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On estimations of dispersions of certain dense block sequences
... a) If y is an increasing point 10 of g(x), n = 1, 2, . . . then by (6) we have g(xy) ...
... (63) 6 3. Now we prove x i x i+1 → 1 as i → ∞. Let i ∈ (n k , n k+1 ) and let e.g. ...
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. ...
... Let a weight function w : N → R + 0 with w(1) > 0 satisfy (1). Then one can easily verify that the function f w defined by f w (0) = w(1) and f w (t) = w(n) for n ∈ N and t ∈ (n − 1, n] belongs to F . ...
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.
... Relations between the density of the block sequence and the dispersion of the block sequence were studied in [11], [1] and [2]. Much more information about the mentioned concepts and their relations can be found in the monograph [5]. ...
... (i) There exists a set X = {x1 < x 2 < . . . } ⊂ N such that D(X) n , d n ∩ N where c 1 = 1, d 1 = 3 and c n = n · d n−1 , and d n = (n − 1) · d 2 n−1 for every n ≥ 2. Then D(X) ≤ lim sup n→∞ c n+1 − d n d n+1 = lim n→∞ (n + 1) · d n − d n n · d · d n d n = ∞. ...
Using new characteristics of an infinite subset of positive integers we give some estimations of the dispersion of the related
block sequence.
... 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]). We prove that the above upper bound for D(X) is also optimal in the remainding case 1 ≤ s < 1+ √ 5 2 , i.e. ...
In this paper, we study the behavior of dispersion of special types of sequences which block sequence is dense.
... 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]. ...
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.
... In Part 3 we prove that (4) does not imply that G(X n ) is singleton. This gives the negative answer to the question recently published in [5, p. 76 (4). This proves that the uniform distribution of (2) implies (4). ...
... Some examples on G(X n ) can be found in [ST1]. In [TMF] the authors introduced and studied the so called dispersion of X n (see also [FT,FT1,FMT]). ...
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.
... 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]). ...
... 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.
In this paper we give necessary and sufficient conditions for the block sequence of the set X = {x
1 < x
2 < … < x
n
< …} ⊂ ℕ to have an asymptotic distribution function in the form x
λ.
Key words and phrasesasymptotic distribution function-uniform distribution-block sequence
Mathematics subject classification number11B05
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