Density of summable subsequences of a sequence and its applications

Rok, strany: 2020, 657 - 666

Let $x=\{x_n\}_n=1^∞$ be a sequence of positive numbers, and $\mathcal{I}_x$ be the collection of all subsets $A\subseteq \mathbb{N}$ such that $∑_{k\in A}x_k<+∞$. 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 $\mathcal{I}_x$, 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 $\mathcal{I}_x$, 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 $\liminf_k\to∞nx_n=0$, 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.

ISO 690:

Hou, B., Xin, Y., Zhang, A. 2020. Density of summable subsequences of a sequence and its applications. In Mathematica Slovaca, vol. 70, no.3, pp. 657-666. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0379

APA:

Hou, B., Xin, Y., Zhang, A. (2020). Density of summable subsequences of a sequence and its applications. Mathematica Slovaca, 70(3), 657-666. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0379

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava