained as dispersion of some subset of N (see [20]). ...
Let 0 ≤ q ≤ 1 and N denotes the set of all positive integers. In this paper we will deal with it too the family U ( x q ) of all regularly distributed set X = { x 1 < x 2 < ⋯ < x n < ⋯ } ⊂ N whose ratio block sequence x 1 x 1 , x 1 x 2 , x 2 x 2 , x 1 x 3 , x 2 x 3 , x 3 x 3 , ⋯ , x 1 x n , x 2 x n , ⋯ , x n x n , ⋯ is asymptotically distributed with distribution function g ( x ) = x q ; x ∈ ( 0 , 1 ] , and we will show that the regular distributed set, regular sequences, regular variation at infinity are equivalent notations. In this paper also we discuss the relationship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion and exponent of convergence.
... The concept of directions sets D k (A) as generalizations of ratio sets was introduced and studied by P. Leonetti and C. Sanna in [11]. [20]). ...
Let and denotes the set of all positive integers. In this paper we will deal with it too the family of all regularly distributed set whose ratio block sequence is asymptotically distributed with distribution function , and we will show that the regular distributed set, regular sequences, regular variation at infinity are equivalent notations. In this paper also we discuss the relation ship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion of sequence and exponent of convergence.
... e) Summing up (13), (16), (17) and (18) we find, for every x ∈ (0, 1), ...
... This kind of block sequences were studied in papers [12], [14] and [3]. Also other kinds of block sequences were studied by several authors, see [1], [6], [7], [9] and [13]. ...
Distribution functions of ratio block sequences formed from sequences of positive integers are investigated in the paper.
We characterize the case when the set of all distribution functions of a ratio block sequence contains c
0, the greatest possible distribution function. Presented results complete some previously published results.
For an increasing sequence xn, n = 1, 2,. .. , of positive integers define the block sequence Xn = (x 1 /xn,. .. , xn/xn). We study the set G(Xn) of all distribution functions of Xn, n = 1, 2,. .. . We find a special xn such that G(Xn) is not connected and we give some criterions for connectivity of G(Xn). We also give an xn such that G(Xn) contains one-step distribution function with step 1 in 1 but does not contain one-step distribution function with step 1 in 0. We prove that if G(X n) is constituted by one-step distribution functions, at least two different, then it contains distribution functions with steps in 0 and 1.
In the paper sufficient conditions for the (R)-density of a set of positive integers in terms of logarithmic densities are given. They differ substantially from those derived previously in terms of asymptotic densities.
Distribution functгons of ratio sequences
- O.-Tóth Strauch
STRAUCH, O.-TÓTH, J. T.: Distribution functгons of ratio sequences, Publ. Math.
Debrecen 58 (2001), 751-778.