Density of summable subsequences of a sequence and its applications

Let x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

For a set A of positive integers a1< a2< · · ·, let d(A), $\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as $\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$. The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, $\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping $\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $, where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension $$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$ where δ is the entropy function $$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$

On Systems of Independent Sets

The classical probability that a randomly chosen number from the set {n ∈ N : n ≤ n0} belongs to a set A ⊆ N can be approximated for large number n0 by the asymptotic density of the set A. We say that the events are independent if the probability of their intersection is equal to the product of their probabilities. By analogy we define the independence of sets. We say that the sets are independent if the asymptotic density of their intersection is equal to the product of their asymptotic densities. In the article is described a generalisation of one of the criteria of independence of sets and one interesting case in which sets are not independent

ON MURATA DENSITIES

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci.56(7) (1980) 351–353; On some fundamental relations among certain asymptotic densities, Math. Rep. Toyama Univ.4(2) (1981) 47–61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).